Logica matematica/Calcolo delle proposizioni - Wikibooks, manuali e libri di testo liberi

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href="#Negazione"> <div class="vector-toc-text"> 2.1.1 Negazione </div> </a> <ul id="toc-Negazione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congiunzione" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Congiunzione"> <div class="vector-toc-text"> 2.1.2 Congiunzione </div> </a> <ul id="toc-Congiunzione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Disgiunzione" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Disgiunzione"> <div class="vector-toc-text"> 2.1.3 Disgiunzione </div> </a> <ul id="toc-Disgiunzione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implicazione" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Implicazione"> <div class="vector-toc-text"> 2.1.4 Implicazione </div> </a> <ul id="toc-Implicazione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Doppia_implicazione" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Doppia_implicazione"> <div class="vector-toc-text"> 2.1.5 Doppia implicazione </div> </a> <ul id="toc-Doppia_implicazione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Assegnazione_booleana" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Assegnazione_booleana"> <div class="vector-toc-text"> 2.1.6 Assegnazione booleana </div> </a> <ul id="toc-Assegnazione_booleana-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Valutazione_booleana" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Valutazione_booleana"> <div class="vector-toc-text"> 2.1.7 Valutazione booleana </div> </a> <ul id="toc-Valutazione_booleana-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_di_equivalenza_delle_valutazioni" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Teorema_di_equivalenza_delle_valutazioni"> <div class="vector-toc-text"> 2.1.8 Teorema di equivalenza delle valutazioni </div> </a> <ul id="toc-Teorema_di_equivalenza_delle_valutazioni-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Soddisfacibilità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Soddisfacibilità"> <div class="vector-toc-text"> 2.2 Soddisfacibilità </div> </a> <ul id="toc-Soddisfacibilità-sublist" class="vector-toc-list"> <li id="toc-Formula_soddisfatta" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Formula_soddisfatta"> <div class="vector-toc-text"> 2.2.1 Formula soddisfatta </div> </a> <ul id="toc-Formula_soddisfatta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formula_soddisfacibile" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Formula_soddisfacibile"> <div class="vector-toc-text"> 2.2.2 Formula soddisfacibile </div> </a> <ul id="toc-Formula_soddisfacibile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tautologia" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tautologia"> <div class="vector-toc-text"> 2.2.3 Tautologia </div> </a> <ul id="toc-Tautologia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contraddizione" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Contraddizione"> <div class="vector-toc-text"> 2.2.4 Contraddizione </div> </a> <ul id="toc-Contraddizione-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tavole_di_verità" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tavole_di_verità"> <div class="vector-toc-text"> 2.2.5 Tavole di verità </div> </a> <ul id="toc-Tavole_di_verità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Teorema"> <div class="vector-toc-text"> 2.2.6 Teorema </div> </a> <ul id="toc-Teorema-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalenza_logica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalenza_logica"> <div class="vector-toc-text"> 2.3 Equivalenza logica </div> </a> <ul id="toc-Equivalenza_logica-sublist" class="vector-toc-list"> <li id="toc-Implicazione_logica" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Implicazione_logica"> <div class="vector-toc-text"> 2.3.1 Implicazione logica </div> </a> <ul id="toc-Implicazione_logica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalenza_logica_(Definizione)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Equivalenza_logica_(Definizione)"> <div class="vector-toc-text"> 2.3.2 Equivalenza logica (Definizione) </div> </a> <ul id="toc-Equivalenza_logica_(Definizione)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primo_teorema_del_rimpiazzamento" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Primo_teorema_del_rimpiazzamento"> <div class="vector-toc-text"> 2.3.3 Primo teorema del rimpiazzamento </div> </a> <ul id="toc-Primo_teorema_del_rimpiazzamento-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Secondo_teorema_del_rimpiazzamento" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Secondo_teorema_del_rimpiazzamento"> <div class="vector-toc-text"> 2.3.4 Secondo teorema del rimpiazzamento </div> </a> <ul id="toc-Secondo_teorema_del_rimpiazzamento-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modelli" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modelli"> <div class="vector-toc-text"> 2.4 Modelli </div> </a> <ul id="toc-Modelli-sublist" class="vector-toc-list"> <li id="toc-Modello_per_una_formula" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modello_per_una_formula"> <div class="vector-toc-text"> 2.4.1 Modello per una formula </div> </a> <ul id="toc-Modello_per_una_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contromodello_per_una_formula" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Contromodello_per_una_formula"> <div class="vector-toc-text"> 2.4.2 Contromodello per una formula </div> </a> <ul id="toc-Contromodello_per_una_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modello_per_un_insieme_di_formule" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modello_per_un_insieme_di_formule"> <div class="vector-toc-text"> 2.4.3 Modello per un insieme di formule </div> </a> <ul id="toc-Modello_per_un_insieme_di_formule-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Implicazione_logica_da_un_insieme_di_formule" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Implicazione_logica_da_un_insieme_di_formule"> <div class="vector-toc-text"> 2.4.4 Implicazione logica da un insieme di formule </div> </a> <ul id="toc-Implicazione_logica_da_un_insieme_di_formule-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Soddisfacibilità_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Soddisfacibilità_2"> <div class="vector-toc-text"> 2.4.5 Soddisfacibilità </div> </a> <ul id="toc-Soddisfacibilità_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Insoddisfacibilità" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Insoddisfacibilità"> <div class="vector-toc-text"> 2.4.6 Insoddisfacibilità </div> </a> <ul id="toc-Insoddisfacibilità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_di_soddisfacibilità" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Teorema_di_soddisfacibilità"> <div class="vector-toc-text"> 2.4.7 Teorema di soddisfacibilità </div> </a> <ul id="toc-Teorema_di_soddisfacibilità-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Compattezza" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compattezza"> <div class="vector-toc-text"> 2.5 Compattezza </div> </a> <ul id="toc-Compattezza-sublist" class="vector-toc-list"> <li id="toc-Insieme_di_formule_finitamente_soddisfacibile" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Insieme_di_formule_finitamente_soddisfacibile"> <div class="vector-toc-text"> 2.5.1 Insieme di formule finitamente soddisfacibile </div> </a> <ul id="toc-Insieme_di_formule_finitamente_soddisfacibile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_di_compattezza_della_soddisfacibilità" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Teorema_di_compattezza_della_soddisfacibilità"> <div class="vector-toc-text"> 2.5.2 Teorema di compattezza della soddisfacibilità </div> </a> <ul id="toc-Teorema_di_compattezza_della_soddisfacibilità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riformulazione_del_teorema_di_compattezza" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Riformulazione_del_teorema_di_compattezza"> <div class="vector-toc-text"> 2.5.3 Riformulazione del teorema di compattezza </div> </a> <ul id="toc-Riformulazione_del_teorema_di_compattezza-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corollario_del_teorema_di_compattezza" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Corollario_del_teorema_di_compattezza"> <div class="vector-toc-text"> 2.5.4 Corollario del teorema di compattezza </div> </a> <ul id="toc-Corollario_del_teorema_di_compattezza-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Teorema_di_deduzione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorema_di_deduzione"> <div class="vector-toc-text"> 2.6 Teorema di deduzione </div> </a> <ul id="toc-Teorema_di_deduzione-sublist" class="vector-toc-list"> <li id="toc-Dimostrazione" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Dimostrazione"> <div class="vector-toc-text"> 2.6.1 Dimostrazione </div> </a> <ul id="toc-Dimostrazione-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Completezza_di_insiemi_di_connettivi" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Completezza_di_insiemi_di_connettivi"> <div class="vector-toc-text"> 3 Completezza di insiemi di connettivi </div> </a> <button aria-controls="toc-Completezza_di_insiemi_di_connettivi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Completezza di insiemi di connettivi </button> <ul id="toc-Completezza_di_insiemi_di_connettivi-sublist" class="vector-toc-list"> <li id="toc-Funzione_di_verità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Funzione_di_verità"> <div class="vector-toc-text"> 3.1 Funzione di verità </div> </a> <ul id="toc-Funzione_di_verità-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivazione_di_connettivi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivazione_di_connettivi"> <div class="vector-toc-text"> 3.2 Derivazione di connettivi </div> </a> <ul id="toc-Derivazione_di_connettivi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Insieme_funzionalmente_completo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Insieme_funzionalmente_completo"> <div class="vector-toc-text"> 3.3 Insieme funzionalmente completo </div> </a> <ul id="toc-Insieme_funzionalmente_completo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Decidibilità" 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href="#I_principali_sistemi_di_dimostrazione"> <div class="vector-toc-text"> 5.2 I principali sistemi di dimostrazione </div> </a> <ul id="toc-I_principali_sistemi_di_dimostrazione-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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href="/wiki/Logica_matematica/Sistemi_formali" title="Logica matematica/Sistemi formali">Sistemi formali</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Sistemi formali</a></li> <li><a class="mw-selflink selflink">Calcolo delle proposizioni</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Calcolo delle proposizioni</a> <ul><li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Sistema_di_Hilbert" title="Logica matematica/Calcolo delle proposizioni/Sistema di Hilbert">Il sistema di Frege-Hilbert</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Calcolo delle proposizioni/Sistema di Hilbert</a></li> <li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/La_deduzione_naturale" title="Logica matematica/Calcolo delle proposizioni/La deduzione naturale">La deduzione naturale</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Calcolo delle proposizioni/La deduzione naturale</a></li> <li><a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni/Il_calcolo_dei_sequenti&action=edit&redlink=1" class="new" title="Logica matematica/Calcolo delle proposizioni/Il calcolo dei sequenti (la pagina non esiste)">Il calcolo dei sequenti</a></li> <li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/I_tableaux_semantici" title="Logica matematica/Calcolo delle proposizioni/I tableaux semantici">I tableaux semantici</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Calcolo delle proposizioni/I tableaux semantici</a></li> <li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/La_risoluzione" title="Logica matematica/Calcolo delle proposizioni/La risoluzione">La risoluzione</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Calcolo delle proposizioni/La risoluzione</a></li> <li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Esercizi_su_tavole_di_verit%C3%A0" title="Logica matematica/Calcolo delle proposizioni/Esercizi su tavole di verità">Esercizi su tavole di verità</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Calcolo delle proposizioni/Esercizi su tavole di verità</a></li></ul></li> <li><a href="/wiki/Logica_matematica/Calcolo_dei_predicati" title="Logica matematica/Calcolo dei predicati">Calcolo dei predicati</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Calcolo dei predicati</a></li> <li><a href="/wiki/Logica_matematica/Teoria_dei_modelli" title="Logica matematica/Teoria dei modelli">Teoria dei modelli</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Teoria dei modelli</a></li> <li><a href="/w/index.php?title=Logica_matematica/Computabilit%C3%A0&action=edit&redlink=1" class="new" title="Logica matematica/Computabilità (la pagina non esiste)">Computabilità</a></li> <li><a href="/wiki/Logica_matematica/Incompletezza" title="Logica matematica/Incompletezza">Incompletezza</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Incompletezza</a> <ul><li><a href="/wiki/Logica_matematica/Incompletezza/Teoremi_di_incompletezza_di_G%C3%B6del" title="Logica matematica/Incompletezza/Teoremi di incompletezza di Gödel">Teoremi di incompletezza di Gödel</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Incompletezza/Teoremi di incompletezza di Gödel</a></li></ul></li> <li><a href="/wiki/Logica_matematica/Intermezzo_Euclide" title="Logica matematica/Intermezzo Euclide">Intermezzo: Euclide</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Intermezzo Euclide</a></li> <li><a href="/wiki/Logica_matematica/Intermezzo_paradossi" title="Logica matematica/Intermezzo paradossi">Intermezzo: i paradossi</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Intermezzo paradossi</a></li> <li><a href="/wiki/Logica_matematica/Risorse_internet" title="Logica matematica/Risorse internet">Risorse internet</a><a href="/wiki/Aiuto:Fasi_di_sviluppo" title="Aiuto:Fasi di sviluppo">Logica matematica/Risorse internet</a></li></ul> <a class="external text" href="https://it.wikibooks.org/w/index.php?title=Template%3ALogica_matematica&action=edit">Modifica il sommario</a> </div> <div class="sommario-v-footer c-azzurro"> <div class="mw-inputbox-centered" style=""><form name="searchbox" class="searchbox mw-inputbox-form" action="/wiki/Speciale:Ricerca"><div class="cdx-text-input"><input class="mw-inputbox-input mw-searchInput searchboxInput cdx-text-input__input" name="search" placeholder="" size="24" dir="ltr" /></div><input type="hidden" value="Logica matematica" name="prefix" /> <input type="submit" name="fulltext" value="Cerca" class="cdx-button" /><input type="hidden" value="Search" name="fulltext" /></form></div></div></div> Partiremo dal punto più semplice: la logica proposizionale, ovvero relativa alle proposizioni. In questo contesto si studiano funzionamento e proprietà dei connettivi logici, e non si analizza come si arriva al valore di verità delle formule atomiche. Il potere espressivo di questo linguaggio è limitato: in esso, infatti, possiamo esprimere proposizioni che sono vere o false, ma non proprietà che possono valere o meno su uno/molti/tutti gli individui di una certa classe. Col basso potere espressivo si coniuga la decidibilità di questa logica. Presentiamo la sintassi e la semantica della logica proposizionale. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sintassi">Sintassi</h2>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=1" title="Modifica la sezione Sintassi" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=1" title="Edit section's source code: Sintassi">modifica sorgente</a>]</div> Introduciamo la nozione di linguaggio proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{\Sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\Sigma }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ac257e8923595977f8c20d6b4b61ae41235f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.023ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{\Sigma }}"> costruito su un alfabeto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }">. Iniziamo con la definizione di alfabeto. <div class="mw-heading mw-heading3"><h3 id="Alfabeto">Alfabeto</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=2" title="Modifica la sezione Alfabeto" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=2" title="Edit section's source code: Alfabeto">modifica sorgente</a>]</div> <style data-mw-deduplicate="TemplateStyles:r465526">.mw-parser-output .box-title{font-weight:bolder;color:#595959}.mw-parser-output .obiettivi-text{background-color:#f2f2f2;border:1px solid #d9d9d9;padding:15px;margin:-20px 0 10px 0;max-width:500px}.mw-parser-output .box-trama{border-left:3px solid #6699ff;background-color:#f2f2f2;padding:15px;margin:-20px 0 10px 0;font-size:.9em}.mw-parser-output .box-sfondo-grigio{background-color:#f2f2f2;border:1px solid #d9d9d9;padding:15px;margin:-20px 0 10px 0}.mw-parser-output .box-sfondo-giallo{background-color:#ffffcc;border:1px solid #ffbf00;padding:15px;margin:-20px 0 10px 0}.mw-parser-output .box-sfondo-beige{background-color:#f5f5dc;border:1px solid #e9e9af;padding:15px;margin:-20px 0 10px 0}.mw-parser-output .box-sfondo-generico{background-color:#f2f2f2;border:1px solid #d9d9d9;padding:15px;margin:15px 0}.mw-parser-output .box-centrato{margin:0 auto;width:80%}.mw-parser-output .box-destra{float:right;max-width:350px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .box-title{color:#fff}html.skin-theme-clientpref-night .mw-parser-output .box-sfondo-generico,html.skin-theme-clientpref-night .mw-parser-output .box-sfondo-grigio,html.skin-theme-clientpref-night .mw-parser-output .box-sfondo-giallo,html.skin-theme-clientpref-night .mw-parser-output .box-sfondo-beige{background:#1f1f23}html.skin-theme-clientpref-night .mw-parser-output .box-trama{border-left:3px solid #858593;background-color:#1f1f23}html.skin-theme-clientpref-night .mw-parser-output .obiettivi-text{background-color:#1f1f23;border:1px solid #858593}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .box-title{color:#fff}html.skin-theme-clientpref-os .mw-parser-output .box-sfondo-generico,html.skin-theme-clientpref-os .mw-parser-output .box-sfondo-grigio,html.skin-theme-clientpref-os .mw-parser-output .box-sfondo-giallo,html.skin-theme-clientpref-os .mw-parser-output .box-sfondo-beige{background:#1f1f23}html.skin-theme-clientpref-os .mw-parser-output .box-trama{border-left:3px solid #858593;background-color:#1f1f23}html.skin-theme-clientpref-os .mw-parser-output .obiettivi-text{background-color:#1f1f23;border:1px solid #858593}}</style> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Un alfabeto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"> è costituito da: <ul><li>i connettivi proposizionali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"> (unario) e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land ,\lor ,\to ,\leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land ,\lor ,\to ,\leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a90c332534b2007a1252f56b93a9f7cb170156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.85ex; height:2.343ex;" alt="{\displaystyle \land ,\lor ,\to ,\leftrightarrow }"> (binari);</li> <li>le costanti proposizionali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top ,\bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> <mo>,</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top ,\bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b53c32848b090c9dd55ca0b0659ea5f9fab8859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.65ex; height:2.509ex;" alt="{\displaystyle \top ,\bot }"> (per denotare il vero e il falso);</li> <li>un insieme non vuoto (finito o numerabile) di simboli proposizionali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}=\{A,B,C,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}=\{A,B,C,...\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0a17793e5e3228dce817659abac3d872670b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.604ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}=\{A,B,C,...\}}">;</li> <li>i simboli separatori <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle '('}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">(</mo> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle '('}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe08ee329398c6b8c1bc50322dd3d33e686454e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.274ex; height:3.009ex;" alt="{\displaystyle '('}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ')'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mo>′</mo> </msup> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ')'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3519e266377ca589936aaccd10cc6057ea8771d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.274ex; height:3.009ex;" alt="{\displaystyle ')'}">.</li></ul></div> Nel seguito scriveremo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"> quando <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"> è chiaro dal contesto. Definiamo ora le formule di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}">. <div class="mw-heading mw-heading3"><h3 id="Formule_ben_formate">Formule ben formate</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=3" title="Modifica la sezione Formule ben formate" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=3" title="Edit section's source code: Formule ben formate">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> L'insieme PROP delle formule ben formate o formule del linguaggio proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"> è l'insieme definito induttivamente come segue: <ol><li>le costanti e i simboli proposizionali sono formule;</li> <li>se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una formula, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\neg A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f712edae73fb05b058dd66129a7f0b9b5f8f2657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.103ex; height:2.843ex;" alt="{\displaystyle (\neg A)}"> è una formula;</li> <li>se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"> è un connettivo binario (cioè <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c76f8e7901565e8a93ba0e159a1e3a86fb2176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}">) e se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> sono due formule, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\circ B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∘</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\circ B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5446ca714cb1d3da609400be73dbd3009a6112a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.511ex; height:2.843ex;" alt="{\displaystyle (A\circ B)}"> è una formula.</li></ol> Le costanti e i simboli proposizionali sono anche detti atomi, le loro negazioni sono dette atomi negati. Gli atomi e gli atomi negati sono anche detti letterali. Gli atomi negati sono talvolta detti letterali negativi. Una formula del linguaggio proposizionale è anche detta proposizione o enunciato proposizionale. Data una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> di PROP, ogni formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> che appare come componente di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è detta sottoformula di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. La definizione di sottoformula, di connettivo principale e di sottoformula immediata è la seguente.</div> <div class="mw-heading mw-heading4"><h4 id="Sottoformule">Sottoformule</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=4" title="Modifica la sezione Sottoformule" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=4" title="Edit section's source code: Sottoformule">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> una formula di PROP, l'insieme delle sottoformule di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è definito come segue: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> stessa è una sottoformula di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">;</li> <li>se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una formula del tipo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\neg B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47831e246cf49ce11b27d5c0618cc44a8f91497c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.124ex; height:2.843ex;" alt="{\displaystyle (\neg B)}">, allora le sottoformule di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> sono anche sottoformule di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"> è detto connettivo principale e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> sottoformula immediata di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">;</li> <li>se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una formula del tipo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\circ C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∘</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\circ C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/554a0772fe15566be3507b7216966fa8aff5e841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.725ex; height:2.176ex;" alt="{\displaystyle B\circ C}">, dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"> è un connettivo binario (cioè <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\lor ,\land ,\rightarrow ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∨</mo> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\lor ,\land ,\rightarrow ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d4558fced122a2bac5e826e77abd0e81213d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\lor ,\land ,\rightarrow ,\leftrightarrow \}}">), e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> due formule, le sottoformule di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> sono anche sottoformule di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"> è detto connettivo principale; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> sottoformule immediate di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.</li></ol></div> Per evitare un uso eccessivo delle parentesi, introduciamo anche delle regole di precedenza tra i connettivi. Per le formule proposizionali, si usa la seguente convenzione: la massima precedenza a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }">, poi, nell'ordine, ai connettivi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land ,\lor ,\to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land ,\lor ,\to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dcdeb858c0eb8050e48828beffe957fe061b70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.492ex; height:2.343ex;" alt="{\displaystyle \land ,\lor ,\to }"> e infine a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }">. Questo significa che, in assenza di parentesi, una formula ben formata va parentesizzata privilegiando le sottoformule i cui connettivi principali hanno precedenza più alta. A parità di precedenza, cioè se siamo in presenza di più occorrenze dello stesso connettivo, si assume che esso associ a destra, cioè la parentesizzazione va eseguita a partire dall'ultima occorrenza a destra del connettivo. <div class="mw-heading mw-heading4"><h4 id="Simboli_proposizionali_occorrenti_in_una_formula">Simboli proposizionali occorrenti in una formula</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=5" title="Modifica la sezione Simboli proposizionali occorrenti in una formula" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=5" title="Edit section's source code: Simboli proposizionali occorrenti in una formula">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> La funzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb:\mathrm {PROP} \mapsto \wp {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">P</mi> </mrow> <mo stretchy="false">↦</mo> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb:\mathrm {PROP} \mapsto \wp {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/883ebb2a87338c624b0f70939e05f68f3f452da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.348ex; height:2.676ex;" alt="{\displaystyle simb:\mathrm {PROP} \mapsto \wp {\mathcal {P}}}"> su PROP nell'inseme potenza di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"> è definita ricorsivamente come segue: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb(A)=\{A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb(A)=\{A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7200b1401096d8f7091c8b2db68e03732375afe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.65ex; height:2.843ex;" alt="{\displaystyle simb(A)=\{A\}}"> se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5e1e51c194308f16c0001472782a8b6c22523c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.287ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {P}}}">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb(\top )=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">⊤</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb(\top )=\emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f264bd9f5608e2960dc034288c65b4b482ce21a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.809ex; height:2.843ex;" alt="{\displaystyle simb(\top )=\emptyset }">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb(\bot )=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb(\bot )=\emptyset }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6747600adf48509eef1127d8043011ac066a72cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.809ex; height:2.843ex;" alt="{\displaystyle simb(\bot )=\emptyset }">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb(\neg A)=simb(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb(\neg A)=simb(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321728e4b3c817cb9ffab04e13de2e21e1208a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.615ex; height:2.843ex;" alt="{\displaystyle simb(\neg A)=simb(A)}">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb(A\circ B)=simb(A)\cup simb(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∘</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb(A\circ B)=simb(A)\cup simb(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747c9195afc16718df36ef954d9a0c8df09ca120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.11ex; height:2.843ex;" alt="{\displaystyle simb(A\circ B)=simb(A)\cup simb(B)}">, dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c76f8e7901565e8a93ba0e159a1e3a86fb2176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}">.</li></ul></div> La funzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0ad754f88ab0f0341738c149d570cbd29eeb3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.931ex; height:2.176ex;" alt="{\displaystyle simb}"> restituisce l'insieme dei simboli proposizionali che occorrono in una formula. Essa è definita in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wp {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wp {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c487f87a81a9bc058b15ac66da183645e8b21b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.182ex; height:2.676ex;" alt="{\displaystyle \wp {\mathcal {P}}}">, cioè nell'inseme dei sottoinsiemi di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}">, dato che restituisce sempre un sottoinsieme di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}">. <div class="mw-heading mw-heading3"><h3 id="Esempi">Esempi</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=6" title="Modifica la sezione Esempi" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=6" title="Edit section's source code: Esempi">modifica sorgente</a>]</div> Le regole appena esposte indicano come formare le frasi del calcolo proposizionale. Le frasi ben formate nascono applicando a formule atomiche i connettivi logici come indicato, se non è possibile risalire a queste regole la frase non è una frase ben formata. Le formule atomiche rappresentano frasi del tipo: <ul><li>"Oggi splende il sole"</li> <li>"Mi chiamo Giovanni"</li> <li>"Tre è dispari"</li></ul> cioè frasi che possono essere vere o false. Da esse verranno poi costruite le formule ben formate. Esempi di frasi ben formate: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((A\to B)\land (C\to D))\to ((A\land C)\to (B\land D))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∧</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((A\to B)\land (C\to D))\to ((A\land C)\to (B\land D))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82d94e7972bdf3b119c9577b47ef5bc6fca17a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.455ex; height:2.843ex;" alt="{\displaystyle ((A\to B)\land (C\to D))\to ((A\land C)\to (B\land D))}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\to \neg \neg \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\to \neg \neg \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/818886b01a36a7fc8c8cac34800bf1b1e116c97b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.302ex; height:2.176ex;" alt="{\displaystyle \neg A\to \neg \neg \neg A}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow B\lor C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo>∨</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B\lor C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eca48482b839de424efd03ff7a81e8f9bd06991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.47ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B\lor C}"></li></ol> mentre frasi invalide sono: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1034681c691406a1afbcd8ca380b428cc02300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.316ex; height:2.843ex;" alt="{\displaystyle A(B)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c40b25b72ada119df42ce7cf8b0af7aa25fcb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.09ex; height:2.176ex;" alt="{\displaystyle \to A\to B}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to \land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to \land B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9994a10c94d7a143144763c1d81302cc609136c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.672ex; height:2.176ex;" alt="{\displaystyle A\to \land B}"></li></ol> <a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Connettivi" title="Logica matematica/Calcolo delle proposizioni/Connettivi">I vari connettivi</a> Queste regole definiscono la sintassi che utilizzeremo per il calcolo delle proposizioni. Ora dobbiamo dare un significato a queste frasi. <div class="mw-heading mw-heading2"><h2 id="Semantica">Semantica</h2>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=7" title="Modifica la sezione Semantica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=7" title="Edit section's source code: Semantica">modifica sorgente</a>]</div> Dobbiamo definire come operano questi connettivi e lo faremo usando le tavole di verità. In questo modo potremo creare una semantica, ovvero dare un significato all'operazione che viene eseguita da ogni operatore. Ci serviremo di variabili atomiche, ovvero indivisibili, che possono assumere solo valori di verità (vero o falso rispettivamente indicati con V o F o, se ci basiamo sulla lingua inglese T o F). <div class="mw-heading mw-heading3"><h3 id="Sistema_di_valutazione">Sistema di valutazione</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=8" title="Modifica la sezione Sistema di valutazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=8" title="Edit section's source code: Sistema di valutazione">modifica sorgente</a>]</div> <style data-mw-deduplicate="TemplateStyles:r465528">.mw-parser-output .vedi-table{margin-bottom:.5em;border:1px solid #CCC;text-align:left;font-size:95%;background:#F9F9F9}.mw-parser-output .vedi-icona{padding:0 .5em}.mw-parser-output .vedi-td{width:100%}.mw-parser-output .vedi-libro{clear:right;border:1px solid #aaa;margin:0 0 .5em .5em;font-size:90%;background:#f9f9f9;width:250px;padding:4px;text-align:left;float:right}.mw-parser-output .vedi-quiz{margin:0em .5em;border:1px solid #ffd9b3;background:#fffae6;padding:.2em}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vedi-table,html.skin-theme-clientpref-night .mw-parser-output .vedi-libro,html.skin-theme-clientpref-night .mw-parser-output .vedi-quiz{background:#1f1f23}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vedi-table,html.skin-theme-clientpref-os .mw-parser-output .vedi-libro,html.skin-theme-clientpref-os .mw-parser-output .vedi-quiz{background:#1f1f23}}</style> <table class="vedi-table"> <tbody><tr> <td class="vedi-icona"><a href="/wiki/File:Searchtool.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Searchtool.svg/18px-Searchtool.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Searchtool.svg/27px-Searchtool.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Searchtool.svg/36px-Searchtool.svg.png 2x" data-file-width="512" data-file-height="512" /></a> </td> <td class="vedi-td">Per approfondire, vedi <a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Tutti_i_connettivi" title="Logica matematica/Calcolo delle proposizioni/Tutti i connettivi">Tutti i connettivi</a>. </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Il sistema di valutazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}=\langle {\mathcal {B}},{\mathcal {T}},{\mathcal {Op}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}=\langle {\mathcal {B}},{\mathcal {T}},{\mathcal {Op}}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70cf09e636782acf4337018a20873dd98954faff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.967ex; height:2.843ex;" alt="{\displaystyle {\mathcal {S}}=\langle {\mathcal {B}},{\mathcal {T}},{\mathcal {Op}}\rangle }"> della logica proposizionale è definito da <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\displaystyle {\mathcal {B}}=\{\mathrm {T} ,\mathrm {F} \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle {\mathcal {B}}=\{\mathrm {T} ,\mathrm {F} \}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf1df669b5b4249488c1c4c55bbf62de94be519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.197ex; height:2.843ex;" alt="{\displaystyle {\displaystyle {\mathcal {B}}=\{\mathrm {T} ,\mathrm {F} \}}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}=\{\mathrm {T} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}=\{\mathrm {T} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ddb9dc600f6aeeae4db5e640e242d71938b1e5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.037ex; height:2.843ex;" alt="{\displaystyle {\mathcal {T}}=\{\mathrm {T} \}}">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}=\{{\mathcal {Op}}_{\neg },{\mathcal {Op}}_{\land },{\mathcal {Op}}_{\lor },{\mathcal {Op}}_{\to },{\mathcal {Op}}_{\leftrightarrow }\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∧</mo> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∨</mo> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↔</mo> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}=\{{\mathcal {Op}}_{\neg },{\mathcal {Op}}_{\land },{\mathcal {Op}}_{\lor },{\mathcal {Op}}_{\to },{\mathcal {Op}}_{\leftrightarrow }\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a7b1704dc89877401b3b6c4da2918bcc966729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.412ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}=\{{\mathcal {Op}}_{\neg },{\mathcal {Op}}_{\land },{\mathcal {Op}}_{\lor },{\mathcal {Op}}_{\to },{\mathcal {Op}}_{\leftrightarrow }\}}">, uno per ognuno dei connettivi del linguaggio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\neg ,\land ,\lor ,\rightarrow ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\neg ,\land ,\lor ,\rightarrow ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d0ddd038763c8722ac403aac32c779b4d62a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.759ex; height:2.843ex;" alt="{\displaystyle \{\neg ,\land ,\lor ,\rightarrow ,\leftrightarrow \}}">, con: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }:{\mathcal {B}}\mapsto {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }:{\mathcal {B}}\mapsto {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e1fafa3f0af194f9c00106d5620bf4c298b0813" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.453ex; margin-bottom: -0.385ex; width:12.986ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }:{\mathcal {B}}\mapsto {\mathcal {B}}}">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\circ }:{\mathcal {B}}\times {\mathcal {B}}\mapsto {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\circ }:{\mathcal {B}}\times {\mathcal {B}}\mapsto {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc2444b45a24f30ada4a8596ba216979bd479e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.453ex; margin-bottom: -0.385ex; width:17.095ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\circ }:{\mathcal {B}}\times {\mathcal {B}}\mapsto {\mathcal {B}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c76f8e7901565e8a93ba0e159a1e3a86fb2176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}">.</li></ul></li></ol></div> <div class="mw-heading mw-heading4"><h4 id="Negazione">Negazione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=9" title="Modifica la sezione Negazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=9" title="Edit section's source code: Negazione">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> La funzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620dd1765fa0c24d4a4a6041389a68ecd448aab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.453ex; margin-bottom: -0.385ex; width:4.348ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }}"> della logica proposizionale è definita come segue: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }(\mathrm {T} )=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }(\mathrm {T} )=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5dc6ab92aba931d2794e8b79e31b82569ae8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.452ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }(\mathrm {T} )=\mathrm {F} }"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }(\mathrm {F} )=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }(\mathrm {F} )=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee0205f9be3c5d912846a5ca4f0fc1e6596ce91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.452ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }(\mathrm {F} )=\mathrm {T} }"></li></ul></div> Questo è di solito espresso in maniera concisa come: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\mathrm {T} )=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (\mathrm {T} )=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779e0e128db6470e31a5e1db79a72064ad05cedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \neg (\mathrm {T} )=\mathrm {F} }"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\mathrm {F} )=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (\mathrm {F} )=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/541046104a873b8b1a29fa2e57380e6c67898455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \neg (\mathrm {F} )=\mathrm {T} }"></li></ul> cioè, la funzione di valutazione associata a un connettivo viene indicata col simbolo stesso del connettivo, e viene definita in forma tabellare mediante la tavola o tabella dei valori di verità per il connettivo come segue: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{|c|c|}A&\neg A\\\hline \mathrm {F} &\mathrm {T} \\\mathrm {T} &\mathrm {F} \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="left right"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{|c|c|}A&\neg A\\\hline \mathrm {F} &\mathrm {T} \\\mathrm {T} &\mathrm {F} \end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde9022d29150a5cdc5bc3c91ad513c440b33771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:9.541ex; height:10.843ex;" alt="{\displaystyle {\begin{array}{|c|c|}A&\neg A\\\hline \mathrm {F} &\mathrm {T} \\\mathrm {T} &\mathrm {F} \end{array}}}"> Tale tabella definisce la semantica dell'operazione di negazione. Se la frase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è vera, la sua negazione è falsa; se la frase <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è falsa, la sua negazione è vera. Si tratta dell'unico connettivo unario, in quanto agisce su un'unica sentenza. Evitando di presentare le funzioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189e0b507984d7a54480638b15ab9ee84bf50843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.453ex; margin-bottom: -0.385ex; width:4.074ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\circ }}"> della logica proposizionale per i connettivi binari, le presentiamo subito in forma di tabella dei valori di verità. <div class="mw-heading mw-heading4"><h4 id="Congiunzione">Congiunzione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=10" title="Modifica la sezione Congiunzione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=10" title="Edit section's source code: Congiunzione">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{|c|c|c|}A&B&A\land B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {F} \\\mathrm {F} &\mathrm {T} &\mathrm {F} \\\mathrm {T} &\mathrm {F} &\mathrm {F} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="left right"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> <mtd> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{|c|c|c|}A&B&A\land B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {F} \\\mathrm {F} &\mathrm {T} &\mathrm {F} \\\mathrm {T} &\mathrm {F} &\mathrm {F} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0201c1d98c7ca20b9a7a2858bcbdc570babc8c34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:16.424ex; height:17.176ex;" alt="{\displaystyle {\begin{array}{|c|c|c|}A&B&A\land B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {F} \\\mathrm {F} &\mathrm {T} &\mathrm {F} \\\mathrm {T} &\mathrm {F} &\mathrm {F} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}"> Il connettivo di congiunzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"> viene definito in modo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"> è vero se sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> (i due congiunti) sono veri, quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\land }=min}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∧</mo> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\land }=min}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58022c78ace486266aeb7147312d27384f38fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.684ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\land }=min}">. </div> <div class="mw-heading mw-heading4"><h4 id="Disgiunzione">Disgiunzione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=11" title="Modifica la sezione Disgiunzione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=11" title="Edit section's source code: Disgiunzione">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{|c|c|c|}A&B&A\lor B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {F} \\\mathrm {F} &\mathrm {T} &\mathrm {T} \\\mathrm {T} &\mathrm {F} &\mathrm {T} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="left right"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> <mtd> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{|c|c|c|}A&B&A\lor B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {F} \\\mathrm {F} &\mathrm {T} &\mathrm {T} \\\mathrm {T} &\mathrm {F} &\mathrm {T} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ecab58e55dcda4d77249916ecab812b7aee0d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:16.424ex; height:17.176ex;" alt="{\displaystyle {\begin{array}{|c|c|c|}A&B&A\lor B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {F} \\\mathrm {F} &\mathrm {T} &\mathrm {T} \\\mathrm {T} &\mathrm {F} &\mathrm {T} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}"> Il connettivo di disgiunzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"> viene definito in modo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"> è vero se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> (uno dei due disgiunti) sono veri, quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\lor }=max}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∨</mo> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\lor }=max}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5679bc4b548cdec190aaabe56406f865a2a0c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.046ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\lor }=max}">. Si noti che, con questa definizione, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"> è la disgiunzione inclusiva, cioè corrisponde al "vel" latino e non all'"aut". </div> <div class="mw-heading mw-heading4"><h4 id="Implicazione">Implicazione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=12" title="Modifica la sezione Implicazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=12" title="Edit section's source code: Implicazione">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\to }(A,B)={\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(A),B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∨</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\to }(A,B)={\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(A),B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61cf24e35c934503167f262f4b74e6a3e81b3364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.199ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\to }(A,B)={\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(A),B)}"> La definizione della semantica del connettivo di implicazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"> (detta implicazione materiale, in cui <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è detto premessa e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> conseguenza) è, in un certo senso, meno intuitiva. Innanzi tutto si noti che, con la definizione data, si ha che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fbf720da5a9387e23c628079fbc3e021399c911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle A\to A}"> è sempre vero, qualunque sia il valore di verità di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">; questo corrisponde alla nostra intuizione. Possiamo quindi accettare il fatto che, affinché <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"> sia vero, basta che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> sia vero, indipendentemente dal valore di verità di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. Questo di fatto ci dice che, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> è vero e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5c92411f29ae67df761f2d1500ce5e79e4517c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.142ex; height:2.176ex;" alt="{\displaystyle B\to B}"> è vero, possiamo "rafforzare" la premessa sostituendo a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> un qualunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> e l'implicazione resta vera. In sostanza, l'implicazione è falsa nel solo caso in cui la premessa è vera e la conseguenza è falsa. </div> <div class="mw-heading mw-heading4"><h4 id="Doppia_implicazione">Doppia implicazione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=13" title="Modifica la sezione Doppia implicazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=13" title="Edit section's source code: Doppia implicazione">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{|c|c|c|}A&B&A\leftrightarrow B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {T} \\\mathrm {F} &\mathrm {T} &\mathrm {F} \\\mathrm {T} &\mathrm {F} &\mathrm {F} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="left right"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> <mtd> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{|c|c|c|}A&B&A\leftrightarrow B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {T} \\\mathrm {F} &\mathrm {T} &\mathrm {F} \\\mathrm {T} &\mathrm {F} &\mathrm {F} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a865d6ad025e94bdc01a2957fde3ca7593e983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:17.456ex; height:17.176ex;" alt="{\displaystyle {\begin{array}{|c|c|c|}A&B&A\leftrightarrow B\\\hline \mathrm {F} &\mathrm {F} &\mathrm {T} \\\mathrm {F} &\mathrm {T} &\mathrm {F} \\\mathrm {T} &\mathrm {F} &\mathrm {F} \\\mathrm {T} &\mathrm {T} &\mathrm {T} \\\end{array}}}"> La definizione della semantica del connettivo di doppia implicazione è del tutto intuitiva: il valore di verità di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}"> è vero se i valori di verità di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> coincidono. </div> <div class="mw-heading mw-heading4"><h4 id="Assegnazione_booleana">Assegnazione booleana</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=14" title="Modifica la sezione Assegnazione booleana" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=14" title="Edit section's source code: Assegnazione booleana">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Un'assegnazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"> ai simboli proposizionali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"> è una funzione totale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}:{\mathcal {P}}\mapsto {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}:{\mathcal {P}}\mapsto {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3049c07d641abf7d72370983cee31ac2a920ae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.328ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}:{\mathcal {P}}\mapsto {\mathcal {B}}}"> </div> Per il fatto che l'insieme delle formule proposizionali è liberamente generato, ogni assegnazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"> si estende a un'unica interpretazione o valutazione booleana. <div class="mw-heading mw-heading4"><h4 id="Valutazione_booleana">Valutazione booleana</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=15" title="Modifica la sezione Valutazione booleana" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=15" title="Edit section's source code: Valutazione booleana">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Una valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}:\mathrm {PROP} \mapsto {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">P</mi> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}:\mathrm {PROP} \mapsto {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01390f9b13188b6c75909373a7820ba1102ea2a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.116ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}:\mathrm {PROP} \mapsto {\mathcal {B}}}"> è l'estensione a PROP di un'assegnazione booleana, cioè <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)={\mathcal {V}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)={\mathcal {V}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef27be6d50d06462372dc7344258be2c7241605d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.07ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)={\mathcal {V}}(A)}"> se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5e1e51c194308f16c0001472782a8b6c22523c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.287ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {P}}}">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(\top )=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">⊤</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(\top )=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd2fcb6276a75c7aa306bc1147c8e877273d8d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.731ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(\top )=\mathrm {T} }">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(\bot )=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(\bot )=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ced6cab0e751f68ba479ff61bc74c6809fa9a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.571ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(\bot )=\mathrm {F} }">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/586916792812a258e6f1d1dca501acaa29de372e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.584ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))}">;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A\circ B)={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A),I_{\mathcal {V}}(B))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∘</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A\circ B)={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A),I_{\mathcal {V}}(B))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb6c21cb43c2d8e8f0534f811c24d458f5e0d136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.663ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A\circ B)={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A),I_{\mathcal {V}}(B))}">.</li></ul> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c76f8e7901565e8a93ba0e159a1e3a86fb2176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}">. Data un'assegnazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}">, l'esistenza e l'unicità dell'estensione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}"> sono garantite dal <a href="/w/index.php?title=Teorema_di_ricorsione&action=edit&redlink=1" class="new" title="Teorema di ricorsione (la pagina non esiste)">teorema di ricorsione</a>. </div> Il valore di verità di una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> dipende dall'assegnazione booleana a tutti i simboli proposizionali, inclusi quelli che non occorrono in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. Tuttavia, per dare una valutazione booleana di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> basta dire come vengono valutati (interpretati) i simboli proposizionali di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. Infatti, sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> una formula proposizionale e sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271bbf8e62dc1c446f442cc70f3a712fc1f10dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.483ex; height:2.843ex;" alt="{\displaystyle simb(A)}"> l'insieme dei simboli proposizionali che occorrono in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, per essi vale la seguente proprietà. <div class="mw-heading mw-heading4"><h4 id="Teorema_di_equivalenza_delle_valutazioni">Teorema di equivalenza delle valutazioni</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=16" title="Modifica la sezione Teorema di equivalenza delle valutazioni" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=16" title="Edit section's source code: Teorema di equivalenza delle valutazioni">modifica sorgente</a>]</div> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d148fbb70b11d450bedc52f429bce99636bd91d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle {\mathcal {V}}_{1}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98abe4cfe1a454b053b423884cedeb2af2d8d526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle {\mathcal {V}}_{2}}"> sono due assegnazioni che coincidono su <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle simb(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle simb(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271bbf8e62dc1c446f442cc70f3a712fc1f10dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.483ex; height:2.843ex;" alt="{\displaystyle simb(A)}">, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aee64219e3e26b27ecc452d75c79391c9b183e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.392ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}">. Dimostrazione Procediamo per induzione strutturale. <ul><li>(Passo Base) Osserviamo che l'asserto è vero per le formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"> perché non contengono simboli proposizionali. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5e1e51c194308f16c0001472782a8b6c22523c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.287ex; height:2.176ex;" alt="{\displaystyle A\in {\mathcal {P}}}">, allora abbiamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V_{1}}}(A)={\mathcal {V_{2}}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V_{1}}}(A)={\mathcal {V_{2}}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd374a7bd016e261182025fd6bd7d2191574d8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.162ex; height:2.843ex;" alt="{\displaystyle {\mathcal {V_{1}}}(A)={\mathcal {V_{2}}}(A)}"> e, per definizione, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aee64219e3e26b27ecc452d75c79391c9b183e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.392ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}">. Pertanto l'asserto vale per i simboli proposizionali.</li></ul> <ul><li>(Passo Induttivo) Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(\neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(\neg B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed50eef7ffd5a2c021f1a693319ea297343d7f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.965ex; height:2.843ex;" alt="{\displaystyle A=(\neg B)}">. Abbiamo per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5212ad2f25b77f60c6ea477f0226bbc5bd8fd7e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.094ex; height:2.676ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/827f8c8586ff4979d968e922b495acb62f5f7484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.094ex; height:2.676ex;" alt="{\displaystyle I_{\mathcal {V_{2}}}}"> che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(\neg B)={\mathcal {Op}}_{\neg }(I_{\mathcal {V_{1}}}(B))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(\neg B)={\mathcal {Op}}_{\neg }(I_{\mathcal {V_{1}}}(B))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae859843dd6298ba527ee5c7cf7a0b59f1ee31e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.142ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(\neg B)={\mathcal {Op}}_{\neg }(I_{\mathcal {V_{1}}}(B))}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{2}}}(\neg B)={\mathcal {Op}}_{\neg }(I_{\mathcal {V_{2}}}(B))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{2}}}(\neg B)={\mathcal {Op}}_{\neg }(I_{\mathcal {V_{2}}}(B))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf462ecc7a064ef3417a64dda8bb177586a6dcdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.142ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{2}}}(\neg B)={\mathcal {Op}}_{\neg }(I_{\mathcal {V_{2}}}(B))}"> e dunque, poiché per ipotesi induttiva <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(B)=I_{\mathcal {V_{2}}}(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(B)=I_{\mathcal {V_{2}}}(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0fd86b8b591ec79c1b45febfd18655d71c5c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.434ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(B)=I_{\mathcal {V_{2}}}(B)}">, segue che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aee64219e3e26b27ecc452d75c79391c9b183e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.392ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}">. Analogamente, sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=B\circ C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> <mo>∘</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B\circ C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e7889ba78b94715d95139fbe3af75f3929b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.567ex; height:2.176ex;" alt="{\displaystyle A=B\circ C}"> con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c76f8e7901565e8a93ba0e159a1e3a86fb2176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}">; abbiamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(B\circ C)={\mathcal {Op}}_{\circ }(I_{\mathcal {V_{1}}}(B),I_{\mathcal {V_{1}}}(C))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∘</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(B\circ C)={\mathcal {Op}}_{\circ }(I_{\mathcal {V_{1}}}(B),I_{\mathcal {V_{1}}}(C))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d17d05cac01b3a7c2db38725b968e609f56461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.982ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(B\circ C)={\mathcal {Op}}_{\circ }(I_{\mathcal {V_{1}}}(B),I_{\mathcal {V_{1}}}(C))}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{2}}}(B\circ C)={\mathcal {Op}}_{\circ }(I_{\mathcal {V_{2}}}(B),I_{\mathcal {V_{2}}}(C))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∘</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{2}}}(B\circ C)={\mathcal {Op}}_{\circ }(I_{\mathcal {V_{2}}}(B),I_{\mathcal {V_{2}}}(C))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a61fc3902e69404cc31497dc195f7f1375d6e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.982ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{2}}}(B\circ C)={\mathcal {Op}}_{\circ }(I_{\mathcal {V_{2}}}(B),I_{\mathcal {V_{2}}}(C))}"> e, per ipotesi induttiva, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(B)=I_{\mathcal {V_{2}}}(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(B)=I_{\mathcal {V_{2}}}(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0fd86b8b591ec79c1b45febfd18655d71c5c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.434ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(B)=I_{\mathcal {V_{2}}}(B)}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(C)=I_{\mathcal {V_{2}}}(C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(C)=I_{\mathcal {V_{2}}}(C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6571d131907d4c6a66877477fe921e4bb3dd4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.439ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(C)=I_{\mathcal {V_{2}}}(C)}">, segue che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aee64219e3e26b27ecc452d75c79391c9b183e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.392ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V_{1}}}(A)=I_{\mathcal {V_{2}}}(A)}">.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Soddisfacibilità">Soddisfacibilità</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=17" title="Modifica la sezione Soddisfacibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=17" title="Edit section's source code: Soddisfacibilità">modifica sorgente</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Formula_soddisfatta">Formula soddisfatta</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=18" title="Modifica la sezione Formula soddisfatta" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=18" title="Edit section's source code: Formula soddisfatta">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Una formula della logica proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è soddisfatta da una valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}"> sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08254f62ed7568c34584df62a4bef0f20116a76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.666ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }">. </div> <div class="mw-heading mw-heading4"><h4 id="Formula_soddisfacibile">Formula soddisfacibile</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=19" title="Modifica la sezione Formula soddisfacibile" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=19" title="Edit section's source code: Formula soddisfacibile">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Una formula della logica proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è soddisfacibile sse è soddisfatta da una qualche valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}">. </div> <div class="mw-heading mw-heading4"><h4 id="Tautologia">Tautologia</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=20" title="Modifica la sezione Tautologia" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=20" title="Edit section's source code: Tautologia">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Una formula della logica proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è tautologica, ossia è una tautologia, sse è soddisfatta da ogni valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}">. </div> <div class="mw-heading mw-heading4"><h4 id="Contraddizione">Contraddizione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=21" title="Modifica la sezione Contraddizione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=21" title="Edit section's source code: Contraddizione">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Una formula della logica proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è contraddittoria, ossia è una contraddizione, sse non è soddisfatta da alcuna valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}">. </div> <div class="mw-heading mw-heading4"><h4 id="Tavole_di_verità">Tavole di verità</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=22" title="Modifica la sezione Tavole di verità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=22" title="Edit section's source code: Tavole di verità">modifica sorgente</a>]</div> Dal teorema di equivalenza delle valutazioni segue che, per determinare se una formula proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è soddisfacibile, tautologica o contraddittoria, basta costruire una tabella di valori di verità in cui compaiono <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> colonne, dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> è il numero di simboli proposizionali distinti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},...,A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},...,A_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27790c7851ada2f2d78ce102293ac5de0ce0b85b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.929ex; height:2.509ex;" alt="{\displaystyle A_{1},...,A_{n}}"> che compaiono in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> e nella n+1-esima colonna viene riportato il corrispondente valore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f816196f0e7c6c8ad512f66c73fc0cf99b7956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.889ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)}">. La tabella ha <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"> righe, tante quante sono le possibili assegnazioni distinte di valori di verità ai simboli di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{|c|c|c|c|}A_{1}&...&A_{n}&I_{\mathcal {V}}(A)\\\hline \mathrm {F} &...&\mathrm {F} &I_{\mathcal {V_{1}}}(A)\\...&...&...&...\\\mathrm {T} &...&\mathrm {T} &I_{\mathcal {V_{2^{n}}}}(A)\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="left right"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid solid"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn class="MJX-tex-caligraphic" mathvariant="script">1</mn> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn class="MJX-tex-caligraphic" mathvariant="script">2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">n</mi> </mrow> </msup> </mrow> </msub> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{|c|c|c|c|}A_{1}&...&A_{n}&I_{\mathcal {V}}(A)\\\hline \mathrm {F} &...&\mathrm {F} &I_{\mathcal {V_{1}}}(A)\\...&...&...&...\\\mathrm {T} &...&\mathrm {T} &I_{\mathcal {V_{2^{n}}}}(A)\\\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420db1a2e869468487d29c1e1dfa0bb8c7971245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:25.204ex; height:14.509ex;" alt="{\displaystyle {\begin{array}{|c|c|c|c|}A_{1}&...&A_{n}&I_{\mathcal {V}}(A)\\\hline \mathrm {F} &...&\mathrm {F} &I_{\mathcal {V_{1}}}(A)\\...&...&...&...\\\mathrm {T} &...&\mathrm {T} &I_{\mathcal {V_{2^{n}}}}(A)\\\end{array}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una tautologia sse l'ultima colonna contiene solo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cc73e47284b51d2ab60d333176c2366a333e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {T} }">, è soddisfacibile sse essa contiene almeno una <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cc73e47284b51d2ab60d333176c2366a333e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {T} }">, è contraddittoria sse essa contiene tutte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd88ce01cebb719c991531bf9a76dacc204f6d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.518ex; height:2.176ex;" alt="{\displaystyle \mathrm {F} }">. <div class="mw-heading mw-heading4"><h4 id="Teorema">Teorema</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=23" title="Modifica la sezione Teorema" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=23" title="Edit section's source code: Teorema">modifica sorgente</a>]</div> Una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una tautologia sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> è una contraddizione. Dimostrazione (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }">) Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> una tautologia. Per definizione, per ogni valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}"> si ha che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08254f62ed7568c34584df62a4bef0f20116a76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.666ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }">. Ora, per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> si ha che: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))={\mathcal {Op}}_{\neg }(\mathrm {T} )=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))={\mathcal {Op}}_{\neg }(\mathrm {T} )=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e5acba4c66a73bafb36b136301c4fd2a4f7a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.134ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))={\mathcal {Op}}_{\neg }(\mathrm {T} )=\mathrm {F} }"> quindi, per ogni valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}"> si ha che: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(\neg A)=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(\neg A)=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee09de49e86e4cfd9d67c55bb3c7fbaa363a181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.056ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(\neg A)=\mathrm {F} }"> dunque, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> è una contraddizione. Procedendo all'inverso, si dimostra analogamente che, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> è una contraddizione, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una tautologia: (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇐</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftarrow }">) sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> una contraddizione. Per definizione, per ogni valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}"> si ha che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(\neg A)=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(\neg A)=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee09de49e86e4cfd9d67c55bb3c7fbaa363a181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.056ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(\neg A)=\mathrm {F} }">. Ora, per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> si ha che: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/586916792812a258e6f1d1dca501acaa29de372e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.584ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(\neg A)={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))}"> quindi: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf6c96b0f5adbb9f8042390a8263e701b770687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.663ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A))=\mathrm {F} }"> da cui, per la definizione di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620dd1765fa0c24d4a4a6041389a68ecd448aab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.453ex; margin-bottom: -0.385ex; width:4.348ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }}">, si deduce che, per ogni valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08254f62ed7568c34584df62a4bef0f20116a76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.666ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> dunque, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una tautologia. <div class="mw-heading mw-heading3"><h3 id="Equivalenza_logica">Equivalenza logica</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=24" title="Modifica la sezione Equivalenza logica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=24" title="Edit section's source code: Equivalenza logica">modifica sorgente</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Implicazione_logica">Implicazione logica</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=25" title="Modifica la sezione Implicazione logica" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=25" title="Edit section's source code: Implicazione logica">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Diciamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> implica logicamente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> sse ogniqualvolta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08254f62ed7568c34584df62a4bef0f20116a76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.666ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> anche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(B)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(B)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd38437aa80f0694d4d4bdcb3bb10e0583d31ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.687ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(B)=\mathrm {T} }">.</div> <div class="mw-heading mw-heading4"><h4 id="Equivalenza_logica_(Definizione)">Equivalenza logica (Definizione)</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=26" title="Modifica la sezione Equivalenza logica (Definizione)" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=26" title="Edit section's source code: Equivalenza logica (Definizione)">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Diciamo che due proposizioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> sono logicamente equivalenti o tautologicamente equivalenti sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=I_{\mathcal {V}}(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=I_{\mathcal {V}}(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/946cde5333387b2ee6289dceea843ea704f06477" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.898ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=I_{\mathcal {V}}(B)}"> per ogni valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}">. In tal caso scriviamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\equiv B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≡</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\equiv B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b933daba3ef47ec3b4f3097ea6e741b85149707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\equiv B}">. </div> Diamo qui di seguito una lista di formule tautologicamente equivalenti. L'equivalenza può essere facilmente verificata tramite tavole di verità. <table> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0142024d07476b780afdeea24f9a3194df075e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.069ex; height:2.176ex;" alt="{\displaystyle A\land A}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </td> <td>idempotenza della congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31940c78d50bd53d7a2a8e89351754385ec6daeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.069ex; height:2.176ex;" alt="{\displaystyle A\lor A}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </td> <td>idempotenza della disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land (B\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land (B\land C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c377ae1c2100e8f4d45873459a129b5c6c82413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.248ex; height:2.843ex;" alt="{\displaystyle A\land (B\land C)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)\land C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)\land C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bea7811042e8fdd9ecbfc264d161abf016c3e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.248ex; height:2.843ex;" alt="{\displaystyle (A\land B)\land C}"> </td> <td>associatività della congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor (B\lor C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∨</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor (B\lor C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ddedd65808d2fe838e068d4ca1a25fd828fe64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.248ex; height:2.843ex;" alt="{\displaystyle A\lor (B\lor C)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\lor B)\lor C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\lor B)\lor C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87dcaf261c40830dd08f0c07095afbd2790c2bdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.248ex; height:2.843ex;" alt="{\displaystyle (A\lor B)\lor C}"> </td> <td>associatività della disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow (B\leftrightarrow C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow (B\leftrightarrow C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3837a80384c25055f7d7cbc862508b0075cddd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.311ex; height:2.843ex;" alt="{\displaystyle A\leftrightarrow (B\leftrightarrow C)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\leftrightarrow B)\leftrightarrow C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow B)\leftrightarrow C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8471e31b8aeb7acb8ca9edc8bb4c2c3ebfd931d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.311ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow B)\leftrightarrow C}"> </td> <td>associatività della doppia implicazione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\land A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∧</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\land A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5920298dbda4592b71e53822ce01829cd77f4190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle B\land A}"> </td> <td>commutatività della congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\lor A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\lor A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22877766a9d54721b6be140f7b16d9db0bad0b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle B\lor A}"> </td> <td>commutatività della disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\leftrightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">↔</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\leftrightarrow A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7af99ff4fc686b0e512ce505333c779a8b639ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle B\leftrightarrow A}"> </td> <td>commutatività della doppia implicazione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land (B\lor C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∨</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land (B\lor C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df394b2737c9b42f114b26dbce84dfa4917b1a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.248ex; height:2.843ex;" alt="{\displaystyle A\land (B\lor C)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)\lor (A\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)\lor (A\land C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f564623cc81c802fdbd0f0d4cdf60d4d7a474922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.383ex; height:2.843ex;" alt="{\displaystyle (A\land B)\lor (A\land C)}"> </td> <td>distributività della congiunzione sulla disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor (B\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor (B\land C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ea6fe8908dde92eef2fa400b6be28aee1f06f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.248ex; height:2.843ex;" alt="{\displaystyle A\lor (B\land C)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\lor B)\land (A\lor C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\lor B)\land (A\lor C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d860c5c0c6f76e2352d9a5075575a94bffc8c88f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.383ex; height:2.843ex;" alt="{\displaystyle (A\lor B)\land (A\lor C)}"> </td> <td>distributività della disgiunzione sulla congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land (A\lor B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land (A\lor B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f12fe311811fd2b575514e32336729d5ea05b4b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.225ex; height:2.843ex;" alt="{\displaystyle A\land (A\lor B)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </td> <td>assorbimento della congiunzione sulla disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor (A\land B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12bb8488352b9416283e3cff2da9b74637adae5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.225ex; height:2.843ex;" alt="{\displaystyle A\lor (A\land B)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </td> <td>assorbimento della disgiunzione sulla congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512837c8be6d69e76981db615488da00a7c167e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.134ex; height:2.176ex;" alt="{\displaystyle A\land \top }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </td> <td>neutralità del vero nella congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8c6b3ef3b282c66aedf38984327a46cee96887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.134ex; height:2.176ex;" alt="{\displaystyle A\lor \top }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"> </td> <td>assorbimento del vero nella disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3777a0d975a3a933bd8ff45d1d7bf401fde50690" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.134ex; height:2.176ex;" alt="{\displaystyle A\lor \bot }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </td> <td>neutralità del falso nella disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b26356dd4c49b4fff1ebf051874b01089b33a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.134ex; height:2.176ex;" alt="{\displaystyle A\land \bot }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"> </td> <td>assorbimento del falso nella congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17b69b336b49d0b40a58dd18025feda76c73a1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.844ex; height:2.176ex;" alt="{\displaystyle \neg \neg A}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </td> <td>doppia negazione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (A\land B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (A\land B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ba373711ac586009d3c4d0117525fbe04a1a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.449ex; height:2.843ex;" alt="{\displaystyle \neg (A\land B)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\lor \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\lor \neg B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1020d4923bd093b4d10a73c88d3db0b3211b4ec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.19ex; height:2.176ex;" alt="{\displaystyle \neg A\lor \neg B}"> </td> <td>legge di De Morgan per la congiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (A\lor B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (A\lor B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4be2de29b87215d778d98ab2d26c4e166056ca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.449ex; height:2.843ex;" alt="{\displaystyle \neg (A\lor B)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\land \neg B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\land \neg B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5008fff96d0c52ff4bfd0361d07b622fd23a8f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.19ex; height:2.176ex;" alt="{\displaystyle \neg A\land \neg B}"> </td> <td>legge di De Morgan per la disgiunzione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59bfe2f8010f7f758cae49d2b8fa7b2c4f812b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.619ex; height:2.176ex;" alt="{\displaystyle A\lor \neg A}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }"> </td> <td>terzo escluso </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695d9f9ae765ace242db86b6f7cdc322cfa99145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.619ex; height:2.176ex;" alt="{\displaystyle A\land \neg A}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"> </td> <td>contraddizione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg B\to \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg B\to \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed8b28f42f63937db9b96078fbbcf92a8fc1cd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.222ex; height:2.176ex;" alt="{\displaystyle \neg B\to \neg A}"> </td> <td>contrapposizione </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\lor B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13321f163366a0912c15b09d401b33e29a1ad46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle \neg A\lor B}"> </td> <td>trasformazione dell'implicazione in disgiunzione </td></tr> </tbody></table> <a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Esercizi_su_algebra_delle_proposizioni" title="Logica matematica/Calcolo delle proposizioni/Esercizi su algebra delle proposizioni">Esercizi ed esempi</a> Relativamente alla legge del terzo escluso, il fatto che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59bfe2f8010f7f758cae49d2b8fa7b2c4f812b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.619ex; height:2.176ex;" alt="{\displaystyle A\lor \neg A}"> sia sempre vero sembra del tutto intuitivo. In effetti, questa è una caratteristica della logica classica; in altre logiche, per esempio nella logica intuizionista, questo fatto non vale. Intuitivamente ci si aspetta che, se una proposizione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è logicamente equivalente a una proposizione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, sostituendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> al posto di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> in una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">, il valore di verità di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> non cambi. Cerchiamo ora di rendere più precisa questa nozione. Indichiamo con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[p]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38f8a6d48ca352c71da6f0955c23e9be49548f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.204ex; height:2.843ex;" alt="{\displaystyle F[p]}"> una formula proposizionale che può contenere delle occorrenze del simbolo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> (detto metavariabile o parametro proposizionale, cioè <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> sta ad indicare un atomo). Con la notazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[X/p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[X/p]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b4799480945ddafe7c0a9f901665d3f81032956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.346ex; height:2.843ex;" alt="{\displaystyle F[X/p]}"> indicheremo la formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26dc049157db08b423b1c6916eba0c81b1bdc18b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.014ex; height:2.843ex;" alt="{\displaystyle F[X]}">, in cui tutte le occorrenze di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> sono state uniformemente e simultaneamente sostituite con occorrenze di una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. Chiameremo quest'operazione sostituzione uniforme. <div class="mw-heading mw-heading4"><h4 id="Primo_teorema_del_rimpiazzamento">Primo teorema del rimpiazzamento</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=27" title="Modifica la sezione Primo teorema del rimpiazzamento" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=27" title="Edit section's source code: Primo teorema del rimpiazzamento">modifica sorgente</a>]</div> Siano <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[p]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38f8a6d48ca352c71da6f0955c23e9be49548f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.204ex; height:2.843ex;" alt="{\displaystyle F[p]}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> formule proposizionali e sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}"> una valutazione booleana. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36c9f891ce715292a61aeb738736fbd44910212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.144ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}">, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1092db6b851f7bccab60e639686b3c53d06aa648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.877ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}">. Dimostrazione Per induzione strutturale. Osserviamo innanzitutto che, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> non occorre in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}">, nessuna sostituzione è stata fatta, e quindi l'asserto è banalmente verificato. (Passo Base) Supponiamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[p]=p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[p]=p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b301a25bc3164dc8f3c9489ef32e36bf80ed304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.472ex; height:2.843ex;" alt="{\displaystyle F[p]=p}">, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[X/p]=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[X/p]=X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e82cf35669584077d227ce387dc64c09a8184a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.425ex; height:2.843ex;" alt="{\displaystyle F[X/p]=X}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[Y/p]=Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[Y/p]=Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e7d301f2c6b83977eea40c97d78397c8d74392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.011ex; height:2.843ex;" alt="{\displaystyle F[Y/p]=Y}">; per ipotesi, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36c9f891ce715292a61aeb738736fbd44910212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.144ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}">, e dunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)=I_{\mathcal {V}}(F[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)=I_{\mathcal {V}}(F[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c4e6e7d5ffbb5c0e19ff244d04af8ab011554e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.119ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)=I_{\mathcal {V}}(F[Y/p])}">. (Passo Induttivo) Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[p]=\neg A[p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[p]=\neg A[p]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caae6c630a10c3ef96afe6280083f432d2359623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.059ex; height:2.843ex;" alt="{\displaystyle F[p]=\neg A[p]}">. Osserviamo che, per definizione di valutazione booleana: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(F[X/p])={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[X/p]))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(F[X/p])={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[X/p]))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df46c991fceb25a1bbdf0bfb8d8558d7309ce29a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.243ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(F[X/p])={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[X/p]))}">. Per ipotesi induttiva: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A[X/p])=I_{\mathcal {V}}(A[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A[X/p])=I_{\mathcal {V}}(A[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86ba7f422cd51cfc577ccc14d78aea277f927a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.881ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A[X/p])=I_{\mathcal {V}}(A[Y/p])}">. Quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[X/p]))={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[Y/p]))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[X/p]))={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[Y/p]))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d592c805e5cd5febae3bd86464bd2a92c0347b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.196ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[X/p]))={\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[Y/p]))}">. Di nuovo, per definizione di valutazione booleana: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[Y/p]))=I_{\mathcal {V}}(F[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[Y/p]))=I_{\mathcal {V}}(F[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc752c3d88c913304a8613096fb97c1c3b4a04ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.829ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A[Y/p]))=I_{\mathcal {V}}(F[Y/p])}">. Dunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1092db6b851f7bccab60e639686b3c53d06aa648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.877ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}">. Analogamente, sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[p]=A[p]\circ B[p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> <mo>∘</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[p]=A[p]\circ B[p]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/413ff329e6e2e5b3ce1825d1f8a5e9d3f525f16b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.93ex; height:2.843ex;" alt="{\displaystyle F[p]=A[p]\circ B[p]}">, con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c76f8e7901565e8a93ba0e159a1e3a86fb2176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}">. Osserviamo che, per definizione di valutazione booleana: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(F[X/p])={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[X/p]),I_{\mathcal {V}}(B[X/p]))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(F[X/p])={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[X/p]),I_{\mathcal {V}}(B[X/p]))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700b3d643821a2428ee3fa3c30bb0a89ed1b6b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.518ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(F[X/p])={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[X/p]),I_{\mathcal {V}}(B[X/p]))}">. Per ipotesi induttiva: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A[X/p])=I_{\mathcal {V}}(A[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A[X/p])=I_{\mathcal {V}}(A[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86ba7f422cd51cfc577ccc14d78aea277f927a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.881ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A[X/p])=I_{\mathcal {V}}(A[Y/p])}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(B[X/p])=I_{\mathcal {V}}(B[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(B[X/p])=I_{\mathcal {V}}(B[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed7d54f5fb9dab2a8b0be815e79cfa4ccd783d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.923ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(B[X/p])=I_{\mathcal {V}}(B[Y/p])}">. Quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[X/p]),I_{\mathcal {V}}(B[X/p]))={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[Y/p]),I_{\mathcal {V}}(B[Y/p]))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[X/p]),I_{\mathcal {V}}(B[X/p]))={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[Y/p]),I_{\mathcal {V}}(B[Y/p]))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed676ba9c042404f32c92c559c0156cee93900e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.54ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[X/p]),I_{\mathcal {V}}(B[X/p]))={\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[Y/p]),I_{\mathcal {V}}(B[Y/p]))}">. Di nuovo, per definizione di valutazione booleana: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[Y/p]),I_{\mathcal {V}}(B[Y/p]))=I_{\mathcal {V}}(F[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[Y/p]),I_{\mathcal {V}}(B[Y/p]))=I_{\mathcal {V}}(F[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef85cfd21b542d37f7258dfd58c36bcd81967e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.898ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\circ }(I_{\mathcal {V}}(A[Y/p]),I_{\mathcal {V}}(B[Y/p]))=I_{\mathcal {V}}(F[Y/p])}">. Dunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1092db6b851f7bccab60e639686b3c53d06aa648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.877ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}">. <div class="mw-heading mw-heading4"><h4 id="Secondo_teorema_del_rimpiazzamento">Secondo teorema del rimpiazzamento</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=28" title="Modifica la sezione Secondo teorema del rimpiazzamento" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=28" title="Edit section's source code: Secondo teorema del rimpiazzamento">modifica sorgente</a>]</div> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\equiv Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>≡</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\equiv Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6a3a2b54357e4997530130050a25c8ce86c888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle X\equiv Y}">, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[X/p]\equiv F[Y/p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo>≡</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[X/p]\equiv F[Y/p]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5381924e00d01f6acfcc51b77fd38e22f0066206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.584ex; height:2.843ex;" alt="{\displaystyle F[X/p]\equiv F[Y/p]}">. Dimostrazione Essendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\equiv Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>≡</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\equiv Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6a3a2b54357e4997530130050a25c8ce86c888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle X\equiv Y}"> un'equivalenza logica, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36c9f891ce715292a61aeb738736fbd44910212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.144ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(X)=I_{\mathcal {V}}(Y)}"> vale per ogni valutazione booleana <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352ea412171100a3b6aea130b24ab1d683bbf3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle I_{\mathcal {V}}}">; per il primo teorema del rimpiazzamento, si ha che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1092db6b851f7bccab60e639686b3c53d06aa648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.877ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(F[X/p])=I_{\mathcal {V}}(F[Y/p])}"> vale per ogni valutazione booleana. Ne consegue che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[X/p]\equiv F[Y/p]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> <mo>≡</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[X/p]\equiv F[Y/p]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5381924e00d01f6acfcc51b77fd38e22f0066206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.584ex; height:2.843ex;" alt="{\displaystyle F[X/p]\equiv F[Y/p]}">. <div class="mw-heading mw-heading3"><h3 id="Modelli">Modelli</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=29" title="Modifica la sezione Modelli" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=29" title="Edit section's source code: Modelli">modifica sorgente</a>]</div> Possiamo fornire una nozione d'interpretazione basata sulla relazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }">. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"> un'assegnazione booleana. L'insieme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> associato a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"> è l'insieme dei simboli proposizionali tali che il loro valore di verità in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"> sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cc73e47284b51d2ab60d333176c2366a333e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathrm {T} }">. Più formalmente: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}=\{p|p\in {\mathcal {P}},{\mathcal {V}}(p)=\mathrm {T} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}=\{p|p\in {\mathcal {P}},{\mathcal {V}}(p)=\mathrm {T} \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768e3e21385bd6918ec677026d2c7705f5e37aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.376ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}=\{p|p\in {\mathcal {P}},{\mathcal {V}}(p)=\mathrm {T} \}}"> Definiamo la relazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models \subseteq (\wp {\mathcal {P}}\times {\mathcal {L}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨⊆</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">℘</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models \subseteq (\wp {\mathcal {P}}\times {\mathcal {L}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc1466bac35e23d8580e7842f9f2f9bee3fe0bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.904ex; height:2.843ex;" alt="{\displaystyle \models \subseteq (\wp {\mathcal {P}}\times {\mathcal {L}})}"> come segue: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0208f6d685882af30ec3730a498811316c026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.153ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A}"> sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08254f62ed7568c34584df62a4bef0f20116a76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.666ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {T} }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d562a51604f9969b8938b16bb7141b5d67cf99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.506ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {F} }"> </div> Osserviamo che, nella definizione di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }">, stiamo sottintendendo il linguaggio proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}">. Nel caso in cui non sia chiaro dal contesto a quale linguaggio ci si riferisce, allora si usa indicare <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models _{\mathcal {L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>⊨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models _{\mathcal {L}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b10facfba7c558983d4ea338ca3a333b433742c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.381ex; height:2.843ex;" alt="{\displaystyle \models _{\mathcal {L}}}">. <div class="mw-heading mw-heading4"><h4 id="Modello_per_una_formula">Modello per una formula</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=30" title="Modifica la sezione Modello per una formula" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=30" title="Edit section's source code: Modello per una formula">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> una formula. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0208f6d685882af30ec3730a498811316c026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.153ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A}"> diciamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> è un modello di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, ovvero che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> rende vera <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. </div> <div class="mw-heading mw-heading4"><h4 id="Contromodello_per_una_formula">Contromodello per una formula</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=31" title="Modifica la sezione Contromodello per una formula" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=31" title="Edit section's source code: Contromodello per una formula">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> una formula. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> diciamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> è un contromodello per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, ovvero che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> rende falsa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. </div> <div class="mw-heading mw-heading4"><h4 id="Modello_per_un_insieme_di_formule">Modello per un insieme di formule</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=32" title="Modifica la sezione Modello per un insieme di formule" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=32" title="Edit section's source code: Modello per un insieme di formule">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> rende vere tutte le formule di un insieme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">, cioè se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0208f6d685882af30ec3730a498811316c026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.153ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A}"> per ogni formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">, diciamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> è un modello per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> e indichiamo ciò con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9730319814c04890506d1efce8718d628c98e7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }">. </div> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una tautologia, possiamo scrivere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e16fd9aa2a895f487bd2117a08fdbcae64c02aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.403ex; height:2.843ex;" alt="{\displaystyle \vDash A}">, in quanto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è vera in ogni modello, compreso quello vuoto. <div class="mw-heading mw-heading4"><h4 id="Implicazione_logica_da_un_insieme_di_formule">Implicazione logica da un insieme di formule</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=33" title="Modifica la sezione Implicazione logica da un insieme di formule" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=33" title="Edit section's source code: Implicazione logica da un insieme di formule">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Dato un insieme di formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> e una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, diciamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> soddisfa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, o implica logicamente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, e scriviamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}">, sse (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac83e52f60777901ac486ff44c24e3bffa93d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.268ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0208f6d685882af30ec3730a498811316c026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.153ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A}">).</div> <div class="mw-heading mw-heading4"><h4 id="Soddisfacibilità_2">Soddisfacibilità</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=34" title="Modifica la sezione Soddisfacibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=34" title="Edit section's source code: Soddisfacibilità">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0208f6d685882af30ec3730a498811316c026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.153ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A}"> per qualche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}">, allora diciamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è soddisfacibile. </div> <div class="mw-heading mw-heading4"><h4 id="Insoddisfacibilità">Insoddisfacibilità</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=35" title="Modifica la sezione Insoddisfacibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=35" title="Edit section's source code: Insoddisfacibilità">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Se per nessun insieme di simboli proposizionali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> è verificato che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd0208f6d685882af30ec3730a498811316c026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.153ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A}">, allora diciamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è insoddisfacibile. </div>Quindi una formula è insoddisfacibile sse per essa non esiste alcun modello. Il simbolo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }"> è giustificato quando si considerano le nozioni di implicazione logica e di equivalenza logica, perché permettono una semplificazione notazionale. Infatti, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> implica logicamente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> scriviamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vDash A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vDash A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3dd78f4f2bb5f59773691ab01e215fbd0843c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.781ex; height:2.843ex;" alt="{\displaystyle \vDash A\to B}"> e se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è logicamente equivalente a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, cioè se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\equiv B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≡</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\equiv B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b933daba3ef47ec3b4f3097ea6e741b85149707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\equiv B}">, scriviamo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vDash A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vDash A\leftrightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9a10af182400342d2f9849c5b9b390c375199d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.781ex; height:2.843ex;" alt="{\displaystyle \vDash A\leftrightarrow B}">. Il seguente teorema lega le nozioni di implicazione logica e di insoddisfacibilità: <div class="mw-heading mw-heading4"><h4 id="Teorema_di_soddisfacibilità">Teorema di soddisfacibilità</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=36" title="Modifica la sezione Teorema di soddisfacibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=36" title="Edit section's source code: Teorema di soddisfacibilità">modifica sorgente</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}"> sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120ba861a8d6344ab9cd2efc9464975398857b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \Gamma \cup \{\neg A\}}"> è insoddisfacibile. Dimostrazione (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }">) Supponiamo che valga <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}">, ma <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120ba861a8d6344ab9cd2efc9464975398857b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \Gamma \cup \{\neg A\}}"> sia soddisfacibile; allora esiste un modello <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9730319814c04890506d1efce8718d628c98e7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07723341c46d2857fd0f0411ddfedc624d89ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.703ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \neg A}">, cioè <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9730319814c04890506d1efce8718d628c98e7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}">, dunque non vale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}">, contraddizione. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇐</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftarrow }">) Supponiamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120ba861a8d6344ab9cd2efc9464975398857b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \Gamma \cup \{\neg A\}}"> sia insoddisfacibile, ma non valga <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}">; allora esiste un modello <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9730319814c04890506d1efce8718d628c98e7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}">, cioè <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9730319814c04890506d1efce8718d628c98e7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.862ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \Gamma }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07723341c46d2857fd0f0411ddfedc624d89ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.703ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \neg A}">, quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models \Gamma \cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi mathvariant="normal">Γ</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models \Gamma \cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8159d37d924f4474c1691c36cc45fadc050a7672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.063ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models \Gamma \cup \{\neg A\}}">, dunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120ba861a8d6344ab9cd2efc9464975398857b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \Gamma \cup \{\neg A\}}"> è soddisfacibile, contraddizione. La proprietà appena enunciata è quella che giustifica le dimostrazioni per assurdo: se si vuole provare che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> segue da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">, basta assumere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> e provare che questo contraddice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">. <div class="mw-heading mw-heading3"><h3 id="Compattezza">Compattezza</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=37" title="Modifica la sezione Compattezza" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=37" title="Edit section's source code: Compattezza">modifica sorgente</a>]</div> Segue quanto abbiamo già notato per le valutazioni booleane: quando consideriamo i modelli di una formula, basta prendere in considerazione solo quelli in cui compaiono i simboli proposizionali della formula. È chiaro che, se in una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> occorrono <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> simboli proposizionali distinti, ovvero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |simb(A)|=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>m</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |simb(A)|=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e51e62b04f40d1b88b64854e919954270231af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.27ex; height:2.843ex;" alt="{\displaystyle |simb(A)|=n}">, allora il numero degli insiemi di simboli che ci interessa verificare per conoscere il valore di verità di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è al più <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}">. Tuttavia, è lecito chiedersi cosa succede se abbiamo un insieme infinito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> di proposizioni e vogliamo sapere se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}">. Il teorema di compattezza della soddisfacibilità fornisce una risposta a questa domanda. <div class="mw-heading mw-heading4"><h4 id="Insieme_di_formule_finitamente_soddisfacibile">Insieme di formule finitamente soddisfacibile</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=38" title="Modifica la sezione Insieme di formule finitamente soddisfacibile" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=38" title="Edit section's source code: Insieme di formule finitamente soddisfacibile">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Un insieme di formule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> si dice finitamente soddisfacibile sse ogni suo sottoinsieme finito è soddisfacibile. </div> <div class="mw-heading mw-heading4"><h4 id="Teorema_di_compattezza_della_soddisfacibilità">Teorema di compattezza della soddisfacibilità</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=39" title="Modifica la sezione Teorema di compattezza della soddisfacibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=39" title="Edit section's source code: Teorema di compattezza della soddisfacibilità">modifica sorgente</a>]</div> Un insieme di proposizioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> è soddisfacibile sse è finitamente soddisfacibile. Dimostrazione Per contraddizione. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }">) Supponiamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> sia soddisfacibile, ma che non sia finitamente soddisfacibile. Allora, esiste un sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> tale che, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}">, con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309072a341fcd77ba3c58cc6bdd214bbf697c416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.176ex;" alt="{\displaystyle A\in \Delta }">. Essendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }">, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3abcc1fad26b26a99f22baeb403e38f7f1f278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle A\in \Gamma }">, quindi, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> esiste una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3abcc1fad26b26a99f22baeb403e38f7f1f278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle A\in \Gamma }"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}">. Dunque, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> non è soddisfacibile. Contraddizione. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }">) Supponiamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> sia finitamente soddisfacibile, ma che non sia soddisfacibile. Allora, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}"> esiste una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b3abcc1fad26b26a99f22baeb403e38f7f1f278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle A\in \Gamma }"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}">. Quindi, esiste un sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> tale che, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}">, con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309072a341fcd77ba3c58cc6bdd214bbf697c416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.176ex;" alt="{\displaystyle A\in \Delta }">. Perciò, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"> non è soddisfacibile, dunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> non è finitamente soddisfacibile. Contraddizione. Quella qui esposta è una dimostrazione non costruttiva, cioè non fornisce un esempio concreto di ciò che dimostra, ma suppone che la tesi sia falsa e arriva ad una contraddizione. Per una dimostrazione costruttiva, si veda la <a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Dimostrazione_del_teorema_di_compattezza" title="Logica matematica/Calcolo delle proposizioni/Dimostrazione del teorema di compattezza">dimostrazione del teorema di compattezza</a>. <div class="mw-heading mw-heading4"><h4 id="Riformulazione_del_teorema_di_compattezza">Riformulazione del teorema di compattezza</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=40" title="Modifica la sezione Riformulazione del teorema di compattezza" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=40" title="Edit section's source code: Riformulazione del teorema di compattezza">modifica sorgente</a>]</div> Il teorema di compattezza può essere enunciato in modo equivalente come segue: un insieme di proposizioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> è insoddisfacibile sse esiste un sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> che è insoddisfacibile. Dimostrazione Per contraddizione. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }">) Supponiamo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> sia insoddisfacibile, ma che non esista un sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> insoddisfacibile. Allora, ogni sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> è soddisfacibile, quindi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> è finitamente soddisfacibile. Ma, per il teorema di compattezza, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> è soddisfacibile. Contraddizione. (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇐</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftarrow }">) Supponiamo che esista un sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> insoddisfacibile, ma che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> sia soddisfacibile. Allora, per il teorema di compattezza, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> è finitamente soddisfacibile, quindi ogni sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> è soddisfacibile, dunque non può esistere un sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \subseteq \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \subseteq \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a986f6e9aee3dc67b8e32a0585e7a3308af0e83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.487ex; height:2.343ex;" alt="{\displaystyle \Delta \subseteq \Gamma }"> insoddisfacibile. Contraddizione. Dal teorema di compattezza seguono delle proprietà interessanti, per esempio il seguente corollario. <div class="mw-heading mw-heading4"><h4 id="Corollario_del_teorema_di_compattezza">Corollario del teorema di compattezza</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=41" title="Modifica la sezione Corollario del teorema di compattezza" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=41" title="Edit section's source code: Corollario del teorema di compattezza">modifica sorgente</a>]</div> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}"> sse esiste un sottoinsieme finito <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}"> di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81747c0600f60684fbb15de2f2fe30b85e9f2ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.555ex; height:2.843ex;" alt="{\displaystyle \Gamma _{0}\models A}">. Dimostrazione Per contraddizione. Supponiamo che, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}"> finito, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceaeefb3b15d37ef339071927efb6931f1e9b6e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.961ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}\nvDash A}">, e consideriamo le seguenti trasformazioni: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceaeefb3b15d37ef339071927efb6931f1e9b6e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.961ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}\nvDash A}">, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}"> finito, ipotesi;</li> <li>sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}\cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}\cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679b9cdb962c8f3d0f120128f01ea5009de60cf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.708ex; height:2.843ex;" alt="{\displaystyle \Gamma _{0}\cup \{\neg A\}}"> è soddisfacibile per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f27c3fa0660b68ef8fa747442140655cff65cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0}}"> finito (da 1, per il teorema di soddisfacibilità);</li> <li>sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120ba861a8d6344ab9cd2efc9464975398857b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \Gamma \cup \{\neg A\}}"> è finitamente soddisfacibile (da 2, per definizione di insieme finitamente soddisfacibile);</li> <li>sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \cup \{\neg A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \cup \{\neg A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/120ba861a8d6344ab9cd2efc9464975398857b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.654ex; height:2.843ex;" alt="{\displaystyle \Gamma \cup \{\neg A\}}"> è soddisfacibile (da 3, per il teorema di compattezza);</li> <li>sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/051adc82d3fff0da75d1fad0dd36006d6f091f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.906ex; height:2.176ex;" alt="{\displaystyle \Gamma \nvDash A}"> (da 4, per il teorema di soddisfacibilità).</li></ol> Contraddizione. <div class="mw-heading mw-heading3"><h3 id="Teorema_di_deduzione">Teorema di deduzione</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=42" title="Modifica la sezione Teorema di deduzione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=42" title="Edit section's source code: Teorema di deduzione">modifica sorgente</a>]</div> Definito il concetto di derivazione semantica, vediamo un importante teorema che permette di ridurre le ipotesi di cui abbiamo bisogno o di sfruttare come ipotesi una parte della sentenza che stiamo analizzando: il teorema di deduzione. Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> un insieme di formule e siano <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> due formule, allora <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma ,A\models B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi>A</mi> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma ,A\models B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27980838a20f7db39469c605cd9f44efecf18e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.299ex; height:2.843ex;" alt="{\displaystyle \Gamma ,A\models B}"> sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0183958708f19578a239389eba7bbdbcbf304243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.879ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A\to B}"></dd></dl> cioè, il teorema dice che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> soddisfano <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> se e solo se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> soddisfa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}">. Se la cosa può sembrare a prima vista sorprendente, ad una analisi più attenta si può notare come sia la semantica profonda dell'implicazione. Vediamo con un semplice esempio in linguaggio naturale: da Oggi piove posso derivare (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }">) prendo l' ombrello equivale a: senza nessuna precondizione posso derivare (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }">) se Oggi piove allora (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }">) prendo l' ombrello Questo teorema sarà particolarmente utile nelle dimostrazioni, perché ci permette di spostare a destra o a sinistra del simbolo di derivazione l' antecedente dell' implicazione. Quindi, se dobbiamo dimostrare che A implica B, la strada più semplice sarà supporre A come ipotesi e quindi dimostrare B. <div class="mw-heading mw-heading4"><h4 id="Dimostrazione">Dimostrazione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=43" title="Modifica la sezione Dimostrazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=43" title="Edit section's source code: Dimostrazione">modifica sorgente</a>]</div> Per definizione di implicazione logica su un insieme di formule, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma ,A\models B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi>A</mi> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma ,A\models B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27980838a20f7db39469c605cd9f44efecf18e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.299ex; height:2.843ex;" alt="{\displaystyle \Gamma ,A\models B}"> sse, per ogni modello <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0e9af92e491685e1d861edd5f929f9bdcae6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}}">, (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac83e52f60777901ac486ff44c24e3bffa93d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.268ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c31121af20d4a933b82f2dae03d1a0c8b9b02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models B}">). Per definizione di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }">, (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd40a30cdd56228ff46f5e1ebb311e1ae83f936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash A}"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c31121af20d4a933b82f2dae03d1a0c8b9b02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models B}">) sse (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A)=\mathrm {F} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d562a51604f9969b8938b16bb7141b5d67cf99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.506ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A)=\mathrm {F} }"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(B)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(B)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd38437aa80f0694d4d4bdcb3bb10e0583d31ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.687ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(B)=\mathrm {T} }">) sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A)),I_{\mathcal {V}}(B))=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∨</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A)),I_{\mathcal {V}}(B))=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87648434055e3f54cef60652f9448aee6aadf35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.924ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A)),I_{\mathcal {V}}(B))=\mathrm {T} }">. Per definizione di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\to }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\to }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2051c67217559c0c716084be531ea9938de093cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.895ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\to }}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A)),I_{\mathcal {V}}(B))={\mathcal {Op}}_{\to }(I_{\mathcal {V}}(A),I_{\mathcal {V}}(B))=I_{\mathcal {V}}(A\to B)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∨</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A)),I_{\mathcal {V}}(B))={\mathcal {Op}}_{\to }(I_{\mathcal {V}}(A),I_{\mathcal {V}}(B))=I_{\mathcal {V}}(A\to B)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4ab5214dfb5ffe134d4dbf6f03b7906c9edbed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.926ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Op}}_{\lor }({\mathcal {Op}}_{\neg }(I_{\mathcal {V}}(A)),I_{\mathcal {V}}(B))={\mathcal {Op}}_{\to }(I_{\mathcal {V}}(A),I_{\mathcal {V}}(B))=I_{\mathcal {V}}(A\to B)=\mathrm {T} }">. Per definizione di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \models }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dbad9a523091069a540122aeb15e41d1fe18b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathcal {V}}(A\to B)=\mathrm {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\mathcal {V}}(A\to B)=\mathrm {T} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4d89fe358fae51d89ddf5a41c4f1dbaffa172e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.044ex; height:2.843ex;" alt="{\displaystyle I_{\mathcal {V}}(A\to B)=\mathrm {T} }"> sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8528604fa1959970a7d06fee01d6549f68250f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.531ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A\to B}">. Dunque, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma ,A\models B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>,</mo> <mi>A</mi> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma ,A\models B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27980838a20f7db39469c605cd9f44efecf18e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.299ex; height:2.843ex;" alt="{\displaystyle \Gamma ,A\models B}"> sse (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊭</mo> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac83e52f60777901ac486ff44c24e3bffa93d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.268ex; height:2.509ex;" alt="{\displaystyle {\mathcal {M_{V}}}\nvDash \Gamma }"> oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M_{V}}}\models A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </msub> </mrow> </mrow> <mo>⊨</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M_{V}}}\models A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8528604fa1959970a7d06fee01d6549f68250f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.531ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M_{V}}}\models A\to B}">) sse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0183958708f19578a239389eba7bbdbcbf304243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.879ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A\to B}">. <div class="mw-heading mw-heading2"><h2 id="Completezza_di_insiemi_di_connettivi">Completezza di insiemi di connettivi</h2>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=44" title="Modifica la sezione Completezza di insiemi di connettivi" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=44" title="Edit section's source code: Completezza di insiemi di connettivi">modifica sorgente</a>]</div> I connettivi definiscono delle funzioni da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}"> (nel caso del connettivo unario <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }">), oppure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}\times {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}\times {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5bbce9db5408e10ada8a9bd6f77786cb97161a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}\times {\mathcal {B}}}"> (nel caso dei connettivi binari) in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}">. Ciascuna di queste funzioni, dette funzioni booleane, è definita attraverso la relativa tavola dei valori di verità, oppure, equivalentemente, attraverso le funzioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\neg }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\neg }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620dd1765fa0c24d4a4a6041389a68ecd448aab2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.453ex; margin-bottom: -0.385ex; width:4.348ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\neg }}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Op}}_{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Op}}_{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189e0b507984d7a54480638b15ab9ee84bf50843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.453ex; margin-bottom: -0.385ex; width:4.074ex; height:2.676ex;" alt="{\displaystyle {\mathcal {Op}}_{\circ }}">, con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo stretchy="false">→</mo> <mo>,</mo> <mo stretchy="false">↔</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c76f8e7901565e8a93ba0e159a1e3a86fb2176f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.178ex; height:2.843ex;" alt="{\displaystyle \circ \in \{\land ,\lor ,\to ,\leftrightarrow \}}">. Più in generale, ogni formula proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> contenente simboli proposizionali distinti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},...,A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},...,A_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27790c7851ada2f2d78ce102293ac5de0ce0b85b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.929ex; height:2.509ex;" alt="{\displaystyle A_{1},...,A_{n}}"> definisce una funzione booleana da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156f1b8f0bce731b00d5ee938b9209b44936ec02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.343ex;" alt="{\displaystyle {\mathcal {B}}^{n}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5622de88a69f68340f8dcb43d0b8bd443ba9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.543ex; height:2.176ex;" alt="{\displaystyle {\mathcal {B}}}">. <div class="mw-heading mw-heading3"><h3 id="Funzione_di_verità">Funzione di verità</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=45" title="Modifica la sezione Funzione di verità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=45" title="Edit section's source code: Funzione di verità">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Sia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> una formula della logica proposizionale contenente esattamente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> simboli proposizionali distinti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1},...,A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},...,A_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27790c7851ada2f2d78ce102293ac5de0ce0b85b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.929ex; height:2.509ex;" alt="{\displaystyle A_{1},...,A_{n}}">; la funzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{A}:{\mathcal {B}}^{n}\mapsto {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{A}:{\mathcal {B}}^{n}\mapsto {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfd54c1e5a81d2d38798b5baf14b57b97585cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.465ex; height:2.676ex;" alt="{\displaystyle f_{A}:{\mathcal {B}}^{n}\mapsto {\mathcal {B}}}"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{A}(v_{1},...,v_{n})=I_{\mathcal {V}}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{A}(v_{1},...,v_{n})=I_{\mathcal {V}}(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d500b216de6615996c4bb6e32286da9a2ba7fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.099ex; height:2.843ex;" alt="{\displaystyle f_{A}(v_{1},...,v_{n})=I_{\mathcal {V}}(A)}">, dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d69f309b6deb2e5008f6130ee11e09bbabd7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.529ex; height:2.176ex;" alt="{\displaystyle {\mathcal {V}}}"> è l'interpretazione per cui <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {V}}(A_{i})=v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">V</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {V}}(A_{i})=v_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da59657683c25be202fc88970154cdf8960cf740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.907ex; height:2.843ex;" alt="{\displaystyle {\mathcal {V}}(A_{i})=v_{i}}"> per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,...,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,...,n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d127f409797d26e216ecc405a9a15d914801ef74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.628ex; height:2.509ex;" alt="{\displaystyle i=1,...,n}"> è detta la funzione di verità associata ad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. </div> Quindi, ogni formula del calcolo proposizionale definisce una funzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-aria (possiamo anche chiamarla connettivo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-ario), dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> è il numero di atomi distinti che in essa compaiono. Si può facilmente verificare che, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, esistono <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bcc9417763ad5d68870290ddaa2ca025ffdaf85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.182ex; height:2.676ex;" alt="{\displaystyle 2^{2^{n}}}"> funzioni booleane distinte (cioè, tante quanti sono i sottoinsiemi di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156f1b8f0bce731b00d5ee938b9209b44936ec02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.343ex;" alt="{\displaystyle {\mathcal {B}}^{n}}">). Quindi, nel caso di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"> esistono 16 connettivi distinti. Qui ne sono stati introdotti 4 e, come alcune equivalenze logiche introdotte precedentemente hanno mostrato, essi non sono indipendenti, nel senso che alcuni sono esprimibili in termini di altri. <div class="mw-heading mw-heading3"><h3 id="Derivazione_di_connettivi">Derivazione di connettivi</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=46" title="Modifica la sezione Derivazione di connettivi" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=46" title="Edit section's source code: Derivazione di connettivi">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Dato un insieme di connettivi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"> e un connettivo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\not \in {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\not \in {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb6bbf0a6ad54aac7f419de24e740ca213f0c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.087ex; height:2.676ex;" alt="{\displaystyle c\not \in {\mathcal {C}}}"> per cui si abbia una funzione di verità <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{c}={\mathcal {Op}}_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{c}={\mathcal {Op}}_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2f7afdf2b5801cc23520cc7b6dc3f818020806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.145ex; height:2.676ex;" alt="{\displaystyle f_{c}={\mathcal {Op}}_{c}}">, si dice che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> si deriva da (oppure si definisce in termini dei) connettivi di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"> se esiste una formula proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> costruita usando solo i connettivi di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{c}\equiv f_{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>≡</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{c}\equiv f_{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727bd705324538ba93d9d7165af907c765e40125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.784ex; height:2.509ex;" alt="{\displaystyle f_{c}\equiv f_{F}}">. </div> Nella logica proposizionale valgono le seguenti equivalenze logiche. <table> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\lor B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13321f163366a0912c15b09d401b33e29a1ad46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle \neg A\lor B}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b0af8bc3d53e1e686ffc609aec72cd6f38bc7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.672ex; height:2.176ex;" alt="{\displaystyle \neg A\to B}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\neg A\land \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (\neg A\land \neg B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696c49cd32630d1ba04a71894737d517a1fc5d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.55ex; height:2.843ex;" alt="{\displaystyle \neg (\neg A\land \neg B)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\neg A\lor \neg B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (\neg A\lor \neg B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdffb25cda99ba7533af46896d8471612e831b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.55ex; height:2.843ex;" alt="{\displaystyle \neg (\neg A\lor \neg B)}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (((A\to \bot )\to \bot )\to (B\to \bot ))\to \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (((A\to \bot )\to \bot )\to (B\to \bot ))\to \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec449221a4b99001df7badfe239462c0b7b46a05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.047ex; height:2.843ex;" alt="{\displaystyle (((A\to \bot )\to \bot )\to (B\to \bot ))\to \bot }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195aae731102b36b14a902a091d04ac5c6a5af49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.176ex;" alt="{\displaystyle \neg A}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5937bb9f895e7ee11c0b362eb7b197fa36c6925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.165ex; height:2.176ex;" alt="{\displaystyle A\to \bot }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land \neg A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land \neg A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695d9f9ae765ace242db86b6f7cdc322cfa99145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.619ex; height:2.176ex;" alt="{\displaystyle A\land \neg A}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightarrow B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">↔</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\leftrightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936ab098710910e69e56ec2734dd89063ce21efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\leftrightarrow B}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≡</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \equiv }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5c34250859b6f6d2a77b4e8a2ceaa90638076d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.081ex; margin-bottom: -0.253ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \equiv }"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to B)\land (B\to A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to B)\land (B\to A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21459944695dadd2c3234fee78fa67e2f2752be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.444ex; height:2.843ex;" alt="{\displaystyle (A\to B)\land (B\to A)}"> </td></tr></tbody></table> Si noti che la costante proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }"> (così come la costante proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊤</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \top }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf12e436fef2365e76fcb1034a51179d8328bb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }">) può essere considerata come un connettivo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}">-ario. Un insieme di connettivi logici è completo se mediante i suoi connettivi si può esprimere qualunque funzione booleana. Più formalmente diciamo che: <div class="mw-heading mw-heading3"><h3 id="Insieme_funzionalmente_completo">Insieme funzionalmente completo</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=47" title="Modifica la sezione Insieme funzionalmente completo" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=47" title="Edit section's source code: Insieme funzionalmente completo">modifica sorgente</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465528"> <table class="vedi-table"> <tbody><tr> <td class="vedi-icona"><a href="/wiki/File:Searchtool.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Searchtool.svg/18px-Searchtool.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Searchtool.svg/27px-Searchtool.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Searchtool.svg/36px-Searchtool.svg.png 2x" data-file-width="512" data-file-height="512" /></a> </td> <td class="vedi-td">Per approfondire, vedi <a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Dimostrazione_di_completezza_di_insiemi_di_connettivi" title="Logica matematica/Calcolo delle proposizioni/Dimostrazione di completezza di insiemi di connettivi">Dimostrazione di completezza di insiemi di connettivi</a>. </td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r465526"> <a href="/wiki/File:Viewmag.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/15/Viewmag.png" decoding="async" width="50" height="50" class="mw-file-element" data-file-width="48" data-file-height="48" /></a> Definizione <div class="box-sfondo-grigio" style=""> Un insieme di connettivi logici <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"> si dice funzionalmente completo sse, data una qualunque <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:{\mathcal {B}}^{n}\mapsto {\mathcal {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:{\mathcal {B}}^{n}\mapsto {\mathcal {B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ffd4ad25302dab843598069c7e274c138c6f66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.14ex; height:2.676ex;" alt="{\displaystyle f:{\mathcal {B}}^{n}\mapsto {\mathcal {B}}}"> esiste una formula proposizionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> costruita mediante i connettivi dell'insieme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}"> tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\equiv f_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>≡</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\equiv f_{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0101939d14c77c1e1371d5e227ff0b91833b04dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.981ex; height:2.509ex;" alt="{\displaystyle f\equiv f_{A}}">. </div> In particolare, un insieme di connettivi logici è completo se mediante i suoi connettivi si può esprimere qualunque altro connettivo. È facile <a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Dimostrazione_di_completezza_di_insiemi_di_connettivi" title="Logica matematica/Calcolo delle proposizioni/Dimostrazione di completezza di insiemi di connettivi">verificare</a> che l'insieme dei connettivi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\neg ,\land ,\lor \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\neg ,\land ,\lor \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b8dad6246e3a501c7ebb4be34b16512ce5b4e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.044ex; height:2.843ex;" alt="{\displaystyle \{\neg ,\land ,\lor \}}"> è completo, così come lo sono <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\neg ,\land \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mo>,</mo> <mo>∧</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\neg ,\land \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b2e0e8e5fe88fcc33dfbff0de89f3caed4c9de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle \{\neg ,\land \}}"> e anche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\neg ,\lor \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mo>,</mo> <mo>∨</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\neg ,\lor \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b3f8704f6779c32673e4261594c27ee58c7894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle \{\neg ,\lor \}}">. Questo ci porta a concludere che nella trattazione della logica potremmo ridurci a considerare due soli connettivi. Si può in effetti anche dimostrare che un solo connettivo binario può bastare per definire tutti gli altri; in particolare, un connettivo a scelta tra il "nand" o il "nor" costituisce un insieme completo. Si può anche dimostrare che nand e nor sono gli unici due connettivi binari completi. Usare un minor numero di connettivi nelle formule porterebbe sicuramente a una maggiore stringatezza nelle dimostrazioni date per casi sui singoli connettivi, ma nuocerebbe gravemente alla leggibilità delle formule stesse. Le equivalenze sopra mostrate ne forniscono evidenza. Si è quindi preferito la leggibilità alla concisione. <div class="mw-heading mw-heading2"><h2 id="Decidibilità">Decidibilità</h2>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=48" title="Modifica la sezione Decidibilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=48" title="Edit section's source code: Decidibilità">modifica sorgente</a>]</div> La logica proposizionale è decidibile. In effetti, la logica proposizionale si chiama calcolo delle proposizioni perché esiste una procedura effettiva che stabilisce se una proposizione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una tautologia o meno. Nel caso del calcolo proposizionale, possiamo sempre essere in grado di calcolare le formule vere del linguaggio. In verità, come si vedrà nel caso della logica del primo ordine, la dizione calcolo si usa anche qualora si abbia una logica semidecidibile. Per verificare la decidibilità del calcolo proposizionale siamo interessati a decidere se una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> del calcolo proposizionale è una tautologia o meno. Un altro interessante problema di decisione per il calcolo proposizionale consiste nello stabilire se una formula è soddisfacibile o meno (questo problema è di solito indicato con SAT). Abbiamo visto che, per decidere se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è una tautologia, bisogna verificare le tavole di verità per ogni assegnazione ai simboli proposizionali che occorrono in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">; pertanto, se in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> occorrono <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> simboli proposizionali, sarà sufficiente verificare <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"> casi. Diciamo che un insieme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> di formule è enumerabile se può essere trasformato in una sequenza. Un insieme di formule è effettivamente enumerabile se esiste un metodo per determinare l'<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-esimo elemento della sequenza, per ogni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }">. <div class="mw-heading mw-heading3"><h3 id="Lemma_di_enumerabilità">Lemma di enumerabilità</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=49" title="Modifica la sezione Lemma di enumerabilità" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=49" title="Edit section's source code: Lemma di enumerabilità">modifica sorgente</a>]</div> Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> è effettivamente enumerabile, allora l'insieme delle conseguenze tautologiche di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">, ovvero l'insieme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A|\Gamma \models A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A|\Gamma \models A\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7a8658eace207f86bfc5f4c6639514fa159364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.216ex; height:2.843ex;" alt="{\displaystyle \{A|\Gamma \models A\}}"> è effettivamente enumerabile. <div class="mw-heading mw-heading4"><h4 id="Dimostrazione_2">Dimostrazione</h4>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=50" title="Modifica la sezione Dimostrazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=50" title="Edit section's source code: Dimostrazione">modifica sorgente</a>]</div> Dobbiamo mostrare che, data una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}">, allora esiste una procedura che risponde SÌ. Consideriamo un'enumerazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0},\Gamma _{1},\Gamma _{2},...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0},\Gamma _{1},\Gamma _{2},...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a8808b3f4e6ce9f57dec7a0e9886e237675cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.338ex; height:2.509ex;" alt="{\displaystyle \Gamma _{0},\Gamma _{1},\Gamma _{2},...}"> dei sottoinsiemi finiti di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }">. Data una formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, consideriamo la lista dei sottoinsiemi finiti di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> nel seguente modo: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vDash A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vDash A,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed5a91cb7cb5c29f287a601bbf5922885363788" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.05ex; height:2.843ex;" alt="{\displaystyle \vDash A,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{0}\models A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊨</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{0}\models A,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e8bc69c32083b091b6571df026a15b5f665af48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.202ex; height:2.843ex;" alt="{\displaystyle \Gamma _{0}\models A,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{1}\models A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊨</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{1}\models A,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc773f3a5e86907c7155414b8fed194be646018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.202ex; height:2.843ex;" alt="{\displaystyle \Gamma _{1}\models A,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad ...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad ...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e0734ac2ffd037bba1bae94b0673a38c11012a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.36ex; height:0.843ex;" alt="{\displaystyle \qquad ...}"> Per il <a class="mw-selflink-fragment" href="#Corollario_del_teorema_di_compattezza">corollario del teorema di compattezza</a>, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa86ac99eb85f7140163c9221d5f9f4d25b6ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \Gamma \models A}"> esiste un <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4497c779a32aa954cc5706fb7f4c2abbe1176dec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.252ex; height:2.509ex;" alt="{\displaystyle \Gamma _{i}}"> finito tale che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{i}\models A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊨</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{i}\models A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff88ffed6e4d393bb6ada049af0a98f43dca89a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.301ex; height:2.843ex;" alt="{\displaystyle \Gamma _{i}\models A}">, e dunque tale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4497c779a32aa954cc5706fb7f4c2abbe1176dec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.252ex; height:2.509ex;" alt="{\displaystyle \Gamma _{i}}"> comparirà nell'enumerazione. <div class="mw-heading mw-heading2"><h2 id="Teoria_della_dimostrazione">Teoria della dimostrazione</h2>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=51" title="Modifica la sezione Teoria della dimostrazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=51" title="Edit section's source code: Teoria della dimostrazione">modifica sorgente</a>]</div> Come abbiamo visto nel paragrafo precedente, il calcolo delle proposizioni è decidibile ed abbiamo un algoritmo (le tavole di verità) per verificare se una sentenza è soddisfacibile. Ora cercheremo dei metodi che ci permettano di dimostrare che una sentenza è valida. Il metodo delle tabelle di verità è un metodo "semantico", cioè che esplora tutte le possibili configurazioni dell'universo per verificarne lo stato, ora vogliamo costruire dei metodi basati più sulla deduzione che non sull'esplorazione e che si basino sull'attività di costruire derivazioni e dimostrazioni, cioè sul ragionamento e non sulla forza bruta. Svilupperemo quindi dei metodi puramente sintattici, cioè dei calcoli che trasformano simboli in altri simboli, che ci permetteranno di creare dimostrazioni sulle sentenze logiche. Introduciamo un nuovo simbolo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c0d30cf8cb7dba179e317fcde9583d842e80f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }">, per indicare che da una sentenza possiamo derivare sintatticamente un'altra sentenza. Abbiamo così due concetti: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\models B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊨</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\models B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e557884001c37bf193b635f9492c944d449cd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.812ex; height:2.843ex;" alt="{\displaystyle A\models B}"> significa che semanticamente se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è vera anche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> è vera, mentre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\vdash B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊢</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\vdash B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45b0d9ce57211fcb763fc21a7313db94189a85b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.218ex; height:2.176ex;" alt="{\displaystyle A\vdash B}"> significa che se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> è vera possiamo costruire una dimostrazione di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. Il grande risultato della logica moderna è stato legare tra loro questi due concetti nel teorema di completezza, che vedremo più avanti. Esploreremo i maggiori sistemi di dimostrazione, tutti con peculiari caratteristiche. Il sistema di Hilbert è quello più simile alla definizione del metodo assiomatico, i sistemi di Gentzen (deduzione naturale e sequenti) tentano di mimare l'attività mentale di un matematico, i tableaux sfruttano la semantica e la risoluzione è il sistema più semplice da realizzare meccanicamente, tanto da essere alla base del linguaggio di programmazione PROLOG. Il sistema di Hilbert si basa su un insieme di assiomi logici e un insieme di regole di inferenza; i tableaux e la risoluzione invece non fanno uso di assiomi logici, ma solo di regole di inferenza e sono basati sulla refutazione (cioè, per dimostrare che una formula è un teorema si dimostra in effetti che <a class="mw-selflink-fragment" href="#Teorema">la sua negata è una contraddizione</a>). Tra questi metodi, solo la risoluzione richiede che le formule siano scritte in una forma normale. Tutti i sistemi deduttivi presentati sono per certi aspetti equivalenti, per tutti infatti vale un teorema di correttezza e completezza, quindi l'insieme di teoremi che si può derivare con ciascuno di essi è lo stesso. <div class="mw-heading mw-heading3"><h3 id="Correttezza_e_completezza_di_un_sistema">Correttezza e completezza di un sistema</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=52" title="Modifica la sezione Correttezza e completezza di un sistema" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=52" title="Edit section's source code: Correttezza e completezza di un sistema">modifica sorgente</a>]</div> Ogni sistema di dimostrazione, per essere un buon sistema, deve avere due proprietà: essere corretto ed essere completo. La proprietà di correttezza è: se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash _{s}\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>⊢</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash _{s}\Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6918f5275be5540c3b1fcbedc5004a334d6da6d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.747ex; height:2.509ex;" alt="{\displaystyle \vdash _{s}\Phi }"> allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vDash \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vDash \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0284578b0737ee805873f74c634ed4bb18d971f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.338ex; height:2.843ex;" alt="{\displaystyle \vDash \Phi }">. cioè, se posso dimostrare <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> con il sistema <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> è una tautologia. In poche parole, tutto quello che può essere provato da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> è vero. La proprietà di completezza è: se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vDash \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊨</mo> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vDash \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0284578b0737ee805873f74c634ed4bb18d971f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.338ex; height:2.843ex;" alt="{\displaystyle \vDash \Phi }"> allora <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash _{s}\Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>⊢</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash _{s}\Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6918f5275be5540c3b1fcbedc5004a334d6da6d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.747ex; height:2.509ex;" alt="{\displaystyle \vdash _{s}\Phi }">. cioè, se una proposizione è una tautologia, posso dimostrarla utilizzando <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">. <div class="mw-heading mw-heading3"><h3 id="I_principali_sistemi_di_dimostrazione">I principali sistemi di dimostrazione</h3>[<a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&veaction=edit&section=53" title="Modifica la sezione I principali sistemi di dimostrazione" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni&action=edit&section=53" title="Edit section's source code: I principali sistemi di dimostrazione">modifica sorgente</a>]</div> <ul><li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/Sistema_di_Hilbert" title="Logica matematica/Calcolo delle proposizioni/Sistema di Hilbert">Il sistema di Frege-Hilbert</a></li> <li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/La_deduzione_naturale" title="Logica matematica/Calcolo delle proposizioni/La deduzione naturale">La deduzione naturale</a></li> <li><a href="/w/index.php?title=Logica_matematica/Calcolo_delle_proposizioni/Il_calcolo_dei_sequenti&action=edit&redlink=1" class="new" title="Logica matematica/Calcolo delle proposizioni/Il calcolo dei sequenti (la pagina non esiste)">Il calcolo dei sequenti</a></li> <li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/I_tableaux_semantici" title="Logica matematica/Calcolo delle proposizioni/I tableaux semantici">I tableaux semantici</a></li> <li><a href="/wiki/Logica_matematica/Calcolo_delle_proposizioni/La_risoluzione" title="Logica matematica/Calcolo delle proposizioni/La risoluzione">La risoluzione</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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