Natural deduction - Wikipedia

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class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propositional_language_syntax" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Propositional_language_syntax"> <div class="vector-toc-text"> 3 Propositional language syntax </div> </a> <button aria-controls="toc-Propositional_language_syntax-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Propositional language syntax subsection </button> <ul id="toc-Propositional_language_syntax-sublist" class="vector-toc-list"> <li id="toc-Common_definition_styles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Common_definition_styles"> <div class="vector-toc-text"> 3.1 Common definition styles </div> </a> <ul id="toc-Common_definition_styles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gentzen-style_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gentzen-style_definition"> <div class="vector-toc-text"> 3.2 Gentzen-style definition </div> </a> <ul id="toc-Gentzen-style_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Gentzen-style_propositional_logic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Gentzen-style_propositional_logic"> <div class="vector-toc-text"> 4 Gentzen-style propositional logic </div> </a> <button aria-controls="toc-Gentzen-style_propositional_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Gentzen-style propositional logic subsection </button> <ul id="toc-Gentzen-style_propositional_logic-sublist" class="vector-toc-list"> <li id="toc-Gentzen-style_inference_rules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gentzen-style_inference_rules"> <div class="vector-toc-text"> 4.1 Gentzen-style inference rules </div> </a> <ul id="toc-Gentzen-style_inference_rules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gentzen-style_example_proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gentzen-style_example_proofs"> <div class="vector-toc-text"> 4.2 Gentzen-style example proofs </div> </a> <ul id="toc-Gentzen-style_example_proofs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Suppes–Lemmon-style_propositional_logic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Suppes–Lemmon-style_propositional_logic"> <div class="vector-toc-text"> 5 Suppes–Lemmon-style propositional logic </div> </a> <button aria-controls="toc-Suppes–Lemmon-style_propositional_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Suppes–Lemmon-style propositional logic subsection </button> <ul id="toc-Suppes–Lemmon-style_propositional_logic-sublist" class="vector-toc-list"> <li id="toc-Suppes–Lemmon-style_inference_rules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Suppes–Lemmon-style_inference_rules"> <div class="vector-toc-text"> 5.1 Suppes–Lemmon-style inference rules </div> </a> <ul id="toc-Suppes–Lemmon-style_inference_rules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Suppes–Lemmon-style_example_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Suppes–Lemmon-style_example_proof"> <div class="vector-toc-text"> 5.2 Suppes–Lemmon-style example proof </div> </a> <ul id="toc-Suppes–Lemmon-style_example_proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Consistency,_completeness,_and_normal_forms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Consistency,_completeness,_and_normal_forms"> <div class="vector-toc-text"> 6 Consistency, completeness, and normal forms </div> </a> <ul id="toc-Consistency,_completeness,_and_normal_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First_and_higher-order_extensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#First_and_higher-order_extensions"> <div class="vector-toc-text"> 7 First and higher-order extensions </div> </a> <ul id="toc-First_and_higher-order_extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proofs_and_type_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proofs_and_type_theory"> <div class="vector-toc-text"> 8 Proofs and type theory </div> </a> <button aria-controls="toc-Proofs_and_type_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Proofs and type theory subsection </button> <ul id="toc-Proofs_and_type_theory-sublist" class="vector-toc-list"> <li id="toc-Substitution_theorem" class="vector-toc-list-item vector-toc-level-2"> <a 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id="toc-Classical_and_modal_logics-sublist" class="vector-toc-list"> <li id="toc-Modal_substitution_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modal_substitution_theorem"> <div class="vector-toc-text"> 9.1 Modal substitution theorem </div> </a> <ul id="toc-Modal_substitution_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Comparison_with_sequent_calculus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Comparison_with_sequent_calculus"> <div class="vector-toc-text"> 10 Comparison with sequent calculus </div> </a> <button aria-controls="toc-Comparison_with_sequent_calculus-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Comparison with sequent calculus subsection </button> <ul id="toc-Comparison_with_sequent_calculus-sublist" class="vector-toc-list"> <li id="toc-Cut_(substitution)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cut_(substitution)"> <div class="vector-toc-text"> 10.1 Cut (substitution) </div> </a> <ul id="toc-Cut_(substitution)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 11 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 12 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 13 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-General_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_references"> <div class="vector-toc-text"> 13.1 General references </div> </a> <ul id="toc-General_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inline_citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inline_citations"> <div class="vector-toc-text"> 13.2 Inline citations </div> </a> <ul id="toc-Inline_citations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main 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data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D9%86%D8%AA%D8%A7%D8%AC_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C" title="استنتاج طبیعی – Persian" lang="fa" hreflang="fa" data-title="استنتاج طبیعی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9duction_naturelle" title="Déduction naturelle – French" lang="fr" hreflang="fr" data-title="Déduction naturelle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%97%B0_%EC%97%B0%EC%97%AD" title="자연 연역 – Korean" lang="ko" hreflang="ko" data-title="자연 연역" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Deduzione_naturale" title="Deduzione naturale – Italian" lang="it" hreflang="it" data-title="Deduzione naturale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Natuurlijke_deductie" title="Natuurlijke deductie – Dutch" lang="nl" hreflang="nl" data-title="Natuurlijke deductie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%BC%94%E7%B9%B9" title="自然演繹 – Japanese" lang="ja" hreflang="ja" data-title="自然演繹" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Kind of proof calculus</div> In <a href="/wiki/Logic" title="Logic">logic</a> and <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, natural deduction is a kind of <a href="/wiki/Proof_calculus" title="Proof calculus">proof calculus</a> in which <a href="/wiki/Logical_reasoning" title="Logical reasoning">logical reasoning</a> is expressed by <a href="/wiki/Inference_rules" class="mw-redirect" title="Inference rules">inference rules</a> closely related to the "natural" way of reasoning.<a href="#cite_note-1">[1]</a> This contrasts with <a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert-style systems</a>, which instead use <a href="/wiki/Axiom" title="Axiom">axioms</a> as much as possible to express the logical laws of <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive reasoning</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=1" title="Edit section: History">edit</a>]</div> Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>, and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> (see, e.g., <a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert system</a>). Such axiomatizations were most famously used by <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead</a> in their mathematical treatise <a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a>. Spurred on by a series of seminars in Poland in 1926 by <a href="/wiki/Jan_Lukasiewicz" class="mw-redirect" title="Jan Lukasiewicz">Łukasiewicz</a> that advocated a more natural treatment of logic, <a href="/wiki/Stanis%C5%82aw_Ja%C5%9Bkowski" title="Stanisław Jaśkowski">Jaśkowski</a> made the earliest attempts at defining a more natural deduction, first in 1929 using a diagrammatic notation, and later updating his proposal in a sequence of papers in 1934 and 1935.<a href="#cite_note-2">[2]</a> His proposals led to different notations such as <a href="/wiki/Fitch-style_calculus" class="mw-redirect" title="Fitch-style calculus">Fitch-style calculus</a> (or Fitch's diagrams) or <a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes</a>' method for which <a href="/wiki/John_Lemmon" title="John Lemmon">Lemmon</a> gave a variant now known as <a href="/wiki/Suppes%E2%80%93Lemmon_notation" title="Suppes–Lemmon notation">Suppes–Lemmon notation</a>. Natural deduction in its modern form was independently proposed by the German mathematician <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> in 1933, in a dissertation delivered to the faculty of mathematical sciences of the <a href="/wiki/University_of_G%C3%B6ttingen" title="University of Göttingen">University of Göttingen</a>.<a href="#cite_note-3">[3]</a> The term natural deduction (or rather, its German equivalent natürliches Schließen) was coined in that paper: <style data-mw-deduplicate="TemplateStyles:r1157697682">.mw-parser-output .verse_translation .translated{padding-left:2em!important}@media only screen and (max-width:43.75em){.mw-parser-output .verse_translation.wrap_when_small td{display:block;padding-left:0.5em}.mw-parser-output .verse_translation.wrap_when_small .translated{padding-left:0.5em!important}}</style> <table role="presentation" class="verse_translation" style="margin-left:1em !important"> <tbody><tr style="vertical-align:top"> <td><div style="font-style:italic;text-align:left" lang="de" class="poem"> Ich wollte nun zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein "Kalkül des natürlichen Schließens".<a href="#cite_note-4">[4]</a> </div> </td> <td class="translated"><div style="font-style:roman;text-align:left" lang="" class="poem"> First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction". </div> </td></tr></tbody></table> Gentzen was motivated by a desire to establish the consistency of <a href="/wiki/Number_theory" title="Number theory">number theory</a>. He was unable to prove the main result required for the consistency result, the <a href="/wiki/Cut_elimination_theorem" class="mw-redirect" title="Cut elimination theorem">cut elimination theorem</a>—the Hauptsatz—directly for natural deduction. For this reason he introduced his alternative system, the <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculus</a>, for which he proved the Hauptsatz both for <a href="/wiki/Classical_logic" title="Classical logic">classical</a> and <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. In a series of seminars in 1961 and 1962 <a href="/wiki/Dag_Prawitz" title="Dag Prawitz">Prawitz</a> gave a comprehensive summary of natural deduction calculi, and transported much of Gentzen's work with sequent calculi into the natural deduction framework. His 1965 monograph Natural deduction: a proof-theoretical study<a href="#cite_note-prawitz1965-5">[5]</a> was to become a reference work on natural deduction, and included applications for <a href="/wiki/Modal_logic" title="Modal logic">modal</a> and <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a>. In natural deduction, a <a href="/wiki/Proposition" title="Proposition">proposition</a> is deduced from a collection of premises by applying inference rules repeatedly. The system presented in this article is a minor variation of Gentzen's or Prawitz's formulation, but with a closer adherence to <a href="/wiki/Per_Martin-L%C3%B6f" title="Per Martin-Löf">Martin-Löf</a>'s description of logical judgments and connectives.<a href="#cite_note-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="History_of_notation_styles">History of notation styles</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=2" title="Edit section: History of notation styles">edit</a>]</div> Natural deduction has had a large variety of notation styles,<a href="#cite_note-SEP_NaturalDeduction-7">[7]</a> which can make it difficult to recognize a proof if you're not familiar with one of them. To help with this situation, this article has a <a href="#Notation">§ Notation</a> section explaining how to read all the notation that it will actually use. This section just explains the historical evolution of notation styles, most of which cannot be shown because there are no illustrations available under a <a href="/wiki/Public_copyright_license" title="Public copyright license">public copyright license</a> – the reader is pointed to the <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/spr2024/entries/natural-deduction/">SEP</a> and <a rel="nofollow" class="external text" href="https://iep.utm.edu/natural-deduction/">IEP</a> for pictures. <ul><li><a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen</a> invented natural deduction using tree-shaped proofs – see <a href="#Gentzen's_tree_notation">§ Gentzen's tree notation</a> for details.</li> <li><a href="/wiki/Stanis%C5%82aw_Ja%C5%9Bkowski" title="Stanisław Jaśkowski">Jaśkowski</a> changed this to a notation that used various nested boxes.<a href="#cite_note-SEP_NaturalDeduction-7">[7]</a></li> <li><a href="/wiki/Frederic_Fitch" title="Frederic Fitch">Fitch</a> changed Jaśkowski method of drawing the boxes, creating <a href="/wiki/Fitch_notation" title="Fitch notation">Fitch notation</a>.<a href="#cite_note-SEP_NaturalDeduction-7">[7]</a></li> <li>1940: In a textbook, <a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine</a><a href="#cite_note-8">[8]</a> indicated antecedent dependencies by line numbers in square brackets, anticipating Suppes' 1957 line-number notation.</li> <li>1950: In a textbook, <a href="#CITEREFQuine1982">Quine (1982</a>, pp. 241–255) demonstrated a method of using one or more asterisks to the left of each line of proof to indicate dependencies. This is equivalent to Kleene's vertical bars. (It is not totally clear if Quine's asterisk notation appeared in the original 1950 edition or was added in a later edition.)</li> <li>1957: An introduction to practical logic theorem proving in a textbook by <a href="#CITEREFSuppes1999">Suppes (1999</a>, pp. 25–150). This indicated dependencies (i.e. antecedent propositions) by line numbers at the left of each line.</li> <li>1963: <a href="#CITEREFStoll1979">Stoll (1979</a>, pp. 183–190, 215–219) uses sets of line numbers to indicate antecedent dependencies of the lines of sequential logical arguments based on natural deduction inference rules.</li> <li>1965: The entire textbook by <a href="#CITEREFLemmon1965">Lemmon (1965)</a> is an introduction to logic proofs using a method based on that of <a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes</a>, what is now known as <a href="/wiki/Suppes%E2%80%93Lemmon_notation" title="Suppes–Lemmon notation">Suppes–Lemmon notation</a>.</li> <li>1967: In a textbook, <a href="#CITEREFKleene2002">Kleene (2002</a>, pp. 50–58, 128–130) briefly demonstrated two kinds of practical logic proofs, one system using explicit quotations of antecedent propositions on the left of each line, the other system using vertical bar-lines on the left to indicate dependencies.<a href="#cite_note-9">[a]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=3" title="Edit section: Notation">edit</a>]</div> Here is a table with the most common notational variants for <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a>. <table class="wikitable"> <caption>Notational variants of the connectives<a href="#cite_note-10">[9]</a><a href="#cite_note-:23-11">[10]</a> </caption> <tbody><tr> <th>Connective </th> <th>Symbol </th></tr> <tr> <td><a href="/wiki/Logical_conjunction" title="Logical conjunction">AND</a> </td> <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}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot 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\cdot B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a90e903f21f11a0f4ab3caca1e6943ba7a9849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.186ex; height:2.176ex;" alt="{\displaystyle A\cdot B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}">, <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> <mi mathvariant="normal">&</mi> <mi>B</mi> </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/f65ee34a8896390e0d1f193c137d9eb64815c1a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.315ex; height:2.176ex;" alt="{\displaystyle A\&B}">, <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> <mi mathvariant="normal">&</mi> <mi mathvariant="normal">&</mi> <mi>B</mi> </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/1ab3517dd859d7eaa8f3f1656c8125f99ada1470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.123ex; height:2.176ex;" alt="{\displaystyle A\&\&B}"> </td></tr> <tr> <td><a href="/wiki/Logical_biconditional" title="Logical biconditional">equivalent</a> </td> <td><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}">, <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/ce08fffb4d36ba12921b8b3e06228887015b2b8e" 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}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\leftrightharpoons 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\leftrightharpoons B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8639312e9e120cd65c98fc48a6d5256d57288c6" 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\leftrightharpoons B}"> </td></tr> <tr> <td><a href="/wiki/Material_conditional" title="Material conditional">implies</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\Rightarrow 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\Rightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e560143d45c97e6387c7c3aa90e9d7745002228" 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\Rightarrow B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\supset 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\supset B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee952838d8b3e67045072a8f2b71e7fc0467dea6" 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\supset B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\rightarrow 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\rightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23efef033def56a67de7ded823f14626de26d174" 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\rightarrow B}"> </td></tr> <tr> <td><a href="/wiki/Sheffer_stroke" title="Sheffer stroke">NAND</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\overline {\land }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∧</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\land }}B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0167d56141342887a74d56a036e6fbbad7172b0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\land }}B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mid 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\mid B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A\cdot B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A\cdot B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/225f35bb78e90b9126458f1bc6bf1ed3f0724bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.301ex; height:3.009ex;" alt="{\displaystyle {\overline {A\cdot B}}}"> </td></tr> <tr> <td>nonequivalent </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \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\not \equiv B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0339ed15cd58f7263c5eec8e5628168aa6006200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.676ex;" alt="{\displaystyle A\not \equiv B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\not \Leftrightarrow 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\not \Leftrightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47bfd97641b05a7f7fc0bcd02e83fa6532c62bb4" 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\not \Leftrightarrow B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\nleftrightarrow 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\nleftrightarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926467de6fd4709bdbc59c3168a21298cdf0d26c" 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\nleftrightarrow B}"> </td></tr> <tr> <td><a href="/wiki/Logical_NOR" title="Logical NOR">NOR</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\overline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>∨</mo> <mo accent="false">¯</mo> </mover> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\overline {\lor }}B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591afbca3e984765b18abb189f4bb1b88116c400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle A{\overline {\lor }}B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\downarrow 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\downarrow B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5e77260e67880093dafe958880ea02f5026164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.509ex;" alt="{\displaystyle A\downarrow B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A+B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A+B}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08840f8e2022f127fc459d801a8f8ce93f65f55a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.462ex; height:3.176ex;" alt="{\displaystyle {\overline {A+B}}}"> </td></tr> <tr> <td><a href="/wiki/Negation" title="Negation">NOT</a> </td> <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}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -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 -A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6daf6db742ace65252b589963f7e7a07603ccb56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.551ex; height:2.343ex;" alt="{\displaystyle -A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim 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 \sim A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79bf96c247833282e773fae43602343150c1665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.196ex; height:2.176ex;" alt="{\displaystyle \sim A}"> </td></tr> <tr> <td><a href="/wiki/Logical_disjunction" title="Logical disjunction">OR</a> </td> <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}">, <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>+</mo> <mi>B</mi> </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/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mid 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\mid B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1400973074b4691cc0638a68118716a2b218fce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\parallel 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\parallel B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd0239f9b74f7ef1520ba4e30454b06e695289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.96ex; height:2.843ex;" alt="{\displaystyle A\parallel B}"> </td></tr> <tr> <td><a href="/wiki/XNOR_gate" title="XNOR gate">XNOR</a> </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}"> XNOR <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}"> </td></tr> <tr> <td><a href="/wiki/Exclusive_or" title="Exclusive or">XOR</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\underline {\lor }}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∨</mo> <mo>_</mo> </munder> </mrow> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\underline {\lor }}B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803e1413692912954b90e99694e10c728e27a153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:5.06ex; height:3.176ex;" alt="{\displaystyle A{\underline {\lor }}B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\oplus 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\oplus B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0512d6bdd29ff000dea0bf68b853618dcaabc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A\oplus B}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Gentzen's_tree_notation">Gentzen's tree notation</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=4" title="Edit section: Gentzen's tree notation">edit</a>]</div> <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen</a>, who invented natural deduction, had his own notation style for arguments. This will be exemplified by a simple argument below. Let's say we have a simple example argument in <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional logic</a>, such as, "if it's raining then it's cloudly; it is raining; therefore it's cloudy". (This is in <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>.) Representing this as a list of propositions, as is common, we would have: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1)~P\to Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1)~P\to Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/364be4519e9193004f9f74228e17ea042b054ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.846ex; height:2.843ex;" alt="{\displaystyle 1)~P\to Q}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2)~P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2)~P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c550623bb1b5204d9c52cb5c11cb080318fc8b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.393ex; height:2.843ex;" alt="{\displaystyle 2)~P}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \therefore ~Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∴</mo> <mtext> </mtext> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \therefore ~Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f12a34ab9c5fdc2ffef71226573f458c57f34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.614ex; height:2.509ex;" alt="{\displaystyle \therefore ~Q}"></dd></dl> In Gentzen's notation,<a href="#cite_note-SEP_NaturalDeduction-7">[7]</a> this would be written like this: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {P\to Q,P}{Q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo>,</mo> <mi>P</mi> </mrow> <mi>Q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P\to Q,P}{Q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c54517fd6fbb7e0354e22696003e88326d62397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.813ex; height:5.843ex;" alt="{\displaystyle {\frac {P\to Q,P}{Q}}}"></dd></dl> The premises are shown above a line, called the inference line,<a href="#cite_note-JanVonPlato_Reasoning-12">[11]</a><a href="#cite_note-MiamiLogic-13">[12]</a> separated by a comma, which indicates combination of premises.<a href="#cite_note-SEP_Substructural-14">[13]</a> The conclusion is written below the inference line.<a href="#cite_note-JanVonPlato_Reasoning-12">[11]</a> The inference line represents syntactic consequence,<a href="#cite_note-JanVonPlato_Reasoning-12">[11]</a> sometimes called deductive consequence,<a href="#cite_note-IEP_Compactness-15">[14]</a> which is also symbolized with ⊢.<a href="#cite_note-ColoradoLogic-16">[15]</a><a href="#cite_note-IEP_Compactness-15">[14]</a> So the above can also be written in one line as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to Q,P\vdash Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo>,</mo> <mi>P</mi> <mo>⊢</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to Q,P\vdash Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6b1bf712367f9c2db9774415f397953dda2fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.526ex; height:2.509ex;" alt="{\displaystyle P\to Q,P\vdash Q}">. (The turnstile, for syntactic consequence, is of lower <a href="/wiki/Order_of_operations" title="Order of operations">precedence</a> than the comma, which represents premise combination, which in turn is of lower precedence than the arrow, used for material implication; so no parentheses are needed to interpret this formula.)<a href="#cite_note-SEP_Substructural-14">[13]</a> Syntactic consequence is contrasted with semantic consequence,<a href="#cite_note-17">[16]</a> which is symbolized with ⊧.<a href="#cite_note-ColoradoLogic-16">[15]</a><a href="#cite_note-IEP_Compactness-15">[14]</a> In this case, the conclusion follows syntactically because natural deduction is a <a href="/wiki/Propositional_calculus#Syntactic_proof_systems" title="Propositional calculus">syntactic proof system</a>, which assumes <a href="/wiki/Inference_rules" class="mw-redirect" title="Inference rules">inference rules</a> <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">as primitives</a>. Gentzen's style will be used in much of this article. Gentzen's discharging annotations used to internalise hypothetical judgments can be avoided by representing proofs as a tree of <a href="/wiki/Sequent" title="Sequent">sequents</a> Γ ⊢A instead of a tree of judgments that A (is true). <div class="mw-heading mw-heading3"><h3 id="Suppes–Lemmon_notation">Suppes–Lemmon notation</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=5" title="Edit section: Suppes–Lemmon notation">edit</a>]</div> Many textbooks use <a href="/wiki/Suppes%E2%80%93Lemmon_notation" title="Suppes–Lemmon notation">Suppes–Lemmon notation</a>,<a href="#cite_note-SEP_NaturalDeduction-7">[7]</a> so this article will also give that – although as of now, this is only included for <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional logic</a>, and the rest of the coverage is given only in Gentzen style. A proof, laid out in accordance with the <a href="/wiki/Suppes%E2%80%93Lemmon_notation" title="Suppes–Lemmon notation">Suppes–Lemmon notation</a> style, is a sequence of lines containing sentences,<a href="#cite_note-AllenHand-18">[17]</a> where each sentence is either an assumption, or the result of applying a rule of proof to earlier sentences in the sequence.<a href="#cite_note-AllenHand-18">[17]</a> Each line of proof is made up of a sentence of proof, together with its annotation, its assumption set, and the current line number.<a href="#cite_note-AllenHand-18">[17]</a> The assumption set lists the assumptions on which the given sentence of proof depends, which are referenced by the line numbers.<a href="#cite_note-AllenHand-18">[17]</a> The annotation specifies which rule of proof was applied, and to which earlier lines, to yield the current sentence.<a href="#cite_note-AllenHand-18">[17]</a> Here's an example proof: <table class="wikitable" style="margin:auto;"> <caption>Suppes–Lemmon style proof (first example) </caption> <tbody><tr> <th>Assumption set </th> <th>Line number </th> <th>Sentence of proof </th> <th>Annotation </th></tr> <tr> <td>1 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cad5b2c2991ae1dbded560c5d875fbf49fe8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\to Q}"> </td> <td>A </td></tr> <tr> <td>2 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff459062914e488a99e0f453ee6fe6b1315a34d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.646ex; height:2.509ex;" alt="{\displaystyle -Q}"> </td> <td>A </td></tr> <tr> <td>3 </td> <td>3 </td> <td><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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> </td> <td>A </td></tr> <tr> <td>1, 3 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> </td> <td>1, 3 →E </td></tr> <tr> <td>1, 2 </td> <td>5 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <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/11b61b997d0d9ebae227bd68325103c1a3e0f487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.554ex; height:2.343ex;" alt="{\displaystyle -P}"> </td> <td>2, 4 RAA </td></tr></tbody></table> This proof will become clearer when the inference rules and their appropriate annotations are specified – see <a href="#Propositional_inference_rules_(Suppes–Lemmon_style)">§ Propositional inference rules (Suppes–Lemmon style)</a>. <div class="mw-heading mw-heading2"><h2 id="Propositional_language_syntax">Propositional language syntax</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=6" title="Edit section: Propositional language syntax">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Propositional_calculus#Syntax" title="Propositional calculus">Propositional calculus § Syntax</a></div>This section defines the <a href="/wiki/Formal_syntax" class="mw-redirect" title="Formal syntax">formal syntax</a> for a <a href="/wiki/Propositional_calculus#Language" title="Propositional calculus">propositional logic language</a>, contrasting the common ways of doing so with a Gentzen-style way of doing so. <div class="mw-heading mw-heading3"><h3 id="Common_definition_styles">Common definition styles</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=7" title="Edit section: Common definition styles">edit</a>]</div> The <a href="/wiki/Formal_language" title="Formal language">formal language</a> <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}}}"> of a <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a> is usually defined by a <a href="/wiki/Recursive_definition" title="Recursive definition">recursive definition</a>, such as this one, from <a href="/wiki/David_Bostock_(philosopher)" title="David Bostock (philosopher)">Bostock</a>:<a href="#cite_note-:BostockLogic-19">[18]</a> <ol><li>Each <a href="/wiki/Propositional_variable" title="Propositional variable">sentence-letter</a> is a <a href="/wiki/Well-formed_formula" title="Well-formed formula">formula</a>.</li> <li>"<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 }">" and "<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 }">" are formulae.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> is a formula, so is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c4b627f483f3efdd7c27e708f6f37c20c63502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.07ex; height:2.176ex;" alt="{\displaystyle \neg \varphi }">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> are formulae, so are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varphi \land \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>∧</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi \land \psi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07215375dfda492538534f26e3f4f1769b81c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.425ex; height:2.843ex;" alt="{\displaystyle (\varphi \land \psi )}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varphi \lor \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi \lor \psi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43eb945ebfdc026f7433771ee07cba53f518600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.425ex; height:2.843ex;" alt="{\displaystyle (\varphi \lor \psi )}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varphi \to \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi \to \psi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2e5835e3374498f795beec9875642375ebd5b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.457ex; height:2.843ex;" alt="{\displaystyle (\varphi \to \psi )}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varphi \leftrightarrow \psi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">↔</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varphi \leftrightarrow \psi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e0250093a40ca047e6ad7b94c579c0a1bf2c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.457ex; height:2.843ex;" alt="{\displaystyle (\varphi \leftrightarrow \psi )}">.</li> <li>Nothing else is a formula.</li></ol> There are other ways of doing it, such as the <a href="/wiki/BNF_grammar" class="mw-redirect" title="BNF grammar">BNF grammar</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ::=a_{1},a_{2},\ldots ~|~\neg \phi ~|~\phi ~\land ~\psi ~|~\phi \vee \psi ~|~\phi \rightarrow \psi ~|~\phi \leftrightarrow \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>::=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi mathvariant="normal">¬</mi> <mi>ϕ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>ϕ</mi> <mtext> </mtext> <mo>∧</mo> <mtext> </mtext> <mi>ψ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>ϕ</mi> <mo>∨</mo> <mi>ψ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>ϕ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>ϕ</mi> <mo stretchy="false">↔</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ::=a_{1},a_{2},\ldots ~|~\neg \phi ~|~\phi ~\land ~\psi ~|~\phi \vee \psi ~|~\phi \rightarrow \psi ~|~\phi \leftrightarrow \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2592aba5890d62a43fd210e3342e7d3630065f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.649ex; height:2.843ex;" alt="{\displaystyle \phi ::=a_{1},a_{2},\ldots ~|~\neg \phi ~|~\phi ~\land ~\psi ~|~\phi \vee \psi ~|~\phi \rightarrow \psi ~|~\phi \leftrightarrow \psi }">.<a href="#cite_note-:42-20">[19]</a><a href="#cite_note-:0-21">[20]</a> <div class="mw-heading mw-heading3"><h3 id="Gentzen-style_definition">Gentzen-style definition</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=8" title="Edit section: Gentzen-style definition">edit</a>]</div> A syntax definition can also be given using <a href="#Gentzen's_tree_notation">§ Gentzen's tree notation</a>, by writing well-formed formulas below the inference line and any schematic variables used by those formulas above it.<a href="#cite_note-:0-21">[20]</a> For instance, the equivalent of rules 3 and 4, from Bostock's definition above, is written as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi }{(\neg \varphi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \lor \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \land \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \rightarrow \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \leftrightarrow \psi )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mspace width="1em" /> <mi>ψ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mspace width="1em" /> <mi>ψ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>∧</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mspace width="1em" /> <mi>ψ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>φ</mi> <mspace width="1em" /> <mi>ψ</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">↔</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi }{(\neg \varphi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \lor \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \land \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \rightarrow \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \leftrightarrow \psi )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252c2f1f4f1569b37ee30eefe37d9c1c3d494c34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.115ex; height:6.176ex;" alt="{\displaystyle {\frac {\varphi }{(\neg \varphi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \lor \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \land \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \rightarrow \psi )}}\quad {\frac {\varphi \quad \psi }{(\varphi \leftrightarrow \psi )}}}">.</dd></dl> A different notational convention sees the language's syntax as a <a href="/wiki/Categorial_grammar" title="Categorial grammar">categorial grammar</a> with the single category "formula", denoted by the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <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">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">. So any elements of the syntax are introduced by categorizations, for which the notation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5ae6e295ce36baa598761f6abd69b1935f3843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.384ex; height:2.676ex;" alt="{\displaystyle \varphi :{\mathcal {F}}}">, meaning "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> is an expression for an object in the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <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">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">."<a href="#cite_note-:1-22">[21]</a> The sentence-letters, then, are introduced by categorizations such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P:{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P:{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb17ebb976339164476ecff277663c6f20ae7eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle P:{\mathcal {F}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q:{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q:{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07a1891b45dab948928c318a0c4c641a1a12c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.702ex; height:2.509ex;" alt="{\displaystyle Q:{\mathcal {F}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R:{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R:{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c51d40ffc2b8cee3867344964ffcacd98ee92e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.628ex; height:2.176ex;" alt="{\displaystyle R:{\mathcal {F}}}">, and so on;<a href="#cite_note-:1-22">[21]</a> the connectives, in turn, are defined by statements similar to the above, but using categorization notation, as seen below: <table class="wikitable"> <caption>Connectives defined through a categorial grammar<a href="#cite_note-:1-22">[21]</a> </caption> <tbody><tr> <th style="text-align: center;">Conjunction (&) </th> <th style="text-align: center;">Disjunction (∨) </th> <th style="text-align: center;">Implication (→) </th> <th style="text-align: center;">Negation (¬) </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\&(A)(B):{\mathcal {F}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mspace width="1em" /> <mi>B</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">&</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\&(A)(B):{\mathcal {F}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b3c44203dc8ccfe10065385f521c764269f0ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.393ex; height:6.176ex;" alt="{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\&(A)(B):{\mathcal {F}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\vee (A)(B):{\mathcal {F}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mspace width="1em" /> <mi>B</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\vee (A)(B):{\mathcal {F}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30043dfb4d5d9ebb3c6619b3858e68f8e774eafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.393ex; height:6.176ex;" alt="{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\vee (A)(B):{\mathcal {F}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\supset (A)(B):{\mathcal {F}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mspace width="1em" /> <mi>B</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mo>⊃</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\supset (A)(B):{\mathcal {F}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06b916149065c60029eae99424e39a4d866536f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.393ex; height:6.176ex;" alt="{\displaystyle {\frac {A:{\mathcal {F}}\quad B:{\mathcal {F}}}{\supset (A)(B):{\mathcal {F}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A:{\mathcal {F}}}{\neg (A):{\mathcal {F}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">¬</mi> <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">F</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A:{\mathcal {F}}}{\neg (A):{\mathcal {F}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5057aac407eddcbfdce4a9ed0ffb0e28cbca523a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.803ex; height:6.176ex;" alt="{\displaystyle {\frac {A:{\mathcal {F}}}{\neg (A):{\mathcal {F}}}}}"> </td></tr></tbody></table> In the rest of this article, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :{\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :{\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5ae6e295ce36baa598761f6abd69b1935f3843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.384ex; height:2.676ex;" alt="{\displaystyle \varphi :{\mathcal {F}}}"> categorization notation will be used for any Gentzen-notation statements defining the language's grammar; any other statements in Gentzen notation will be inferences, asserting that a sequent follows rather than that an expression is a well-formed formula. <div class="mw-heading mw-heading2"><h2 id="Gentzen-style_propositional_logic">Gentzen-style propositional logic</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=9" title="Edit section: Gentzen-style propositional logic">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Gentzen-style_inference_rules">Gentzen-style inference rules</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=10" title="Edit section: Gentzen-style inference rules">edit</a>]</div> The following is a complete list of primitive inference rules for natural deduction in classical propositional logic:<a href="#cite_note-:0-21">[20]</a> <table class="wikitable"> <caption>Rules for classical propositional logic </caption> <tbody><tr> <th>Introduction rules </th> <th>Elimination rules </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}\varphi \qquad \psi \\\hline \varphi \wedge \psi \end{array}}(\wedge _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mi>φ</mi> <mspace width="2em" /> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> <mo>∧</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}\varphi \qquad \psi \\\hline \varphi \wedge \psi \end{array}}(\wedge _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40b838f940e6696733e2ba8c2579ae9500841057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.447ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}\varphi \qquad \psi \\\hline \varphi \wedge \psi \end{array}}(\wedge _{i})}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}\varphi \wedge \psi \\\hline \varphi \end{array}}(\wedge _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mi>φ</mi> <mo>∧</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}\varphi \wedge \psi \\\hline \varphi \end{array}}(\wedge _{e})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e14aff68856d6c7ed3ea20d2dcb9ec64356d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.583ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}\varphi \wedge \psi \\\hline \varphi \end{array}}(\wedge _{e})}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}\varphi \\\hline \varphi \vee \psi \end{array}}(\vee _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <msub> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}\varphi \\\hline \varphi \vee \psi \end{array}}(\vee _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35e858fd4c9f766bfe9215a7f9a22fc0f739af7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.385ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}\varphi \\\hline \varphi \vee \psi \end{array}}(\vee _{i})}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\varphi \vee \psi \quad {\begin{matrix}[\varphi ]^{u}\\\vdots \\\chi \end{matrix}}\quad {\begin{matrix}[\psi ]^{v}\\\vdots \\\chi \end{matrix}}}{\chi }}\ \vee _{E^{u,v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">[</mo> <mi>φ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>χ</mi> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">[</mo> <mi>ψ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>χ</mi> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <msub> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\varphi \vee \psi \quad {\begin{matrix}[\varphi ]^{u}\\\vdots \\\chi \end{matrix}}\quad {\begin{matrix}[\psi ]^{v}\\\vdots \\\chi \end{matrix}}}{\chi }}\ \vee _{E^{u,v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f723fa6df4f87f8f0c7240be570424a731d5f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.001ex; height:14.509ex;" alt="{\displaystyle {\cfrac {\varphi \vee \psi \quad {\begin{matrix}[\varphi ]^{u}\\\vdots \\\chi \end{matrix}}\quad {\begin{matrix}[\psi ]^{v}\\\vdots \\\chi \end{matrix}}}{\chi }}\ \vee _{E^{u,v}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}[\varphi ]^{u}\\\vdots \\\psi \\\hline \varphi \to \psi \end{array}}(\to _{i})\ u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none none solid"> <mtr> <mtd> <mo stretchy="false">[</mo> <mi>φ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}[\varphi ]^{u}\\\vdots \\\psi \\\hline \varphi \to \psi \end{array}}(\to _{i})\ u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9134af77b25770df3f913190e8bd88ec48ecc6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:16.1ex; height:14.843ex;" alt="{\displaystyle {\begin{array}{c}[\varphi ]^{u}\\\vdots \\\psi \\\hline \varphi \to \psi \end{array}}(\to _{i})\ u}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}\varphi \qquad \varphi \to \psi \\\hline \psi \end{array}}(\to _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mi>φ</mi> <mspace width="2em" /> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <mi>ψ</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}\varphi \qquad \varphi \to \psi \\\hline \psi \end{array}}(\to _{e})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38f6276d1a021e785899aa852a5b0efed29df548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.554ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}\varphi \qquad \varphi \to \psi \\\hline \psi \end{array}}(\to _{e})}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}[\varphi ]^{u}\\\vdots \\\bot \\\hline \neg \varphi \end{array}}(\neg _{i})\ u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none none solid"> <mtr> <mtd> <mo stretchy="false">[</mo> <mi>φ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">⊥</mi> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">¬</mi> <mi>φ</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}[\varphi ]^{u}\\\vdots \\\bot \\\hline \neg \varphi \end{array}}(\neg _{i})\ u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38af2a2e2ba05c6e1da33e8fb509c6e64f21de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:12.665ex; height:14.843ex;" alt="{\displaystyle {\begin{array}{c}[\varphi ]^{u}\\\vdots \\\bot \\\hline \neg \varphi \end{array}}(\neg _{i})\ u}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}\varphi \qquad \neg \varphi \\\hline \bot \end{array}}(\neg _{e})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mi>φ</mi> <mspace width="2em" /> <mi mathvariant="normal">¬</mi> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">⊥</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}\varphi \qquad \neg \varphi \\\hline \bot \end{array}}(\neg _{e})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819fc50d54cc640c659d62b7c35a7dc38b202ad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.203ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}\varphi \qquad \neg \varphi \\\hline \bot \end{array}}(\neg _{e})}"> </td></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}[\neg \varphi ]^{u}\\\vdots \\\bot \\\hline \varphi \end{array}}({\text{PBC}})\ u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none none solid"> <mtr> <mtd> <mo stretchy="false">[</mo> <mi mathvariant="normal">¬</mi> <mi>φ</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">⊥</mi> </mtd> </mtr> <mtr> <mtd> <mi>φ</mi> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>PBC</mtext> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}[\neg \varphi ]^{u}\\\vdots \\\bot \\\hline \varphi \end{array}}({\text{PBC}})\ u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3750981dee8ceff32a0d05a542086c9ff44a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:16.772ex; height:14.843ex;" alt="{\displaystyle {\begin{array}{c}[\neg \varphi ]^{u}\\\vdots \\\bot \\\hline \varphi \end{array}}({\text{PBC}})\ u}"> </td></tr></tbody></table> This table follows <a href="/wiki/Propositional_calculus#Constants_and_schemata" title="Propositional calculus">the custom of using Greek letters as schemata</a>, which may range over any formulas, rather than only over atomic propositions. The name of a rule is given to the right of its formula tree. For instance, the first introduction rule is named <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6a640048ec39dd86127a7500e94208b909505b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.35ex; height:2.343ex;" alt="{\displaystyle \wedge _{i}}">, which is short for "conjunction introduction". <div class="mw-heading mw-heading3"><h3 id="Gentzen-style_example_proofs">Gentzen-style example proofs</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=11" title="Edit section: Gentzen-style example proofs">edit</a>]</div> As an example of the use of inference rules, consider commutativity of conjunction. If A ∧ B, then B ∧ A; this derivation can be drawn by composing inference rules in such a fashion that premises of a lower inference match the conclusion of the next higher inference. <div style="margin-left: 2em"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {A\wedge B}{B{\hbox{ }}}}\ \wedge _{E2}\qquad {\cfrac {A\wedge B}{A}}\ \wedge _{E1}}{B\wedge A}}\ \wedge _{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> </mtext> </mstyle> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mn>2</mn> </mrow> </msub> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>∧</mo> <mi>A</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {A\wedge B}{B{\hbox{ }}}}\ \wedge _{E2}\qquad {\cfrac {A\wedge B}{A}}\ \wedge _{E1}}{B\wedge A}}\ \wedge _{I}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49758ab3d5ddacdc980d6462c1412ee275bff9ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.439ex; height:10.676ex;" alt="{\displaystyle {\cfrac {{\cfrac {A\wedge B}{B{\hbox{ }}}}\ \wedge _{E2}\qquad {\cfrac {A\wedge B}{A}}\ \wedge _{E1}}{B\wedge A}}\ \wedge _{I}}"> </div> As a second example, consider the derivation of "A ⊃ (B ⊃ (A ∧ B))": <div style="margin-left: 2em"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {{\cfrac {}{A}}\ u\quad {\cfrac {}{B}}\ w}{A\wedge B}}\ \wedge _{I}}{{\cfrac {B\supset \left(A\wedge B\right)}{A\supset \left(B\supset \left(A\wedge B\right)\right)}}\ \supset _{I^{u}}}}\ \supset _{I^{w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>u</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>w</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>⊃</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>⊃</mo> <mrow> <mo>(</mo> <mrow> <mi>B</mi> <mo>⊃</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <msub> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mtext> </mtext> <msub> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {{\cfrac {}{A}}\ u\quad {\cfrac {}{B}}\ w}{A\wedge B}}\ \wedge _{I}}{{\cfrac {B\supset \left(A\wedge B\right)}{A\supset \left(B\supset \left(A\wedge B\right)\right)}}\ \supset _{I^{u}}}}\ \supset _{I^{w}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0cb810f1be9ee299fa1af281e7f400c06c493b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:31.385ex; height:18.009ex;" alt="{\displaystyle {\cfrac {{\cfrac {{\cfrac {}{A}}\ u\quad {\cfrac {}{B}}\ w}{A\wedge B}}\ \wedge _{I}}{{\cfrac {B\supset \left(A\wedge B\right)}{A\supset \left(B\supset \left(A\wedge B\right)\right)}}\ \supset _{I^{u}}}}\ \supset _{I^{w}}}"> </div> This full derivation has no unsatisfied premises; however, sub-derivations are hypothetical. For instance, the derivation of "B ⊃ (A ∧ B)" is hypothetical with antecedent "A" (named u). <div class="mw-heading mw-heading2"><h2 id="Suppes–Lemmon-style_propositional_logic">Suppes–Lemmon-style propositional logic</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=12" title="Edit section: Suppes–Lemmon-style propositional logic">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Suppes–Lemmon-style_inference_rules">Suppes–Lemmon-style inference rules</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=13" title="Edit section: Suppes–Lemmon-style inference rules">edit</a>]</div> Natural deduction inference rules, due ultimately to <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen</a>, are given below.<a href="#cite_note-LemmonLogic-23">[22]</a> There are ten primitive rules of proof, which are the rule assumption, plus four pairs of introduction and elimination rules for the binary connectives, and the rule reductio ad adbsurdum.<a href="#cite_note-AllenHand-18">[17]</a> Disjunctive Syllogism can be used as an easier alternative to the proper ∨-elimination,<a href="#cite_note-AllenHand-18">[17]</a> and MTT and DN are commonly given rules,<a href="#cite_note-LemmonLogic-23">[22]</a> although they are not primitive.<a href="#cite_note-AllenHand-18">[17]</a> <table class="wikitable" style="margin:auto;"> <caption>List of Inference Rules </caption> <tbody><tr> <th>Rule Name </th> <th>Alternative names </th> <th>Annotation </th> <th>Assumption set </th> <th>Statement </th></tr> <tr> <td>Rule of Assumptions<a href="#cite_note-LemmonLogic-23">[22]</a></td> <td>Assumption<a href="#cite_note-AllenHand-18">[17]</a></td> <td>A<a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-AllenHand-18">[17]</a> </td> <td>The current line number.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>At any stage of the argument, introduce a proposition as an assumption of the argument.<a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Conjunction introduction</td> <td>Ampersand introduction,<a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-AllenHand-18">[17]</a> conjunction (CONJ)<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-TWArthurLogic-24">[23]</a></td> <td>m, n &I<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>The union of the assumption sets at lines m and n.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> at lines m and n, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ~\&~\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mtext> </mtext> <mi mathvariant="normal">&</mi> <mtext> </mtext> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ~\&~\psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349658552cb30c288027982baf983d594188b08c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.003ex; height:2.676ex;" alt="{\displaystyle \varphi ~\&~\psi }">.<a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Conjunction elimination</td> <td>Simplification (S),<a href="#cite_note-AllenHand-18">[17]</a> ampersand elimination<a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-AllenHand-18">[17]</a></td> <td>m &E<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>The same as at line m.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ~\&~\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mtext> </mtext> <mi mathvariant="normal">&</mi> <mtext> </mtext> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ~\&~\psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/349658552cb30c288027982baf983d594188b08c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.003ex; height:2.676ex;" alt="{\displaystyle \varphi ~\&~\psi }"> at line m, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }">.<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-LemmonLogic-23">[22]</a> </td></tr> <tr> <td>Disjunction introduction<a href="#cite_note-LemmonLogic-23">[22]</a></td> <td>Addition (ADD)<a href="#cite_note-AllenHand-18">[17]</a></td> <td>m ∨I<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>The same as at line m.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> at line m, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \lor \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \lor \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.676ex;" alt="{\displaystyle \varphi \lor \psi }">, whatever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> may be.<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-LemmonLogic-23">[22]</a> </td></tr> <tr> <td>Disjunction elimination</td> <td>Wedge elimination,<a href="#cite_note-LemmonLogic-23">[22]</a> dilemma (DL)<a href="#cite_note-TWArthurLogic-24">[23]</a></td> <td>j,k,l,m,n ∨E<a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>The lines j,k,l,m,n.<a href="#cite_note-LemmonLogic-23">[22]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \lor \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \lor \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.676ex;" alt="{\displaystyle \varphi \lor \psi }"> at line j, and an assumption of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> at line k, and a derivation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> at line l, and an assumption of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> at line m, and a derivation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> at line n, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.009ex;" alt="{\displaystyle \chi }">.<a href="#cite_note-LemmonLogic-23">[22]</a> </td></tr> <tr> <td>Disjunctive Syllogism </td> <td>Wedge elimination (∨E),<a href="#cite_note-AllenHand-18">[17]</a> modus tollendo ponens (MTP)<a href="#cite_note-AllenHand-18">[17]</a> </td> <td>m,n DS<a href="#cite_note-AllenHand-18">[17]</a> </td> <td>The union of the assumption sets at lines m and n.<a href="#cite_note-AllenHand-18">[17]</a> </td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \lor \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \lor \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.676ex;" alt="{\displaystyle \varphi \lor \psi }"> at line m and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d48bd06914d3df71eb66552fc8f0fdb7af496a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.328ex; height:2.509ex;" alt="{\displaystyle -\varphi }"> at line n, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }">; from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \lor \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∨</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \lor \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53732d566089f41274c3fc138c14cd87ba59febd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.676ex;" alt="{\displaystyle \varphi \lor \psi }"> at line m and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db179e0c31a9ae42d7e993a683a5ca791245ec7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.321ex; height:2.509ex;" alt="{\displaystyle -\psi }"> at line n, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }">.<a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Arrow elimination<a href="#cite_note-AllenHand-18">[17]</a></td> <td>Modus ponendo ponens (MPP),<a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-AllenHand-18">[17]</a> modus ponens (MP),<a href="#cite_note-TWArthurLogic-24">[23]</a><a href="#cite_note-AllenHand-18">[17]</a> conditional elimination</td> <td>m, n →E<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>The union of the assumption sets at lines m and n.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \to \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \to \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8cebc7ec869e87352963afa13d3f7862058f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \varphi \to \psi }"> at line m, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> at line n, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }">.<a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Arrow introduction<a href="#cite_note-AllenHand-18">[17]</a></td> <td>Conditional proof (CP),<a href="#cite_note-TWArthurLogic-24">[23]</a><a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-AllenHand-18">[17]</a> conditional introduction</td> <td>n, →I (m)<a href="#cite_note-AllenHand-18">[17]</a><a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>Everything in the assumption set at line n, excepting m, the line where the antecedent was assumed.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> at line n, following from the assumption of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> at line m, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \to \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \to \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8cebc7ec869e87352963afa13d3f7862058f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \varphi \to \psi }">.<a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Reductio ad absurdum<a href="#cite_note-LemmonLogic-23">[22]</a></td> <td>Indirect Proof (IP),<a href="#cite_note-AllenHand-18">[17]</a> negation introduction (−I),<a href="#cite_note-AllenHand-18">[17]</a> negation elimination (−E)<a href="#cite_note-AllenHand-18">[17]</a></td> <td>m, n RAA (k)<a href="#cite_note-AllenHand-18">[17]</a> </td> <td>The union of the assumption sets at lines m and n, excluding k (the denied assumption).<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From a sentence and its denial<a href="#cite_note-25">[b]</a> at lines m and n, infer the denial of any assumption appearing in the proof (at line k).<a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Double arrow introduction<a href="#cite_note-AllenHand-18">[17]</a></td> <td>Biconditional definition (Df ↔),<a href="#cite_note-LemmonLogic-23">[22]</a> biconditional introduction</td> <td>m, n ↔ I<a href="#cite_note-AllenHand-18">[17]</a> </td> <td>The union of the assumption sets at lines m and n.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \to \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \to \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8cebc7ec869e87352963afa13d3f7862058f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \varphi \to \psi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \to \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \to \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36aefa2940287c7dc1e23df0a900019ea00df615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \psi \to \varphi }"> at lines m and n, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \leftrightarrow \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">↔</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \leftrightarrow \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \varphi \leftrightarrow \psi }">.<a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Double arrow elimination<a href="#cite_note-AllenHand-18">[17]</a></td> <td>Biconditional definition (Df ↔),<a href="#cite_note-LemmonLogic-23">[22]</a> biconditional elimination</td> <td>m ↔ E<a href="#cite_note-AllenHand-18">[17]</a> </td> <td>The same as at line m.<a href="#cite_note-AllenHand-18">[17]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \leftrightarrow \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">↔</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \leftrightarrow \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68c9c0e49df2cdfb5bbb241b0d3f6f10e20aa90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \varphi \leftrightarrow \psi }"> at line m, infer either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \to \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \to \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8cebc7ec869e87352963afa13d3f7862058f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \varphi \to \psi }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \to \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">→</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \to \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36aefa2940287c7dc1e23df0a900019ea00df615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \psi \to \varphi }">.<a href="#cite_note-AllenHand-18">[17]</a> </td></tr> <tr> <td>Double negation<a href="#cite_note-LemmonLogic-23">[22]</a><a href="#cite_note-TWArthurLogic-24">[23]</a></td> <td>Double negation elimination</td> <td>m DN<a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>The same as at line m.<a href="#cite_note-LemmonLogic-23">[22]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle --\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo>−</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle --\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefa96151e4c2e8a8353adfa47aa23232326055d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.169ex; height:2.509ex;" alt="{\displaystyle --\varphi }"> at line m, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }">.<a href="#cite_note-LemmonLogic-23">[22]</a> </td></tr> <tr> <td>Modus tollendo tollens<a href="#cite_note-LemmonLogic-23">[22]</a></td> <td>Modus tollens (MT)<a href="#cite_note-TWArthurLogic-24">[23]</a></td> <td>m, n MTT<a href="#cite_note-LemmonLogic-23">[22]</a> </td> <td>The union of the assumption sets at lines m and n.<a href="#cite_note-LemmonLogic-23">[22]</a></td> <td>From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \to \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">→</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \to \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8cebc7ec869e87352963afa13d3f7862058f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.647ex; height:2.676ex;" alt="{\displaystyle \varphi \to \psi }"> at line m, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db179e0c31a9ae42d7e993a683a5ca791245ec7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.321ex; height:2.509ex;" alt="{\displaystyle -\psi }"> at line n, infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d48bd06914d3df71eb66552fc8f0fdb7af496a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.328ex; height:2.509ex;" alt="{\displaystyle -\varphi }">.<a href="#cite_note-LemmonLogic-23">[22]</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Suppes–Lemmon-style_example_proof">Suppes–Lemmon-style example proof</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=14" title="Edit section: Suppes–Lemmon-style example proof">edit</a>]</div> Recall that an example proof was already given when introducing <a href="#Suppes–Lemmon_notation">§ Suppes–Lemmon notation</a>. This is a second example. <table class="wikitable" style="margin:auto;"> <caption>Suppes–Lemmon style proof (second example) </caption> <tbody><tr> <th>Assumption set </th> <th>Line number </th> <th>Sentence of proof </th> <th>Annotation </th></tr> <tr> <td>1 </td> <td>1 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\lor Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\lor Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2bc60d4b9ff5ec772fec5c2ef72a39536d4323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\lor Q}"> </td> <td>A </td></tr> <tr> <td>2 </td> <td>2 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"> </td> <td>A </td></tr> <tr> <td>3 </td> <td>3 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P\to \neg Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P\to \neg Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f557b1e30ca9d61042833c85784c3e74363caad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.298ex; height:2.509ex;" alt="{\displaystyle \neg P\to \neg Q}"> </td> <td>A </td></tr> <tr> <td>2, 3 </td> <td>4 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad34798abb0bbbc063c906e459f103a09b1660e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle \neg Q}"> </td> <td>2, 3 →E </td></tr> <tr> <td>1, 2, 3 </td> <td>5 </td> <td><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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> </td> <td>1, 4 &E </td></tr> <tr> <td>1, 3 </td> <td>6 </td> <td><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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> </td> <td>2, 5 RAA(2) </td></tr> <tr> <td>2, 3 </td> <td>7 </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"> </td> <td>2, 3 RAA(1) </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Consistency,_completeness,_and_normal_forms">Consistency, completeness, and normal forms</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=15" title="Edit section: Consistency, completeness, and normal forms">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> A <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">theory</a> is said to be consistent if falsehood is not provable (from no assumptions) and is complete if every theorem or its negation is provable using the inference rules of the logic. These are statements about the entire logic, and are usually tied to some notion of a <a href="/wiki/Model_theory" title="Model theory">model</a>. However, there are local notions of consistency and completeness that are purely syntactic checks on the inference rules, and require no appeals to models. The first of these is local consistency, also known as local reducibility, which says that any derivation containing an introduction of a connective followed immediately by its elimination can be turned into an equivalent derivation without this detour. It is a check on the strength of elimination rules: they must not be so strong that they include knowledge not already contained in their premises. As an example, consider conjunctions. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\cfrac {{\cfrac {{\cfrac {}{A\ }}u\qquad {\cfrac {}{B\ }}w}{A\wedge B\ }}\wedge _{I}}{A\ }}\wedge _{E1}\end{aligned}}\quad \Rightarrow \quad {\cfrac {}{A\ }}u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mi>u</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mi>w</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\cfrac {{\cfrac {{\cfrac {}{A\ }}u\qquad {\cfrac {}{B\ }}w}{A\wedge B\ }}\wedge _{I}}{A\ }}\wedge _{E1}\end{aligned}}\quad \Rightarrow \quad {\cfrac {}{A\ }}u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4f7d4519466d37b128b1de4e84d59764ebd333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:35.623ex; height:14.343ex;" alt="{\displaystyle {\begin{aligned}{\cfrac {{\cfrac {{\cfrac {}{A\ }}u\qquad {\cfrac {}{B\ }}w}{A\wedge B\ }}\wedge _{I}}{A\ }}\wedge _{E1}\end{aligned}}\quad \Rightarrow \quad {\cfrac {}{A\ }}u}"></dd></dl> Dually, local completeness says that the elimination rules are strong enough to decompose a connective into the forms suitable for its introduction rule. Again for conjunctions: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {}{A\wedge B\ }}u\quad \Rightarrow \quad {\begin{aligned}{\cfrac {{\cfrac {{\cfrac {}{A\wedge B\ }}u}{A\ }}\wedge _{E1}\qquad {\cfrac {{\cfrac {}{A\wedge B\ }}u}{B\ }}\wedge _{E2}}{A\wedge B\ }}\wedge _{I}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mi>u</mi> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mi>u</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mn>1</mn> </mrow> </msub> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mi>u</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mtext> </mtext> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msub> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {}{A\wedge B\ }}u\quad \Rightarrow \quad {\begin{aligned}{\cfrac {{\cfrac {{\cfrac {}{A\wedge B\ }}u}{A\ }}\wedge _{E1}\qquad {\cfrac {{\cfrac {}{A\wedge B\ }}u}{B\ }}\wedge _{E2}}{A\wedge B\ }}\wedge _{I}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80941a1fe9c037d9d5e51723cb57a94515c92d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:54.036ex; height:14.343ex;" alt="{\displaystyle {\cfrac {}{A\wedge B\ }}u\quad \Rightarrow \quad {\begin{aligned}{\cfrac {{\cfrac {{\cfrac {}{A\wedge B\ }}u}{A\ }}\wedge _{E1}\qquad {\cfrac {{\cfrac {}{A\wedge B\ }}u}{B\ }}\wedge _{E2}}{A\wedge B\ }}\wedge _{I}\end{aligned}}}"></dd></dl> These notions correspond exactly to <a href="/wiki/Lambda_calculus#.CE.B2-reduction" title="Lambda calculus">β-reduction (beta reduction)</a> and <a href="/wiki/Lambda_calculus#.CE.B7-conversion" title="Lambda calculus">η-conversion (eta conversion)</a> in the <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a>, using the <a href="/wiki/Curry%E2%80%93Howard_isomorphism" class="mw-redirect" title="Curry–Howard isomorphism">Curry–Howard isomorphism</a>. By local completeness, we see that every derivation can be converted to an equivalent derivation where the principal connective is introduced. In fact, if the entire derivation obeys this ordering of eliminations followed by introductions, then it is said to be normal. In a normal derivation all eliminations happen above introductions. In most logics, every derivation has an equivalent normal derivation, called a <a href="/wiki/Normal_form_(abstract_rewriting)" title="Normal form (abstract rewriting)">normal form</a>. The existence of normal forms is generally hard to prove using natural deduction alone, though such accounts do exist in the literature, most notably by <a href="/wiki/Dag_Prawitz" title="Dag Prawitz">Dag Prawitz</a> in 1961.<a href="#cite_note-26">[24]</a> It is much easier to show this indirectly by means of a <a href="/wiki/Cut_elimination" class="mw-redirect" title="Cut elimination">cut-free</a> <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculus</a> presentation. <div class="mw-heading mw-heading2"><h2 id="First_and_higher-order_extensions">First and higher-order extensions</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=16" title="Edit section: First and higher-order extensions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:First_order_natural_deduction.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/First_order_natural_deduction.png/220px-First_order_natural_deduction.png" decoding="async" width="220" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/First_order_natural_deduction.png/330px-First_order_natural_deduction.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/First_order_natural_deduction.png/440px-First_order_natural_deduction.png 2x" data-file-width="685" data-file-height="873" /></a><figcaption>Summary of first-order system</figcaption></figure> The logic of the earlier section is an example of a single-sorted logic, i.e., a logic with a single kind of object: propositions. Many extensions of this simple framework have been proposed; in this section we will extend it with a second sort of individuals or <a href="/wiki/Term_(logic)" title="Term (logic)">terms</a>. More precisely, we will add a new category, "term", denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}">. We shall fix a <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> of variables, another countable set <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}"> of function symbols, and construct terms with the following formation rules: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {v\in V}{v:{\mathcal {T}}}}{\hbox{ var}}_{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mrow> <mrow> <mi>v</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> var</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {v\in V}{v:{\mathcal {T}}}}{\hbox{ var}}_{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1867140a281c92fe7a9ca9a722c185ee262004f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.937ex; height:5.509ex;" alt="{\displaystyle {\frac {v\in V}{v:{\mathcal {T}}}}{\hbox{ var}}_{F}}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f\in F\qquad t_{1}:{\mathcal {T}}\qquad t_{2}:{\mathcal {T}}\qquad \cdots \qquad t_{n}:{\mathcal {T}}}{f(t_{1},t_{2},\cdots ,t_{n}):{\mathcal {T}}}}{\hbox{ app}}_{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo>∈</mo> <mi>F</mi> <mspace width="2em" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mspace width="2em" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mspace width="2em" /> <mo>⋯</mo> <mspace width="2em" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> app</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f\in F\qquad t_{1}:{\mathcal {T}}\qquad t_{2}:{\mathcal {T}}\qquad \cdots \qquad t_{n}:{\mathcal {T}}}{f(t_{1},t_{2},\cdots ,t_{n}):{\mathcal {T}}}}{\hbox{ app}}_{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/358ad1b9dbcc09fdddeed40c421817bd80a20d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.031ex; height:6.343ex;" alt="{\displaystyle {\frac {f\in F\qquad t_{1}:{\mathcal {T}}\qquad t_{2}:{\mathcal {T}}\qquad \cdots \qquad t_{n}:{\mathcal {T}}}{f(t_{1},t_{2},\cdots ,t_{n}):{\mathcal {T}}}}{\hbox{ app}}_{F}}"></dd></dl> For propositions, we consider a third countable set P of <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicates</a>, and define atomic predicates over terms with the following formation rule: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\phi \in P\qquad t_{1}:{\mathcal {T}}\qquad t_{2}:{\mathcal {T}}\qquad \cdots \qquad t_{n}:{\mathcal {T}}}{\phi (t_{1},t_{2},\cdots ,t_{n}):{\mathcal {F}}}}{\hbox{ pred}}_{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo>∈</mo> <mi>P</mi> <mspace width="2em" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mspace width="2em" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mspace width="2em" /> <mo>⋯</mo> <mspace width="2em" /> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mrow> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> pred</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\phi \in P\qquad t_{1}:{\mathcal {T}}\qquad t_{2}:{\mathcal {T}}\qquad \cdots \qquad t_{n}:{\mathcal {T}}}{\phi (t_{1},t_{2},\cdots ,t_{n}):{\mathcal {F}}}}{\hbox{ pred}}_{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f98bc3c1386591760a2f9ad03eb4346b72336a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.924ex; height:6.343ex;" alt="{\displaystyle {\frac {\phi \in P\qquad t_{1}:{\mathcal {T}}\qquad t_{2}:{\mathcal {T}}\qquad \cdots \qquad t_{n}:{\mathcal {T}}}{\phi (t_{1},t_{2},\cdots ,t_{n}):{\mathcal {F}}}}{\hbox{ pred}}_{F}}"></dd></dl> The first two rules of formation provide a definition of a term that is effectively the same as that defined in <a href="/wiki/Term_algebra" title="Term algebra">term algebra</a> and <a href="/wiki/Model_theory" title="Model theory">model theory</a>, although the focus of those fields of study is quite different from natural deduction. The third rule of formation effectively defines an <a href="/wiki/Atomic_formula" title="Atomic formula">atomic formula</a>, as in <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, and again in model theory. To these are added a pair of formation rules, defining the notation for <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantified</a> propositions; one for universal (∀) and existential (∃) quantification: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x\in V\qquad A:{\mathcal {F}}}{\forall x.A:{\mathcal {F}}}}\;\forall _{F}\qquad \qquad {\frac {x\in V\qquad A:{\mathcal {F}}}{\exists x.A:{\mathcal {F}}}}\;\exists _{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mspace width="2em" /> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>∈</mo> <mi>V</mi> <mspace width="2em" /> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>.</mo> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x\in V\qquad A:{\mathcal {F}}}{\forall x.A:{\mathcal {F}}}}\;\forall _{F}\qquad \qquad {\frac {x\in V\qquad A:{\mathcal {F}}}{\exists x.A:{\mathcal {F}}}}\;\exists _{F}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3899e191e422b081619811fa74356b385bd3c403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.184ex; height:5.509ex;" alt="{\displaystyle {\frac {x\in V\qquad A:{\mathcal {F}}}{\forall x.A:{\mathcal {F}}}}\;\forall _{F}\qquad \qquad {\frac {x\in V\qquad A:{\mathcal {F}}}{\exists x.A:{\mathcal {F}}}}\;\exists _{F}}"></dd></dl> The <a href="/wiki/Universal_quantifier" class="mw-redirect" title="Universal quantifier">universal quantifier</a> has the introduction and elimination rules: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\begin{array}{c}{\cfrac {}{a:{\mathcal {T}}}}{\text{ u}}\\\vdots \\{}[a/x]A\end{array}}{\forall x.A}}\;\forall _{I^{u,a}}\qquad \qquad {\frac {\forall x.A\qquad t:{\mathcal {T}}}{[t/x]A}}\;\forall _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> u</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">]</mo> <mi>A</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi>A</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>,</mo> <mi>a</mi> </mrow> </msup> </mrow> </msub> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi>A</mi> <mspace width="2em" /> <mi>t</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mrow> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">]</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\begin{array}{c}{\cfrac {}{a:{\mathcal {T}}}}{\text{ u}}\\\vdots \\{}[a/x]A\end{array}}{\forall x.A}}\;\forall _{I^{u,a}}\qquad \qquad {\frac {\forall x.A\qquad t:{\mathcal {T}}}{[t/x]A}}\;\forall _{E}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa20ba2dd507c0b97e8f3820831eaa07bfd51e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.744ex; height:18.509ex;" alt="{\displaystyle {\cfrac {\begin{array}{c}{\cfrac {}{a:{\mathcal {T}}}}{\text{ u}}\\\vdots \\{}[a/x]A\end{array}}{\forall x.A}}\;\forall _{I^{u,a}}\qquad \qquad {\frac {\forall x.A\qquad t:{\mathcal {T}}}{[t/x]A}}\;\forall _{E}}"></dd></dl> The <a href="/wiki/Existential_quantifier" class="mw-redirect" title="Existential quantifier">existential quantifier</a> has the introduction and elimination rules: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {[t/x]A}{\exists x.A}}\;\exists _{I}\qquad \qquad {\cfrac {\begin{array}{cc}&\underbrace {\,{\cfrac {}{a:{\mathcal {T}}}}{\hbox{ u}}\quad {\cfrac {}{[a/x]A}}{\hbox{ v}}\,} \\&\vdots \\\exists x.A\quad &C\\\end{array}}{C}}\exists _{E^{a,u,v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">]</mo> <mi>A</mi> </mrow> <mrow> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>.</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> u</mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">]</mo> <mi>A</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> v</mtext> </mstyle> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>⏟</mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>.</mo> <mi>A</mi> <mspace width="1em" /> </mtd> <mtd> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {[t/x]A}{\exists x.A}}\;\exists _{I}\qquad \qquad {\cfrac {\begin{array}{cc}&\underbrace {\,{\cfrac {}{a:{\mathcal {T}}}}{\hbox{ u}}\quad {\cfrac {}{[a/x]A}}{\hbox{ v}}\,} \\&\vdots \\\exists x.A\quad &C\\\end{array}}{C}}\exists _{E^{a,u,v}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94ac4f9f1480f3290af8775c42cfd26fa28670c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.254ex; height:20.176ex;" alt="{\displaystyle {\frac {[t/x]A}{\exists x.A}}\;\exists _{I}\qquad \qquad {\cfrac {\begin{array}{cc}&\underbrace {\,{\cfrac {}{a:{\mathcal {T}}}}{\hbox{ u}}\quad {\cfrac {}{[a/x]A}}{\hbox{ v}}\,} \\&\vdots \\\exists x.A\quad &C\\\end{array}}{C}}\exists _{E^{a,u,v}}}"></dd></dl> In these rules, the notation [t/x] A stands for the substitution of t for every (visible) instance of x in A, avoiding capture.<a href="#cite_note-27">[c]</a> As before the superscripts on the name stand for the components that are discharged: the term a cannot occur in the conclusion of ∀I (such terms are known as eigenvariables or parameters), and the hypotheses named u and v in ∃E are localised to the second premise in a hypothetical derivation. Although the propositional logic of earlier sections was <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>, adding the quantifiers makes the logic undecidable. So far, the quantified extensions are first-order: they distinguish propositions from the kinds of objects quantified over. <a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order logic</a> takes a different approach and has only a single sort of propositions. The quantifiers have as the domain of quantification the very same sort of propositions, as reflected in the formation rules: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\begin{matrix}{\cfrac {}{p:{\mathcal {F}}}}{\hbox{ u}}\\\vdots \\A:{\mathcal {F}}\\\end{matrix}}{\forall p.A:{\mathcal {F}}}}\;\forall _{F^{u}}\qquad \qquad {\cfrac {\begin{matrix}{\cfrac {}{p:{\mathcal {F}}}}{\hbox{ u}}\\\vdots \\A:{\mathcal {F}}\\\end{matrix}}{\exists p.A:{\mathcal {F}}}}\;\exists _{F^{u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> u</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>p</mi> <mo>.</mo> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mrow> </msub> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> u</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>p</mi> <mo>.</mo> <mi>A</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msub> <mi mathvariant="normal">∃</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\begin{matrix}{\cfrac {}{p:{\mathcal {F}}}}{\hbox{ u}}\\\vdots \\A:{\mathcal {F}}\\\end{matrix}}{\forall p.A:{\mathcal {F}}}}\;\forall _{F^{u}}\qquad \qquad {\cfrac {\begin{matrix}{\cfrac {}{p:{\mathcal {F}}}}{\hbox{ u}}\\\vdots \\A:{\mathcal {F}}\\\end{matrix}}{\exists p.A:{\mathcal {F}}}}\;\exists _{F^{u}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e9a6a36903df1d889eded734e5d1f891a563c75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.93ex; height:18.343ex;" alt="{\displaystyle {\cfrac {\begin{matrix}{\cfrac {}{p:{\mathcal {F}}}}{\hbox{ u}}\\\vdots \\A:{\mathcal {F}}\\\end{matrix}}{\forall p.A:{\mathcal {F}}}}\;\forall _{F^{u}}\qquad \qquad {\cfrac {\begin{matrix}{\cfrac {}{p:{\mathcal {F}}}}{\hbox{ u}}\\\vdots \\A:{\mathcal {F}}\\\end{matrix}}{\exists p.A:{\mathcal {F}}}}\;\exists _{F^{u}}}"></dd></dl> A discussion of the introduction and elimination forms for higher-order logic is beyond the scope of this article. It is possible to be in-between first-order and higher-order logics. For example, <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a> has two kinds of propositions, one kind quantifying over terms, and the second kind quantifying over propositions of the first kind. <div class="mw-heading mw-heading2"><h2 id="Proofs_and_type_theory">Proofs and type theory</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=17" title="Edit section: Proofs and type theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> The presentation of natural deduction so far has concentrated on the nature of propositions without giving a formal definition of a proof. To formalise the notion of proof, we alter the presentation of hypothetical derivations slightly. We label the antecedents with proof variables (from some countable set V of variables), and decorate the succedent with the actual proof. The antecedents or hypotheses are separated from the succedent by means of a <a href="/wiki/Turnstile_(symbol)" title="Turnstile (symbol)">turnstile</a> (⊢). This modification sometimes goes under the name of localised hypotheses. The following diagram summarises the change. <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>──── u1 ──── u2 ... ──── un J1 J2 Jn ⋮ J </pre> </td> <td width="10%" align="center">⇒</td> <td> <pre>u1:J1, u2:J2, ..., un:Jn ⊢ J </pre> </td></tr></tbody></table> The collection of hypotheses will be written as Γ when their exact composition is not relevant. To make proofs explicit, we move from the proof-less judgment "A" to a judgment: "π is a proof of (A)", which is written symbolically as "π : A". Following the standard approach, proofs are specified with their own formation rules for the judgment "π proof". The simplest possible proof is the use of a labelled hypothesis; in this case the evidence is the label itself. <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>u ∈ V ─────── proof-F u proof </pre> </td> <td width="10%"></td> <td> <pre>───────────────────── hyp u:A ⊢ u : A </pre> </td></tr></tbody></table> Let us re-examine some of the connectives with explicit proofs. For conjunction, we look at the introduction rule ∧I to discover the form of proofs of conjunction: they must be a pair of proofs of the two conjuncts. Thus: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>π1 proof π2 proof ──────────────────── pair-F (π1, π2) proof </pre> </td> <td width="10%"></td> <td> <pre>Γ ⊢ π1 : A Γ ⊢ π2 : B ───────────────────────── ∧I Γ ⊢ (π1, π2) : A ∧ B </pre> </td></tr></tbody></table> The elimination rules ∧E1 and ∧E2 select either the left or the right conjunct; thus the proofs are a pair of projections—first (fst) and second (snd). <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>π proof ─────────── fst-F fst π proof </pre> </td> <td width="10%"></td> <td> <pre>Γ ⊢ π : A ∧ B ───────────── ∧E1 Γ ⊢ fst π : A </pre> </td></tr> <tr> <td> <pre>π proof ─────────── snd-F snd π proof </pre> </td> <td width="10%"></td> <td> <pre>Γ ⊢ π : A ∧ B ───────────── ∧E2 Γ ⊢ snd π : B </pre> </td></tr></tbody></table> For implication, the introduction form localises or binds the hypothesis, written using a λ; this corresponds to the discharged label. In the rule, "Γ, u:A" stands for the collection of hypotheses Γ, together with the additional hypothesis u. <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>π proof ──────────── λ-F λu. π proof </pre> </td> <td width="10%"></td> <td> <pre>Γ, u:A ⊢ π : B ───────────────── ⊃I Γ ⊢ λu. π : A ⊃ B </pre> </td></tr> <tr> <td> <pre>π1 proof π2 proof ─────────────────── app-F π1 π2 proof </pre> </td> <td width="10%"></td> <td> <pre>Γ ⊢ π1 : A ⊃ B Γ ⊢ π2 : A ──────────────────────────── ⊃E Γ ⊢ π1 π2 : B </pre> </td></tr></tbody></table> With proofs available explicitly, one can manipulate and reason about proofs. The key operation on proofs is the substitution of one proof for an assumption used in another proof. This is commonly known as a substitution theorem, and can be proved by <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a> on the depth (or structure) of the second judgment. <div class="mw-heading mw-heading3"><h3 id="Substitution_theorem">Substitution theorem</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=18" title="Edit section: Substitution theorem">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <dl><dd>If Γ ⊢ π1 : A and Γ, u:A ⊢ π2 : B, then Γ ⊢ [π1/u] π2 : B.</dd></dl> So far the judgment "Γ ⊢ π : A" has had a purely logical interpretation. In <a href="/wiki/Type_theory" title="Type theory">type theory</a>, the logical view is exchanged for a more computational view of objects. Propositions in the logical interpretation are now viewed as types, and proofs as programs in the <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a>. Thus the interpretation of "π : A" is "the program π has type A". The logical connectives are also given a different reading: conjunction is viewed as <a href="/wiki/Product_type" title="Product type">product</a> (×), implication as the function <a href="/wiki/Function_type" title="Function type">arrow</a> (→), etc. The differences are only cosmetic, however. Type theory has a natural deduction presentation in terms of formation, introduction and elimination rules; in fact, the reader can easily reconstruct what is known as simple type theory from the previous sections. The difference between logic and type theory is primarily a shift of focus from the types (propositions) to the programs (proofs). Type theory is chiefly interested in the convertibility or reducibility of programs. For every type, there are canonical programs of that type which are irreducible; these are known as canonical forms or values. If every program can be reduced to a canonical form, then the type theory is said to be <a href="/wiki/Normalization_property_(abstract_rewriting)" class="mw-redirect" title="Normalization property (abstract rewriting)">normalising</a> (or weakly normalising). If the canonical form is unique, then the theory is said to be strongly normalising. Normalisability is a rare feature of most non-trivial type theories, which is a big departure from the logical world. (Recall that almost every logical derivation has an equivalent normal derivation.) To sketch the reason: in type theories that admit recursive definitions, it is possible to write programs that never reduce to a value; such looping programs can generally be given any type. In particular, the looping program has type ⊥, although there is no logical proof of "⊥". For this reason, the propositions as types; proofs as programs paradigm only works in one direction, if at all: interpreting a type theory as a logic generally gives an inconsistent logic. <div class="mw-heading mw-heading3"><h3 id="Example:_Dependent_Type_Theory">Example: Dependent Type Theory</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=19" title="Edit section: Example: Dependent Type Theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> Like logic, type theory has many extensions and variants, including first-order and higher-order versions. One branch, known as <a href="/wiki/Dependent_type_theory" class="mw-redirect" title="Dependent type theory">dependent type theory</a>, is used in a number of <a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">computer-assisted proof</a> systems. Dependent type theory allows quantifiers to range over programs themselves. These quantified types are written as Π and Σ instead of ∀ and ∃, and have the following formation rules: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Γ ⊢ A type Γ, x:A ⊢ B type ───────────────────────────── Π-F Γ ⊢ Πx:A. B type </pre> </td> <td width="10%"></td> <td> <pre>Γ ⊢ A type Γ, x:A ⊢ B type ──────────────────────────── Σ-F Γ ⊢ Σx:A. B type </pre> </td></tr></tbody></table> These types are generalisations of the arrow and product types, respectively, as witnessed by their introduction and elimination rules. <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Γ, x:A ⊢ π : B ──────────────────── ΠI Γ ⊢ λx. π : Πx:A. B </pre> </td> <td width="10%"></td> <td> <pre>Γ ⊢ π1 : Πx:A. B Γ ⊢ π2 : A ───────────────────────────── ΠE Γ ⊢ π1 π2 : [π2/x] B </pre> </td></tr></tbody></table> <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Γ ⊢ π1 : A Γ, x:A ⊢ π2 : B ───────────────────────────── ΣI Γ ⊢ (π1, π2) : Σx:A. B </pre> </td> <td width="10%"></td> <td> <pre>Γ ⊢ π : Σx:A. B ──────────────── ΣE1 Γ ⊢ fst π : A </pre> </td></tr></tbody></table> <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Γ ⊢ π : Σx:A. B ──────────────────────── ΣE2 Γ ⊢ snd π : [fst π/x] B </pre> </td></tr></tbody></table> Dependent type theory in full generality is very powerful: it is able to express almost any conceivable property of programs directly in the types of the program. This generality comes at a steep price — either typechecking is undecidable (<a href="/wiki/Extensional_type_theory" class="mw-redirect" title="Extensional type theory">extensional type theory</a>), or extensional reasoning is more difficult (<a href="/wiki/Intensional_type_theory" class="mw-redirect" title="Intensional type theory">intensional type theory</a>). For this reason, some dependent type theories do not allow quantification over arbitrary programs, but rather restrict to programs of a given decidable index domain, for example integers, strings, or linear programs. Since dependent type theories allow types to depend on programs, a natural question to ask is whether it is possible for programs to depend on types, or any other combination. There are many kinds of answers to such questions. A popular approach in type theory is to allow programs to be quantified over types, also known as <a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">parametric polymorphism</a>; of this there are two main kinds: if types and programs are kept separate, then one obtains a somewhat more well-behaved system called <a href="/wiki/Predicative_polymorphism" class="mw-redirect" title="Predicative polymorphism">predicative polymorphism</a>; if the distinction between program and type is blurred, one obtains the type-theoretic analogue of higher-order logic, also known as <a href="/wiki/Impredicative_polymorphism" class="mw-redirect" title="Impredicative polymorphism">impredicative polymorphism</a>. Various combinations of dependency and polymorphism have been considered in the literature, the most famous being the <a href="/wiki/Lambda_cube" title="Lambda cube">lambda cube</a> of <a href="/wiki/Henk_Barendregt" title="Henk Barendregt">Henk Barendregt</a>. The intersection of logic and type theory is a vast and active research area. New logics are usually formalised in a general type theoretic setting, known as a <a href="/wiki/Logical_framework" title="Logical framework">logical framework</a>. Popular modern logical frameworks such as the <a href="/wiki/Calculus_of_constructions" title="Calculus of constructions">calculus of constructions</a> and <a href="/wiki/LF_(logical_framework)" class="mw-redirect" title="LF (logical framework)">LF</a> are based on higher-order dependent type theory, with various trade-offs in terms of decidability and expressive power. These logical frameworks are themselves always specified as natural deduction systems, which is a testament to the versatility of the natural deduction approach. <div class="mw-heading mw-heading2"><h2 id="Classical_and_modal_logics">Classical and modal logics</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=20" title="Edit section: Classical and modal logics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> For simplicity, the logics presented so far have been <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic</a>. <a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a> extends intuitionistic logic with an additional <a href="/wiki/Axiom" title="Axiom">axiom</a> or principle of <a href="/wiki/Excluded_middle" class="mw-redirect" title="Excluded middle">excluded middle</a>: <dl><dd>For any proposition p, the proposition p ∨ ¬p is true.</dd></dl> This statement is not obviously either an introduction or an elimination; indeed, it involves two distinct connectives. Gentzen's original treatment of excluded middle prescribed one of the following three (equivalent) formulations, which were already present in analogous forms in the systems of <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a> and <a href="/wiki/Arend_Heyting" title="Arend Heyting">Heyting</a>: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>────────────── XM1 A ∨ ¬A </pre> </td> <td width="5%"></td> <td> <pre>¬¬A ────────── XM2 A </pre> </td> <td width="5%"></td> <td> <pre>──────── u ¬A ⋮ p ────── XM3u, p A </pre> </td></tr></tbody></table> (XM3 is merely XM2 expressed in terms of E.) This treatment of excluded middle, in addition to being objectionable from a purist's standpoint, introduces additional complications in the definition of normal forms. A comparatively more satisfactory treatment of classical natural deduction in terms of introduction and elimination rules alone was first proposed by <a href="/w/index.php?title=Michel_Parigot&action=edit&redlink=1" class="new" title="Michel Parigot (page does not exist)">Parigot</a> in 1992 in the form of a classical <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> called <a href="/wiki/Lambda-mu_calculus" title="Lambda-mu calculus">λμ</a>. The key insight of his approach was to replace a truth-centric judgment A with a more classical notion, reminiscent of the <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculus</a>: in localised form, instead of Γ ⊢ A, he used Γ ⊢ Δ, with Δ a collection of propositions similar to Γ. Γ was treated as a conjunction, and Δ as a disjunction. This structure is essentially lifted directly from classical <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculi</a>, but the innovation in λμ was to give a computational meaning to classical natural deduction proofs in terms of a <a href="/wiki/Callcc" class="mw-redirect" title="Callcc">callcc</a> or a throw/catch mechanism seen in <a href="/wiki/LISP" class="mw-redirect" title="LISP">LISP</a> and its descendants. (See also: <a href="/wiki/First_class_control" class="mw-redirect" title="First class control">first class control</a>.) Another important extension was for <a href="/wiki/Modal_logic" title="Modal logic">modal</a> and other logics that need more than just the basic judgment of truth. These were first described, for the alethic modal logics <a href="/wiki/S4_(modal_logic)" class="mw-redirect" title="S4 (modal logic)">S4</a> and <a href="/wiki/S5_(modal_logic)" title="S5 (modal logic)">S5</a>, in a natural deduction style by <a href="/wiki/Dag_Prawitz" title="Dag Prawitz">Prawitz</a> in 1965,<a href="#cite_note-prawitz1965-5">[5]</a> and have since accumulated a large body of related work. To give a simple example, the modal logic S4 requires one new judgment, "A valid", that is categorical with respect to truth: <dl><dd>If "A" (is true) under no assumption that "B" (is true), then "A valid".</dd></dl> This categorical judgment is internalised as a unary connective ◻A (read "necessarily A") with the following introduction and elimination rules: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>A valid ──────── ◻I ◻ A </pre> </td> <td width="5%"></td> <td> <pre>◻ A ──────── ◻E A </pre> </td></tr></tbody></table> Note that the premise "A valid" has no defining rules; instead, the categorical definition of validity is used in its place. This mode becomes clearer in the localised form when the hypotheses are explicit. We write "Ω;Γ ⊢ A" where Γ contains the true hypotheses as before, and Ω contains valid hypotheses. On the right there is just a single judgment "A"; validity is not needed here since "Ω ⊢ A valid" is by definition the same as "Ω;⋅ ⊢ A". The introduction and elimination forms are then: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Ω;⋅ ⊢ π : A ──────────────────── ◻I Ω;⋅ ⊢ box π : ◻ A </pre> </td> <td width="5%"></td> <td> <pre>Ω;Γ ⊢ π : ◻ A ────────────────────── ◻E Ω;Γ ⊢ unbox π : A </pre> </td></tr></tbody></table> The modal hypotheses have their own version of the hypothesis rule and substitution theorem. <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>─────────────────────────────── valid-hyp Ω, u: (A valid) ; Γ ⊢ u : A </pre> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Modal_substitution_theorem">Modal substitution theorem</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=21" title="Edit section: Modal substitution theorem">edit</a>]</div> <dl><dt><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table></dt> <dd>If Ω;⋅ ⊢ π1 : A and Ω, u: (A valid) ; Γ ⊢ π2 : C, then Ω;Γ ⊢ [π1/u] π2 : C.</dd></dl> This framework of separating judgments into distinct collections of hypotheses, also known as multi-zoned or polyadic contexts, is very powerful and extensible; it has been applied for many different modal logics, and also for <a href="/wiki/Linear_logic" title="Linear logic">linear</a> and other <a href="/wiki/Substructural_logic" title="Substructural logic">substructural logics</a>, to give a few examples. However, relatively few systems of modal logic can be formalised directly in natural deduction. To give proof-theoretic characterisations of these systems, extensions such as labelling or systems of deep inference. The addition of labels to formulae permits much finer control of the conditions under which rules apply, allowing the more flexible techniques of <a href="/wiki/Analytic_tableau" class="mw-redirect" title="Analytic tableau">analytic tableaux</a> to be applied, as has been done in the case of <a href="/w/index.php?title=Labelled_deduction&action=edit&redlink=1" class="new" title="Labelled deduction (page does not exist)">labelled deduction</a>. Labels also allow the naming of worlds in Kripke semantics; <a href="#CITEREFSimpson1993">Simpson (1993)</a> presents an influential technique for converting frame conditions of modal logics in Kripke semantics into inference rules in a natural deduction formalisation of <a href="/wiki/Hybrid_logic" title="Hybrid logic">hybrid logic</a>. <a href="#CITEREFStouppa2004">Stouppa (2004)</a> surveys the application of many proof theories, such as Avron and Pottinger's <a href="/wiki/Hypersequent" title="Hypersequent">hypersequents</a> and Belnap's <a href="/wiki/Display_logic" class="mw-redirect" title="Display logic">display logic</a> to such modal logics as S5 and B. <div class="mw-heading mw-heading2"><h2 id="Comparison_with_sequent_calculus">Comparison with sequent calculus</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=22" title="Edit section: Comparison with sequent calculus">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></div> The sequent calculus is the chief alternative to natural deduction as a foundation of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>. In natural deduction the flow of information is bi-directional: elimination rules flow information downwards by deconstruction, and introduction rules flow information upwards by assembly. Thus, a natural deduction proof does not have a purely bottom-up or top-down reading, making it unsuitable for automation in proof search. To address this fact, <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen</a> in 1935 proposed his <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculus</a>, though he initially intended it as a technical device for clarifying the consistency of <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a>. <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene</a>, in his seminal 1952 book Introduction to Metamathematics, gave the first formulation of the sequent calculus in the modern style.<a href="#cite_note-28">[25]</a> In the sequent calculus all inference rules have a purely bottom-up reading. Inference rules can apply to elements on both sides of the <a href="/wiki/Turnstile_(symbol)" title="Turnstile (symbol)">turnstile</a>. (To differentiate from natural deduction, this article uses a double arrow ⇒ instead of the right tack ⊢ for sequents.) The introduction rules of natural deduction are viewed as right rules in the sequent calculus, and are structurally very similar. The elimination rules on the other hand turn into left rules in the sequent calculus. To give an example, consider disjunction; the right rules are familiar: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Γ ⇒ A ───────── ∨R1 Γ ⇒ A ∨ B </pre> </td> <td></td> <td> <pre>Γ ⇒ B ───────── ∨R2 Γ ⇒ A ∨ B </pre> </td></tr></tbody></table> On the left: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Γ, u:A ⇒ C Γ, v:B ⇒ C ─────────────────────────── ∨L Γ, w: (A ∨ B) ⇒ C </pre> </td></tr></tbody></table> Recall the ∨E rule of natural deduction in localised form: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>Γ ⊢ A ∨ B Γ, u:A ⊢ C Γ, v:B ⊢ C ─────────────────────────────────────── ∨E Γ ⊢ C </pre> </td></tr></tbody></table> The proposition A ∨ B, which is the succedent of a premise in ∨E, turns into a hypothesis of the conclusion in the left rule ∨L. Thus, left rules can be seen as a sort of inverted elimination rule. This observation can be illustrated as follows: <table align="center"> <tbody><tr> <th>natural deduction </th> <td> </td> <th>sequent calculus </th></tr> <tr> <td><pre> ────── hyp | | elim. rules | ↓ ────────────────────── ↑↓ meet ↑ | | intro. rules | conclusion</pre> </td> <td width="20%" align="center">⇒</td> <td><pre> ─────────────────────────── init ↑ ↑ | | | left rules | right rules | | conclusion</pre> </td></tr></tbody></table> In the sequent calculus, the left and right rules are performed in lock-step until one reaches the initial sequent, which corresponds to the meeting point of elimination and introduction rules in natural deduction. These initial rules are superficially similar to the hypothesis rule of natural deduction, but in the sequent calculus they describe a transposition or a handshake of a left and a right proposition: <table style="margin-left: 2em;"> <tbody><tr> <td> <pre>────────── init Γ, u:A ⇒ A </pre> </td></tr></tbody></table> The correspondence between the sequent calculus and natural deduction is a pair of soundness and completeness theorems, which are both provable by means of an inductive argument. <dl><dt>Soundness of ⇒ wrt. ⊢</dt> <dd>If Γ ⇒ A, then Γ ⊢ A.</dd> <dt>Completeness of ⇒ wrt. ⊢</dt> <dd>If Γ ⊢ A, then Γ ⇒ A.</dd></dl> It is clear by these theorems that the sequent calculus does not change the notion of truth, because the same collection of propositions remain true. Thus, one can use the same proof objects as before in sequent calculus derivations. As an example, consider the conjunctions. The right rule is virtually identical to the introduction rule <table style="margin-left: 2em;"> <tbody><tr> <th>sequent calculus </th> <td> </td> <th>natural deduction </th></tr> <tr> <td> <pre>Γ ⇒ π1 : A Γ ⇒ π2 : B ─────────────────────────── ∧R Γ ⇒ (π1, π2) : A ∧ B </pre> </td> <td width="20%"></td> <td> <pre>Γ ⊢ π1 : A Γ ⊢ π2 : B ───────────────────────── ∧I Γ ⊢ (π1, π2) : A ∧ B </pre> </td></tr></tbody></table> The left rule, however, performs some additional substitutions that are not performed in the corresponding elimination rules. <table style="margin-left: 2em;"> <tbody><tr> <th>sequent calculus </th> <td> </td> <th>natural deduction </th></tr> <tr> <td> <pre>Γ, u:A ⇒ π : C ──────────────────────────────── ∧L1 Γ, v: (A ∧ B) ⇒ [fst v/u] π : C </pre> </td> <td width="20%"></td> <td> <pre>Γ ⊢ π : A ∧ B ───────────── ∧E1 Γ ⊢ fst π : A </pre> </td></tr> <tr> <td> <pre>Γ, u:B ⇒ π : C ──────────────────────────────── ∧L2 Γ, v: (A ∧ B) ⇒ [snd v/u] π : C </pre> </td> <td width="20%"></td> <td> <pre>Γ ⊢ π : A ∧ B ───────────── ∧E2 Γ ⊢ snd π : B </pre> </td></tr></tbody></table> The kinds of proofs generated in the sequent calculus are therefore rather different from those of natural deduction. The sequent calculus produces proofs in what is known as the β-normal η-long form, which corresponds to a canonical representation of the normal form of the natural deduction proof. If one attempts to describe these proofs using natural deduction itself, one obtains what is called the intercalation calculus (first described by John Byrnes), which can be used to formally define the notion of a normal form for natural deduction. The substitution theorem of natural deduction takes the form of a <a href="/wiki/Structural_rule" title="Structural rule">structural rule</a> or structural theorem known as cut in the sequent calculus. <div class="mw-heading mw-heading3"><h3 id="Cut_(substitution)">Cut (substitution)</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=23" title="Edit section: Cut (substitution)">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Natural_deduction" title="Special:EditPage/Natural deduction">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (May 2024) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <dl><dd>If Γ ⇒ π1 : A and Γ, u:A ⇒ π2 : C, then Γ ⇒ [π1/u] π2 : C.</dd></dl> In most well behaved logics, cut is unnecessary as an inference rule, though it remains provable as a <a href="/wiki/Meta-theorem" class="mw-redirect" title="Meta-theorem">meta-theorem</a>; the superfluousness of the cut rule is usually presented as a computational process, known as cut elimination. This has an interesting application for natural deduction; usually it is extremely tedious to prove certain properties directly in natural deduction because of an unbounded number of cases. For example, consider showing that a given proposition is not provable in natural deduction. A simple inductive argument fails because of rules like ∨E or E which can introduce arbitrary propositions. However, we know that the sequent calculus is complete with respect to natural deduction, so it is enough to show this unprovability in the sequent calculus. Now, if cut is not available as an inference rule, then all sequent rules either introduce a connective on the right or the left, so the depth of a sequent derivation is fully bounded by the connectives in the final conclusion. Thus, showing unprovability is much easier, because there are only a finite number of cases to consider, and each case is composed entirely of sub-propositions of the conclusion. A simple instance of this is the global consistency theorem: "⋅ ⊢ ⊥" is not provable. In the sequent calculus version, this is manifestly true because there is no rule that can have "⋅ ⇒ ⊥" as a conclusion! Proof theorists often prefer to work on cut-free sequent calculus formulations because of such properties. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=24" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" /><a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a></li> <li><a href="/wiki/System_L" class="mw-redirect" title="System L">System L</a> (tabular natural deduction)</li> <li><a href="/wiki/Argument_map" title="Argument map">Argument map</a>, the general term for tree-like logic notation</li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=25" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><a href="#cite_ref-9">^</a> A particular advantage of Kleene's tabular natural deduction systems is that he proves the validity of the inference rules for both propositional calculus and predicate calculus. See <a href="#CITEREFKleene2002">Kleene 2002</a>, pp. 44–45, 118–119. </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> To simplify the statement of the rule, the word "denial" here is used in this way: the denial of a formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> that is not a negation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d48bd06914d3df71eb66552fc8f0fdb7af496a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.328ex; height:2.509ex;" alt="{\displaystyle -\varphi }">, whereas a negation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d48bd06914d3df71eb66552fc8f0fdb7af496a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.328ex; height:2.509ex;" alt="{\displaystyle -\varphi }">, has two denials, viz., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle --\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo>−</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle --\varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefa96151e4c2e8a8353adfa47aa23232326055d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.169ex; height:2.509ex;" alt="{\displaystyle --\varphi }">.<a href="#cite_note-AllenHand-18">[17]</a> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> See the article on <a href="/wiki/Lambda_calculus" title="Lambda calculus">lambda calculus</a> for more detail about the concept of substitution. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=26" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=27" title="Edit section: General references">edit</a>]</div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBarker-PlummerBarwiseEtchemendy2011" class="citation book cs1">Barker-Plummer, Dave; <a href="/wiki/Jon_Barwise" title="Jon Barwise">Barwise, Jon</a>; <a href="/wiki/John_Etchemendy" title="John Etchemendy">Etchemendy, John</a> (2011). Language Proof and Logic (2nd ed.). CSLI Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1575866321" title="Special:BookSources/978-1575866321"><bdi>978-1575866321</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Language+Proof+and+Logic&rft.edition=2nd&rft.pub=CSLI+Publications&rft.date=2011&rft.isbn=978-1575866321&rft.aulast=Barker-Plummer&rft.aufirst=Dave&rft.au=Barwise%2C+Jon&rft.au=Etchemendy%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallier2005" class="citation web cs1"><a href="/wiki/Jean_Gallier" title="Jean Gallier">Gallier, Jean</a> (2005). <a rel="nofollow" class="external text" href="ftp://ftp.cis.upenn.edu/pub/papers/gallier/conslog1.ps">"Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λ-Calculi"</a>. Retrieved 12 June 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Constructive+Logics.+Part+I%3A+A+Tutorial+on+Proof+Systems+and+Typed+%CE%BB-Calculi&rft.date=2005&rft.aulast=Gallier&rft.aufirst=Jean&rft_id=ftp%3A%2F%2Fftp.cis.upenn.edu%2Fpub%2Fpapers%2Fgallier%2Fconslog1.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGentzen1934" class="citation journal cs1"><a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen, Gerhard Karl Erich</a> (1934). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002375508">"Untersuchungen über das logische Schließen. I"</a>. Mathematische Zeitschrift. 39 (2): 176–210. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01201353">10.1007/BF01201353</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121546341">121546341</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Untersuchungen+%C3%BCber+das+logische+Schlie%C3%9Fen.+I&rft.volume=39&rft.issue=2&rft.pages=176-210&rft.date=1934&rft_id=info%3Adoi%2F10.1007%2FBF01201353&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121546341%23id-name%3DS2CID&rft.aulast=Gentzen&rft.aufirst=Gerhard+Karl+Erich&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fresolveppn%2F%3FPPN%3DGDZPPN002375508&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> (English translation Investigations into Logical Deduction in M. E. Szabo. The Collected Works of Gerhard Gentzen. North-Holland Publishing Company, 1969.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGentzen1935" class="citation journal cs1"><a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen, Gerhard Karl Erich</a> (1935). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002375605">"Untersuchungen über das logische Schließen. II"</a>. Mathematische Zeitschrift. 39 (3): 405–431. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01201363">10.1007/bf01201363</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186239837">186239837</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Untersuchungen+%C3%BCber+das+logische+Schlie%C3%9Fen.+II&rft.volume=39&rft.issue=3&rft.pages=405-431&rft.date=1935&rft_id=info%3Adoi%2F10.1007%2Fbf01201363&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186239837%23id-name%3DS2CID&rft.aulast=Gentzen&rft.aufirst=Gerhard+Karl+Erich&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fresolveppn%2F%3FPPN%3DGDZPPN002375605&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGirard1990" class="citation book cs1"><a href="/wiki/Jean-Yves_Girard" title="Jean-Yves Girard">Girard, Jean-Yves</a> (1990). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160704202340/http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html">Proofs and Types</a>. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, England. Archived from <a rel="nofollow" class="external text" href="http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html">the original</a> on 2016-07-04. Retrieved 2006-04-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+and+Types&rft.series=Cambridge+Tracts+in+Theoretical+Computer+Science&rft.pub=Cambridge+University+Press%2C+Cambridge%2C+England&rft.date=1990&rft.aulast=Girard&rft.aufirst=Jean-Yves&rft_id=http%3A%2F%2Fwww.cs.man.ac.uk%2F~pt%2Fstable%2FProofs%2BTypes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> Translated and with appendices by Paul Taylor and Yves Lafont.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaśkowski1934" class="citation book cs1"><a href="/wiki/Stanis%C5%82aw_Ja%C5%9Bkowski" title="Stanisław Jaśkowski">Jaśkowski, Stanisław</a> (1934). On the rules of suppositions in formal logic.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+rules+of+suppositions+in+formal+logic&rft.date=1934&rft.aulast=Ja%C5%9Bkowski&rft.aufirst=Stanis%C5%82aw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> Reprinted in Polish logic 1920–39, ed. Storrs McCall.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene1980" class="citation book cs1"><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a> (1980) [1952]. Introduction to metamathematics (Eleventh ed.). North-Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7204-2103-3" title="Special:BookSources/978-0-7204-2103-3"><bdi>978-0-7204-2103-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+metamathematics&rft.edition=Eleventh&rft.pub=North-Holland&rft.date=1980&rft.isbn=978-0-7204-2103-3&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene2009" class="citation book cs1"><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a> (2009) [1952]. Introduction to metamathematics. Ishi Press International. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-923891-57-2" title="Special:BookSources/978-0-923891-57-2"><bdi>978-0-923891-57-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+metamathematics&rft.pub=Ishi+Press+International&rft.date=2009&rft.isbn=978-0-923891-57-2&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleene2002" class="citation book cs1"><a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a> (2002) [1967]. Mathematical logic. Mineola, New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-42533-7" title="Special:BookSources/978-0-486-42533-7"><bdi>978-0-486-42533-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+logic&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications&rft.date=2002&rft.isbn=978-0-486-42533-7&rft.aulast=Kleene&rft.aufirst=Stephen+Cole&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemmon1965" class="citation book cs1"><a href="/wiki/John_Lemmon" title="John Lemmon">Lemmon, Edward John</a> (1965). Beginning logic. Thomas Nelson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-17-712040-4" title="Special:BookSources/978-0-17-712040-4"><bdi>978-0-17-712040-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beginning+logic&rft.pub=Thomas+Nelson&rft.date=1965&rft.isbn=978-0-17-712040-4&rft.aulast=Lemmon&rft.aufirst=Edward+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin-Löf1996" class="citation journal cs1"><a href="/wiki/Per_Martin-L%C3%B6f" title="Per Martin-Löf">Martin-Löf, Per</a> (1996). <a rel="nofollow" class="external text" href="http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf">"On the meanings of the logical constants and the justifications of the logical laws"</a> (PDF). <a href="/w/index.php?title=Nordic_Journal_of_Philosophical_Logic&action=edit&redlink=1" class="new" title="Nordic Journal of Philosophical Logic (page does not exist)">Nordic Journal of Philosophical Logic</a>. 1 (1): 11–60.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nordic+Journal+of+Philosophical+Logic&rft.atitle=On+the+meanings+of+the+logical+constants+and+the+justifications+of+the+logical+laws&rft.volume=1&rft.issue=1&rft.pages=11-60&rft.date=1996&rft.aulast=Martin-L%C3%B6f&rft.aufirst=Per&rft_id=http%3A%2F%2Fdocenti.lett.unisi.it%2Ffiles%2F4%2F1%2F1%2F6%2Fmartinlof4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> Lecture notes to a short course at Università degli Studi di Siena, April 1983.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPfenningDavies2001" class="citation journal cs1">Pfenning, Frank; Davies, Rowan (2001). <a rel="nofollow" class="external text" href="http://www-2.cs.cmu.edu/~fp/papers/mscs00.pdf">"A judgmental reconstruction of modal logic"</a> (PDF). Mathematical Structures in Computer Science. 11 (4): 511–540. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.1611">10.1.1.43.1611</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0960129501003322">10.1017/S0960129501003322</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16467268">16467268</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Structures+in+Computer+Science&rft.atitle=A+judgmental+reconstruction+of+modal+logic&rft.volume=11&rft.issue=4&rft.pages=511-540&rft.date=2001&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.43.1611%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16467268%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0960129501003322&rft.aulast=Pfenning&rft.aufirst=Frank&rft.au=Davies%2C+Rowan&rft_id=http%3A%2F%2Fwww-2.cs.cmu.edu%2F~fp%2Fpapers%2Fmscs00.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrawitz1965" class="citation book cs1"><a href="/wiki/Dag_Prawitz" title="Dag Prawitz">Prawitz, Dag</a> (1965). <a rel="nofollow" class="external text" href="https://archive.org/details/naturaldeduction0000praw">Natural deduction: A proof-theoretical study</a>. Acta Universitatis Stockholmiensis, Stockholm studies in philosophy 3. Stockholm, Göteborg, Uppsala: Almqvist & Wicksell.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Natural+deduction%3A+A+proof-theoretical+study&rft.place=Stockholm%2C+G%C3%B6teborg%2C+Uppsala&rft.series=Acta+Universitatis+Stockholmiensis%2C+Stockholm+studies+in+philosophy+3&rft.pub=Almqvist+%26+Wicksell&rft.date=1965&rft.aulast=Prawitz&rft.aufirst=Dag&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnaturaldeduction0000praw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrawitz2006" class="citation book cs1"><a href="/wiki/Dag_Prawitz" title="Dag Prawitz">Prawitz, Dag</a> (2006) [1965]. Natural deduction: A proof-theoretical study. Mineola, New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-44655-4" title="Special:BookSources/978-0-486-44655-4"><bdi>978-0-486-44655-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Natural+deduction%3A+A+proof-theoretical+study&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications&rft.date=2006&rft.isbn=978-0-486-44655-4&rft.aulast=Prawitz&rft.aufirst=Dag&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuine1981" class="citation book cs1"><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine, Willard Van Orman</a> (1981) [1940]. Mathematical logic (Revised ed.). Cambridge, Massachusetts: Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-55451-1" title="Special:BookSources/978-0-674-55451-1"><bdi>978-0-674-55451-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+logic&rft.place=Cambridge%2C+Massachusetts&rft.edition=Revised&rft.pub=Harvard+University+Press&rft.date=1981&rft.isbn=978-0-674-55451-1&rft.aulast=Quine&rft.aufirst=Willard+Van+Orman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuine1982" class="citation book cs1"><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine, Willard Van Orman</a> (1982) [1950]. Methods of logic (Fourth ed.). Cambridge, Massachusetts: Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-57176-1" title="Special:BookSources/978-0-674-57176-1"><bdi>978-0-674-57176-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+logic&rft.place=Cambridge%2C+Massachusetts&rft.edition=Fourth&rft.pub=Harvard+University+Press&rft.date=1982&rft.isbn=978-0-674-57176-1&rft.aulast=Quine&rft.aufirst=Willard+Van+Orman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimpson1993" class="citation book cs1">Simpson, Alex (1993). <a rel="nofollow" class="external text" href="https://www.era.lib.ed.ac.uk/bitstream/handle/1842/407/ECS-LFCS-94-308.PDF?sequence=2&isAllowed=y">The proof theory and semantics of intuitionistic modal logic</a> (PDF). University of Edinburgh.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+proof+theory+and+semantics+of+intuitionistic+modal+logic&rft.pub=University+of+Edinburgh&rft.date=1993&rft.aulast=Simpson&rft.aufirst=Alex&rft_id=https%3A%2F%2Fwww.era.lib.ed.ac.uk%2Fbitstream%2Fhandle%2F1842%2F407%2FECS-LFCS-94-308.PDF%3Fsequence%3D2%26isAllowed%3Dy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> PhD thesis.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoll1979" class="citation book cs1">Stoll, Robert Roth (1979) [1963]. Set Theory and Logic. Mineola, New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63829-4" title="Special:BookSources/978-0-486-63829-4"><bdi>978-0-486-63829-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Logic&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications&rft.date=1979&rft.isbn=978-0-486-63829-4&rft.aulast=Stoll&rft.aufirst=Robert+Roth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStouppa2004" class="citation book cs1">Stouppa, Phiniki (2004). The Design of Modal Proof Theories: The Case of S5. University of Dresden. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.140.1858">10.1.1.140.1858</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Design+of+Modal+Proof+Theories%3A+The+Case+of+S5&rft.pub=University+of+Dresden&rft.date=2004&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.140.1858%23id-name%3DCiteSeerX&rft.aulast=Stouppa&rft.aufirst=Phiniki&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> MSc thesis.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuppes1999" class="citation book cs1"><a href="/wiki/Patrick_Suppes" title="Patrick Suppes">Suppes, Patrick Colonel</a> (1999) [1957]. Introduction to logic. Mineola, New York: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-40687-9" title="Special:BookSources/978-0-486-40687-9"><bdi>978-0-486-40687-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+logic&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications&rft.date=1999&rft.isbn=978-0-486-40687-9&rft.aulast=Suppes&rft.aufirst=Patrick+Colonel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Dalen2013" class="citation book cs1"><a href="/wiki/Dirk_van_Dalen" title="Dirk van Dalen">Van Dalen, Dirk</a> (2013) [1980]. Logic and Structure. Universitext (5 ed.). 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Retrieved 2024-05-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Natural+Deduction+%7C+Internet+Encyclopedia+of+Philosophy&rft_id=https%3A%2F%2Fiep.utm.edu%2Fnatural-deduction%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFJaśkowski1934">Jaśkowski 1934</a>. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFGentzen1934">Gentzen 1934</a>, <a href="#CITEREFGentzen1935">Gentzen 1935</a>. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFGentzen1934">Gentzen 1934</a>, p. 176. </li> <li id="cite_note-prawitz1965-5">^ <a href="#cite_ref-prawitz1965_5-0">a</a> <a href="#cite_ref-prawitz1965_5-1">b</a> <a href="#CITEREFPrawitz1965">Prawitz 1965</a>, <a href="#CITEREFPrawitz2006">Prawitz 2006</a>. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFMartin-Löf1996">Martin-Löf 1996</a>. </li> <li id="cite_note-SEP_NaturalDeduction-7">^ <a href="#cite_ref-SEP_NaturalDeduction_7-0">a</a> <a href="#cite_ref-SEP_NaturalDeduction_7-1">b</a> <a href="#cite_ref-SEP_NaturalDeduction_7-2">c</a> <a href="#cite_ref-SEP_NaturalDeduction_7-3">d</a> <a href="#cite_ref-SEP_NaturalDeduction_7-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPelletierHazen2024" class="citation cs2">Pelletier, Francis Jeffry; Hazen, Allen (2024), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/spr2024/entries/natural-deduction/">"Natural Deduction Systems in Logic"</a>, in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-03-22</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Natural+Deduction+Systems+in+Logic&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Spring+2024&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2024&rft.aulast=Pelletier&rft.aufirst=Francis+Jeffry&rft.au=Hazen%2C+Allen&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fspr2024%2Fentries%2Fnatural-deduction%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFQuine1981">Quine (1981)</a>. See particularly pages 91–93 for Quine's line-number notation for antecedent dependencies. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlato2013" class="citation book cs1">Plato, Jan von (2013). Elements of logical reasoning (1. publ ed.). 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Cham: Springer. p. 38. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-08454-7" title="Special:BookSources/978-3-030-08454-7"><bdi>978-3-030-08454-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+formal+philosophy&rft.place=Cham&rft.series=Springer+undergraduate+texts+in+philosophy&rft.pages=38&rft.pub=Springer&rft.date=2018&rft.isbn=978-3-030-08454-7&rft.aulast=Hansson&rft.aufirst=Sven+Ove&rft.au=Hendricks%2C+Vincent+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> </li> <li id="cite_note-:0-21">^ <a href="#cite_ref-:0_21-0">a</a> <a href="#cite_ref-:0_21-1">b</a> <a href="#cite_ref-:0_21-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAyala-Rincónde_Moura2017" class="citation book cs1">Ayala-Rincón, Mauricio; de Moura, Flávio L.C. (2017). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-319-51653-0">Applied Logic for Computer Scientists</a>. Undergraduate Topics in Computer Science. Springer. pp. 2, 20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-51653-0">10.1007/978-3-319-51653-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-51651-6" title="Special:BookSources/978-3-319-51651-6"><bdi>978-3-319-51651-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Logic+for+Computer+Scientists&rft.series=Undergraduate+Topics+in+Computer+Science&rft.pages=2%2C+20&rft.pub=Springer&rft.date=2017&rft_id=info%3Adoi%2F10.1007%2F978-3-319-51653-0&rft.isbn=978-3-319-51651-6&rft.aulast=Ayala-Rinc%C3%B3n&rft.aufirst=Mauricio&rft.au=de+Moura%2C+Fl%C3%A1vio+L.C.&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-319-51653-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> </li> <li id="cite_note-:1-22">^ <a href="#cite_ref-:1_22-0">a</a> <a href="#cite_ref-:1_22-1">b</a> <a href="#cite_ref-:1_22-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlato2013" class="citation book cs1">Plato, Jan von (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QrdEAgAAQBAJ">Elements of Logical Reasoning</a>. Cambridge University Press. pp. 12–13. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-03659-8" title="Special:BookSources/978-1-107-03659-8"><bdi>978-1-107-03659-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Logical+Reasoning&rft.pages=12-13&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1-107-03659-8&rft.aulast=Plato&rft.aufirst=Jan+von&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQrdEAgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> </li> <li id="cite_note-LemmonLogic-23">^ <a href="#cite_ref-LemmonLogic_23-0">a</a> <a href="#cite_ref-LemmonLogic_23-1">b</a> <a href="#cite_ref-LemmonLogic_23-2">c</a> <a href="#cite_ref-LemmonLogic_23-3">d</a> <a href="#cite_ref-LemmonLogic_23-4">e</a> <a href="#cite_ref-LemmonLogic_23-5">f</a> <a href="#cite_ref-LemmonLogic_23-6">g</a> <a href="#cite_ref-LemmonLogic_23-7">h</a> <a href="#cite_ref-LemmonLogic_23-8">i</a> <a href="#cite_ref-LemmonLogic_23-9">j</a> <a href="#cite_ref-LemmonLogic_23-10">k</a> <a href="#cite_ref-LemmonLogic_23-11">l</a> <a href="#cite_ref-LemmonLogic_23-12">m</a> <a href="#cite_ref-LemmonLogic_23-13">n</a> <a href="#cite_ref-LemmonLogic_23-14">o</a> <a href="#cite_ref-LemmonLogic_23-15">p</a> <a href="#cite_ref-LemmonLogic_23-16">q</a> <a href="#cite_ref-LemmonLogic_23-17">r</a> <a href="#cite_ref-LemmonLogic_23-18">s</a> <a href="#cite_ref-LemmonLogic_23-19">t</a> <a href="#cite_ref-LemmonLogic_23-20">u</a> <a href="#cite_ref-LemmonLogic_23-21">v</a> <a href="#cite_ref-LemmonLogic_23-22">w</a> <a href="#cite_ref-LemmonLogic_23-23">x</a> <a href="#cite_ref-LemmonLogic_23-24">y</a> <a href="#cite_ref-LemmonLogic_23-25">z</a> <a href="#cite_ref-LemmonLogic_23-26">aa</a> <a href="#cite_ref-LemmonLogic_23-27">ab</a> <a href="#cite_ref-LemmonLogic_23-28">ac</a> <a href="#cite_ref-LemmonLogic_23-29">ad</a> <a href="#cite_ref-LemmonLogic_23-30">ae</a> <a href="#cite_ref-LemmonLogic_23-31">af</a> <a href="#cite_ref-LemmonLogic_23-32">ag</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemmon1998" class="citation book cs1">Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. passim, especially 39-40. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-38090-7" title="Special:BookSources/978-0-412-38090-7"><bdi>978-0-412-38090-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beginning+logic&rft.place=Boca+Raton%2C+FL&rft.pages=passim%2C+especially+39-40&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=1998&rft.isbn=978-0-412-38090-7&rft.aulast=Lemmon&rft.aufirst=Edward+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> </li> <li id="cite_note-TWArthurLogic-24">^ <a href="#cite_ref-TWArthurLogic_24-0">a</a> <a href="#cite_ref-TWArthurLogic_24-1">b</a> <a href="#cite_ref-TWArthurLogic_24-2">c</a> <a href="#cite_ref-TWArthurLogic_24-3">d</a> <a href="#cite_ref-TWArthurLogic_24-4">e</a> <a href="#cite_ref-TWArthurLogic_24-5">f</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArthur2017" class="citation book cs1">Arthur, Richard T. W. (2017). <a rel="nofollow" class="external text" href="https://www.worldcat.org/title/962129086">An introduction to logic: using natural deduction, real arguments, a little history, and some humour</a> (2nd ed.). Peterborough, Ontario: Broadview Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-55481-332-2" title="Special:BookSources/978-1-55481-332-2"><bdi>978-1-55481-332-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/962129086">962129086</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+logic%3A+using+natural+deduction%2C+real+arguments%2C+a+little+history%2C+and+some+humour&rft.place=Peterborough%2C+Ontario&rft.edition=2nd&rft.pub=Broadview+Press&rft.date=2017&rft_id=info%3Aoclcnum%2F962129086&rft.isbn=978-1-55481-332-2&rft.aulast=Arthur&rft.aufirst=Richard+T.+W.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Ftitle%2F962129086&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> See also his book <a href="#CITEREFPrawitz1965">Prawitz 1965</a>, <a href="#CITEREFPrawitz2006">Prawitz 2006</a>. </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="#CITEREFKleene2009">Kleene 2009</a>, pp. 440–516. See also <a href="#CITEREFKleene1980">Kleene 1980</a>. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Natural_deduction&action=edit&section=29" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaboreo2004" class="citation web cs1">Laboreo, Daniel Clemente (August 2004). <a rel="nofollow" class="external text" href="https://www.danielclemente.com/logica/dn.en.pdf">"Introduction to natural deduction"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Introduction+to+natural+deduction&rft.date=2004-08&rft.aulast=Laboreo&rft.aufirst=Daniel+Clemente&rft_id=https%3A%2F%2Fwww.danielclemente.com%2Flogica%2Fdn.en.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.winterdrache.de/freeware/domino/">"Domino On Acid"</a>. Retrieved 10 December 2023. <q>Natural deduction visualized as a game of dominoes</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Domino+On+Acid&rft_id=http%3A%2F%2Fwww.winterdrache.de%2Ffreeware%2Fdomino%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPelletier" class="citation web cs1">Pelletier, Francis Jeffry. <a rel="nofollow" class="external text" href="https://www.sfu.ca/~jeffpell/papers/pelletierNDtexts.pdf">"A History of Natural Deduction and Elementary Logic Textbooks"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+History+of+Natural+Deduction+and+Elementary+Logic+Textbooks&rft.aulast=Pelletier&rft.aufirst=Francis+Jeffry&rft_id=https%3A%2F%2Fwww.sfu.ca%2F~jeffpell%2Fpapers%2FpelletierNDtexts.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/natural-deduction/">"Natural Deduction Systems in Logic"</a> entry by Pelletier, Francis Jeffry; Hazen, Allen in the <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>, 29 October 2021</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevy" class="citation web cs1">Levy, Michel. <a rel="nofollow" class="external text" href="http://teachinglogic.univ-grenoble-alpes.fr/DN1/">"A Propositional Prover"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Propositional+Prover&rft.aulast=Levy&rft.aufirst=Michel&rft_id=http%3A%2F%2Fteachinglogic.univ-grenoble-alpes.fr%2FDN1%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+deduction" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul 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aria-labelledby="Mathematical_logic" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini 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href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)  and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems">list</a>), <a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories">list</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic systems</a> (<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a class="mw-selflink selflink">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure 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title="Edit this template">e</abbr></a></li></ul></div><div id="Major_topics_in_Foundations_of_Mathematics" style="font-size:114%;margin:0 4em">Major topics in <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of Mathematics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="background:#fdf;;width:1%"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a></li> <li><a href="/wiki/Mathematical_induction" title="Mathematical induction">Mathematical induction</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Formal system</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic system</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert 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Natural deduction - Wikipedia