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Näistä ominaisuuksista keskeisimpiä ovat <a href="/wiki/Totuus" title="Totuus">totuus</a> ja lauseiden väliset <a href="/wiki/P%C3%A4%C3%A4ttely" title="Päättely">päättelysuhteet</a>. <sup id="cite_ref-a_1-0" class="reference"><a href="#cite_note-a-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Propositiosymboleina käytetään formaalikielessä yleensä merkkejä <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f1b08d7d69712872e051c2b33fdfa9f5d42319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{2}}"></span> jne. Eri propositiosymbolien voidaan tulkita edustavan toisistaan riippumattomia asiantiloja. Loogisille konnektiiveille käytetään usein merkkejä kuten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg ,\land ,\lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo>,</mo> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg ,\land ,\lor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd8b055b883e894b17d6129fb6e7851c88ef368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.719ex; height:2.343ex;" alt="{\displaystyle \neg ,\land ,\lor }"></span>. Nämä vastaavat karkeasti ottaen luonnollisen kielen <a href="/w/index.php?title=Lausekonnektiivi&action=edit&redlink=1" class="new" title="Lausekonnektiivi (sivua ei ole)">lausekonnektiiveja</a>, esimerkiksi "ei", "ja" ja "tai". </p><p>Propositiologiikkaa kehittivät ensimmäisenä <a href="/wiki/Stoalaisuus" title="Stoalaisuus">stoalaiset</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="fi" dir="ltr"><h2 id="mw-toc-heading">Sisällys</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Propositiologiikan_syntaksi"><span class="tocnumber">1</span> <span class="toctext">Propositiologiikan syntaksi</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Päättely_propositiologiikassa"><span class="tocnumber">2</span> <span class="toctext">Päättely propositiologiikassa</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Totuusjakauma_ja_propositiolauseen_totuusarvo"><span class="tocnumber">3</span> <span class="toctext">Totuusjakauma ja propositiolauseen totuusarvo</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Tautologia_ja_looginen_seuraus"><span class="tocnumber">4</span> <span class="toctext">Tautologia ja looginen seuraus</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Propositiologiikan_täydellisyys-_ja_eheyslause"><span class="tocnumber">5</span> <span class="toctext">Propositiologiikan täydellisyys- ja eheyslause</span></a> <ul> <li class="toclevel-2 tocsection-6"><a href="#Todistus"><span class="tocnumber">5.1</span> <span class="toctext">Todistus</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-7"><a href="#Konnektiivit_ja_loogiset_portit"><span class="tocnumber">6</span> <span class="toctext">Konnektiivit ja loogiset portit</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Katso_myös"><span class="tocnumber">7</span> <span class="toctext">Katso myös</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Lähteet"><span class="tocnumber">8</span> <span class="toctext">Lähteet</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Kirjallisuutta"><span class="tocnumber">9</span> <span class="toctext">Kirjallisuutta</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Aiheesta_muualla"><span class="tocnumber">10</span> <span class="toctext">Aiheesta muualla</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Propositiologiikan_syntaksi">Propositiologiikan syntaksi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=1" title="Muokkaa osiota Propositiologiikan syntaksi" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=1" title="Muokkaa osion lähdekoodia: Propositiologiikan syntaksi"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Propositiologiikassa <a href="/wiki/Atomilause" title="Atomilause">atomilauseita</a> merkitään propositiosymboleilla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0},p_{1},p_{2},\ldots \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0},p_{1},p_{2},\ldots \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d7ef4b4ce13683f77048f5850b9809f36e3c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:12.973ex; height:2.009ex;" alt="{\displaystyle p_{0},p_{1},p_{2},\ldots \,\!}"></span>. Lausemuuttujina käytetään suuria kirjaimia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,\ldots \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mo>…</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,\ldots \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35999135699802870611dece062602b7e90242d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:11.486ex; height:2.509ex;" alt="{\displaystyle A,B,C,\ldots \,\!}"></span>. Lausemuuttujat kuvaavat mielivaltaisia tai toistaiseksi määrittelemättömiä lauseita. </p><p>Propositiosymboleja voidaan määritellä seuraavaan tapaan: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}=_{df}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}=_{df}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a8f0cbf7c28bfc92e3fd22dc916a2d10586373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; margin-right: -0.387ex; width:7.15ex; height:2.343ex;" alt="{\displaystyle p_{0}=_{df}\,\!}"></span>"<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+4=6\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mn>6</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+4=6\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb05e496726c35a279a5a5f071a91237f8e9912a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:9.813ex; height:2.343ex;" alt="{\displaystyle 2+4=6\,\!}"></span>".</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}=_{df}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}=_{df}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb59fa66b24c5566395832ba32ed1becdec3c8a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; margin-right: -0.387ex; width:7.15ex; height:2.343ex;" alt="{\displaystyle p_{1}=_{df}\,\!}"></span>"Esko ui".</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}=_{df}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}=_{df}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a569dcc3897225fd69cc7e73d0f3630a0457d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; margin-right: -0.387ex; width:7.15ex; height:2.343ex;" alt="{\displaystyle p_{2}=_{df}\,\!}"></span>"Esko kastuu".</li></ul> <p>Propositiosymboleista voidaan rakentaa monimutkaisempia ilmaisuja loogisten operaattoreiden eli <i>konnektiivien</i> avulla. Joskus osa konnektiiveista voidaan korvata määrittelemällä ne muutaman valitun konnektiivin avulla. Yleensä konnektiiveja esitellään seuraavat viisi, mutta on olemassa myös pari muuta konnektiivia: <a href="/wiki/Shefferin_viiva" title="Shefferin viiva">Shefferin viiva</a> ja <a href="/wiki/Peircen_nuoli" title="Peircen nuoli">Peircen nuoli</a>. </p> <table border="1" class="prettytable"> <tbody><tr> <th>Merkitys </th> <th>Merkintä </th> <th>Lukutapa </th></tr> <tr> <td><a href="/wiki/Negaatio" class="mw-redirect" title="Negaatio">negaatio</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {}A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e69dd411f372400c23f6045a45d75f00cc4e18" 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}"></span> </td> <td>"ei A" (<a href="/wiki/Englannin_kieli" title="Englannin kieli">engl.</a> <span lang="en"><i>not A</i></span>) </td></tr> <tr> <td><a href="/wiki/Konjunktio_(logiikka)" title="Konjunktio (logiikka)">konjunktio</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\wedge {}B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\wedge {}B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d538ecf9803a65164c9a7c269107274112313d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\wedge {}B)}"></span> </td> <td>"A ja B" (<a href="/wiki/Englannin_kieli" title="Englannin kieli">engl.</a> <span lang="en"><i>A and B</i></span>) </td></tr> <tr> <td><a href="/wiki/Disjunktio" title="Disjunktio">disjunktio</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\vee {}B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\vee {}B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/543b391a130015f9d8f4acce6f266f30ac03a6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\vee {}B)}"></span> </td> <td>"A tai B" (<a href="/wiki/Englannin_kieli" title="Englannin kieli">engl.</a> <span lang="en"><i>A or B</i></span>) </td></tr> <tr> <td><a href="/wiki/Implikaatio" title="Implikaatio">implikaatio</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to {}B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to {}B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5cb80b61b4ae33ef2f31e7b685812bee48a14f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.93ex; height:2.843ex;" alt="{\displaystyle (A\to {}B)}"></span> </td> <td>"jos A niin B" </td></tr> <tr> <td>ekvivalenssi </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow {}B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22f6a75f362d6dfa992c73abccc9746b94ed58f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.93ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow {}B)}"></span> </td> <td>"A <a href="/wiki/Jos_ja_vain_jos" title="Jos ja vain jos">jos ja vain jos</a> B" </td></tr></tbody></table> <p>Seuraavassa <a href="/wiki/Rekursio" title="Rekursio">rekursiivisessa</a> määritelmässä määritellään kaikki propositiolauseet. </p><p><b>Määritelmä 1</b> Propositiolause<br /> </p> <ol><li>Propositiosymbolit ovat propositiolauseita.</li> <li>Jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> on propositiolause, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {}A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29afb820f3af66bf3b0873aa775f212345c56bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.681ex; height:2.176ex;" alt="{\displaystyle \neg {}A\,\!}"></span> on propositiolause.</li> <li>Jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2670cb2bfb471dca31dd3da10251997439eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}"></span> ovat propositiolauseita, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\wedge {}B)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\wedge {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befb1b997723b72a9ff7c70d97fa522e7470eb38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.286ex; height:2.843ex;" alt="{\displaystyle (A\wedge {}B)\,\!}"></span> on propositiolause.</li> <li>Jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2670cb2bfb471dca31dd3da10251997439eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}"></span> ovat propositiolauseita, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\vee {}B)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\vee {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b33417652f2427b63900f2b935b3aaccb364e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.286ex; height:2.843ex;" alt="{\displaystyle (A\vee {}B)\,\!}"></span> on propositiolause.</li> <li>Jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2670cb2bfb471dca31dd3da10251997439eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}"></span> ovat propositiolauseita, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to {}B)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be5864ccef45fcc6c1662afc734d3dd80fcc5df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle (A\to {}B)\,\!}"></span> on propositiolause.</li> <li>Jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2670cb2bfb471dca31dd3da10251997439eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}"></span> ovat propositiolauseita, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0bcc2ca0a595a8209d974d679bc1707446425b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow {}B)\,\!}"></span> on propositiolause.</li></ol> <p><b>Esimerkki 2</b> Propositiolauseita<br /> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {}p_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {}p_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fa9b9a0a9d9336c284307b0537da624c5962b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:4.161ex; height:2.009ex;" alt="{\displaystyle \neg {}p_{0}\,\!}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{0}\to {}p_{1})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p_{0}\to {}p_{1})\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1826c9ca4f92af0432701970d50958722425f527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:10.258ex; height:2.843ex;" alt="{\displaystyle (p_{0}\to {}p_{1})\,\!}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1}))\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1}))\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a83e10b6ef5328b27937136b8a22e6b15d15df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:25.039ex; height:2.843ex;" alt="{\displaystyle \neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1}))\,\!}"></span></li></ul> <p>Määritelmän 1 perusteella propositiolauseet voidaan purkaa osatekijöikseen yksiselitteisellä tavalla (katso <a href="/w/index.php?title=Propositiologiikan_rakennepuu&action=edit&redlink=1" class="new" title="Propositiologiikan rakennepuu (sivua ei ole)">propositiologiikan rakennepuu</a>). Tällöin edetään vastakkaiseen suuntaan. Jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> on propositiolause, niin se on välttämättä muotoa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9244e1fdd5dbac42cd1b427614f3986c87fbe3b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:2.446ex; height:2.009ex;" alt="{\displaystyle p_{i}\,\!}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {}B\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {}B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048ec2a14d949e80d59c48b1520c36d3c4d93bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.701ex; height:2.176ex;" alt="{\displaystyle \neg {}B\,\!}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\wedge {}C)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>C</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\wedge {}C)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cccba7cc65b3900df9af53227b768642c3790401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.309ex; height:2.843ex;" alt="{\displaystyle (B\wedge {}C)\,\!}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\vee {}C)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>C</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\vee {}C)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ce55070f25b93c3d633d982b9b6b3bee42373d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.309ex; height:2.843ex;" alt="{\displaystyle (B\vee {}C)\,\!}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\to {}C)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>C</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\to {}C)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079bfe9f4ffd81477fb64aadcd0a673f68f09a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.341ex; height:2.843ex;" alt="{\displaystyle (B\to {}C)\,\!}"></span> tai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B\leftrightarrow {}C)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">↔</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>C</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B\leftrightarrow {}C)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a5f634801b47e074ae1e6cd43a023ec08bfa6b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.341ex; height:2.843ex;" alt="{\displaystyle (B\leftrightarrow {}C)\,\!}"></span>. Muussa tapauksessa sitä ei ole muodostettu määritelmän mukaisesti. Tämä mahdollistaa <a href="/wiki/Matemaattinen_induktio" title="Matemaattinen induktio">matemaattisen induktion</a> soveltamisen logiikkaa koskevissa todistuksissa. </p> <div class="mw-heading mw-heading2"><h2 id="Päättely_propositiologiikassa"><span id="P.C3.A4.C3.A4ttely_propositiologiikassa"></span>Päättely propositiologiikassa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=2" title="Muokkaa osiota Päättely propositiologiikassa" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=2" title="Muokkaa osion lähdekoodia: Päättely propositiologiikassa"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="" class="metadata plainlinks ambox ambox-content"> <tbody><tr> <td class="ambox-image"><div style="width:52px;text-align:center"> <span typeof="mw:File"><a href="/wiki/Tiedosto:Noto_Emoji_Oreo_1f3d7.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Noto_Emoji_Oreo_1f3d7.svg/40px-Noto_Emoji_Oreo_1f3d7.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Noto_Emoji_Oreo_1f3d7.svg/60px-Noto_Emoji_Oreo_1f3d7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Noto_Emoji_Oreo_1f3d7.svg/80px-Noto_Emoji_Oreo_1f3d7.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div> </td> <td class="ambox-text"><b>Tämä artikkeli tai osio on keskeneräinen.</b><br /> <small>Voit auttaa Wikipediaa <span class="plainlinks"><a class="external text" href="https://fi.wikipedia.org/w/index.php?title=Propositiologiikka&action=edit">laajentamalla sivua</a></span>. Lisää tietoa saattaa olla <a href="/wiki/Keskustelu:Propositiologiikka" title="Keskustelu:Propositiologiikka">keskustelusivulla</a>. <br /></small> </td></tr></tbody></table> <p>Propositiologiikassa (kuten formaalissa logiikassa muutenkin) voidaan erottaa kaksi päätapaa tutkia päättelyä: <i>Päättelysäännöt</i> (syntaktinen näkökulma) ja <i>totuusarvon laskeminen</i> (semanttinen näkökulma). Päättelysäännöt sinänsä eivät takaa sitä, että päättely säilyttää totuuden. Tämän takaa vasta sellaisten päättelysääntöjen käyttäminen, joiden <a href="/wiki/Eheys" class="mw-disambig" title="Eheys">eheys</a> (katso eheyslause alempana) on todistettu. Päättelysääntöjen sinänsä soveltaminen on ainoastaan uusien lauseiden johtamista jo oletetuista. Sen sijaan, jos tiettyjen päättelysääntöjen eheys on todistettu, voidaan päättelyn pätevyys todistaa jo pelkästään näihin päättelysääntöihin nojautuen. Eheydestä käytetään usein myös nimityksiä <i>validius</i> ja <i>korrektisuus</i>. </p><p>Aksioomat ... </p><p>Päättelysäännöt ... </p><p>Pari esimerkkiä ... </p> <div class="mw-heading mw-heading2"><h2 id="Totuusjakauma_ja_propositiolauseen_totuusarvo">Totuusjakauma ja propositiolauseen totuusarvo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=3" title="Muokkaa osiota Totuusjakauma ja propositiolauseen totuusarvo" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=3" title="Muokkaa osion lähdekoodia: Totuusjakauma ja propositiolauseen totuusarvo"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Määritelmä 3</b> Totuusjakauma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f34217ded608407636238760709b92635f19dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,\!}"></span> on <a href="/wiki/Funktio" title="Funktio">kuvaus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:\mathbb {N} \to \{0,1\}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v:\mathbb {N} \to \{0,1\}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87ad52bad73fa2aac5480bd1daab4f0bb737770" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:14.428ex; height:2.843ex;" alt="{\displaystyle v:\mathbb {N} \to \{0,1\}\,\!}"></span>, jossa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=_{df}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=_{df}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b48a531b7d4b75f91787e1dfdeea7bc48775e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:5.999ex; height:2.843ex;" alt="{\displaystyle 1=_{df}\,\!}"></span>"tosi" ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=_{df}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=_{df}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad3c63b925db8acbf4bb28c01bc0f5a457fd93e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:5.999ex; height:2.843ex;" alt="{\displaystyle 0=_{df}\,\!}"></span>"epätosi". </p><p><b>Määritelmä 4</b> Propositiosymbolien totuusarvo<br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(p_{i})=_{df}v(i)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> <mi>v</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(p_{i})=_{df}v(i)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a672a9d60024c1ae7bbde8d9dd2bacdda7bfc18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.387ex; width:14.127ex; height:3.009ex;" alt="{\displaystyle v(p_{i})=_{df}v(i)\,\!}"></span>. </p><p>Symboli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f34217ded608407636238760709b92635f19dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,\!}"></span> esiintyy määritelmässä 4 kahdessa merkityksessä: totuusjakauman symbolina yhtäläisyysmerkin vasemmalla puolella ja propositiolauseen totuusarvon määrittäjänä oikealla puolella. </p> <table border="0"> <tbody><tr> <td align="left" colspan="2"><b>Esimerkki 5</b> </td></tr> <tr> <td align="left" colspan="2">Olkoon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f34217ded608407636238760709b92635f19dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,\!}"></span> totuusjakauma siten, että </td></tr> <tr> <td align="right" valign="middle" rowspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(i)=_{df}{\Bigg \{}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>f</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">{</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(i)=_{df}{\Bigg \{}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51eb6de74b4996d10e157b211955a249601c386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.387ex; width:11.094ex; height:7.509ex;" alt="{\displaystyle v(i)=_{df}{\Bigg \{}\,\!}"></span> </td> <td valign="bottom" align="left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>, jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b7ee14307e6fd5d5a8a594a11e35db07d1d747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i<2}"></span> </td></tr> <tr> <td valign="middle" align="left"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> muuten. </td></tr></tbody></table> <p>Nyt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(p_{0})=1,v(p_{1})=1,v(p_{2})=0,v(p_{3})=0,\ldots \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(p_{0})=1,v(p_{1})=1,v(p_{2})=0,v(p_{3})=0,\ldots \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcdd808a719141e3123a1046057339eea10b995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:44.932ex; height:2.843ex;" alt="{\displaystyle v(p_{0})=1,v(p_{1})=1,v(p_{2})=0,v(p_{3})=0,\ldots \,\!}"></span> </p><p>Propositiologiikan konnektiivit ovat <i>totuusfunktionaalisia</i>. Kutakin funktion määrittelyjoukon totuusarvoa, kaksipaikkaisten konnektiivien tapauksessa totuusarvoparia, vastaa arvojoukossa täsmälleen yksi totuusarvo. Selkeä tapa määritellä täsmällisesti konnektiivien merkitykset on <i>totuusarvotaulukko</i>. </p><p><b>Määritelmä 6</b> Konnektiivien totuusarvotaulukot </p> <table class="wikitable"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2670cb2bfb471dca31dd3da10251997439eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>A</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {}A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29afb820f3af66bf3b0873aa775f212345c56bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.681ex; height:2.176ex;" alt="{\displaystyle \neg {}A\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {}B\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {}B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048ec2a14d949e80d59c48b1520c36d3c4d93bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.701ex; height:2.176ex;" alt="{\displaystyle \neg {}B\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\wedge {}B)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\wedge {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befb1b997723b72a9ff7c70d97fa522e7470eb38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.286ex; height:2.843ex;" alt="{\displaystyle (A\wedge {}B)\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\vee {}B)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\vee {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b33417652f2427b63900f2b935b3aaccb364e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.286ex; height:2.843ex;" alt="{\displaystyle (A\vee {}B)\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to {}B)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be5864ccef45fcc6c1662afc734d3dd80fcc5df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle (A\to {}B)\,\!}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">↔</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow {}B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0bcc2ca0a595a8209d974d679bc1707446425b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle (A\leftrightarrow {}B)\,\!}"></span> </td></tr> <tr> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>1 </td> <td>1 </td> <td>1 </td></tr> <tr> <td>1 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td>0 </td></tr> <tr> <td>0 </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <td>0 </td> <td>0 </td> <td>1 </td> <td>1 </td> <td>0 </td> <td>0 </td> <td>1 </td> <td>1 </td></tr></tbody></table> <p><b>Negaatio</b> vastaa luonnollisen kielen sanaa <i>ei</i>. Se määrittää lauseen vastakohdan. Lauseen A negaatio <i>ei A</i> on tosi jos (jos ja vain jos) lause A on epätosi. </p><p><b>Konjunktio</b> vastaa luonnollisen kielen sanaa <i>ja</i>. Lauseiden A ja B konjunktio <i>A ja B</i> on tosi vain, jos molemmat sen yhdistämät ilmaisut eli lauseet A ja B ovat tosia. </p><p>Luonnollisen kielen 'tai'-sana on kaksiselitteinen. Joskus sitä käytetään <i>inklusiivisesti</i>, toisinaan taas <i>eksklusiivisesti</i>. Inklusiivisen 'tai'-sanan sisältävä ilmaisu on tosi, jos toinen tai molemmat vaihtoehdoista ovat tosia. Nykykielessä tällöin käytetään joskus sanontaa ”<a href="/wiki/Ja/tai" title="Ja/tai">ja/tai</a>”. Eksklusiivinen 'tai'-ilmaisu on tosi, jos vain toinen ilmaisuista on tosi mutta eivät molemmat. Logiikassa tällaista tulkinnanvaraisuutta ei ole, koska konnektiivien merkitykset määritellään täsmällisesti. Yleisempää on käyttää inklusiivista disjunktiota. <a href="/wiki/Eksklusiivinen_disjunktio" title="Eksklusiivinen disjunktio">Eksklusiivinen disjunktio</a> voidaan kuitenkin määritellä seuraavasti: '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((A\vee B)\wedge \neg (A\wedge B))\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∨</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((A\vee B)\wedge \neg (A\wedge B))\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e90c9f5bf950df50903e0bfd5badb58c2302d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:22.127ex; height:2.843ex;" alt="{\displaystyle ((A\vee B)\wedge \neg (A\wedge B))\,\!}"></span>'. </p><p><b>Implikaatiolla</b> ilmaistaan totuuden <i>riittävää</i> tai <i>välttämätöntä</i> edellytystä. Lauseen '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to B)\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\to B)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a7153cb3b7d8dabf8ed627acf4b8dc959d96d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.318ex; height:2.843ex;" alt="{\displaystyle (A\to B)\,\!}"></span>' (luetaan: <i>jos A niin B</i>) mukaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2670cb2bfb471dca31dd3da10251997439eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}"></span>:n <a href="/wiki/Riitt%C3%A4v%C3%A4_ehto" class="mw-redirect" title="Riittävä ehto">riittävä edellytys</a> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2670cb2bfb471dca31dd3da10251997439eed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.151ex; height:2.176ex;" alt="{\displaystyle B\,\!}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b041434092d1f0042c2b4c7ab32ea84d462cb53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,\!}"></span>:n <a href="/wiki/V%C3%A4ltt%C3%A4m%C3%A4t%C3%B6n_ehto" class="mw-redirect" title="Välttämätön ehto">välttämätön edellytys</a>. </p><p>Ekvivalenssi on tosi jos sen yhdistämien ilmaisujen totuusarvot ovat samat. </p><p>Määritelmien 4 ja 6 perusteella voidaan laskea minkä tahansa propositiolauseen totuusarvo millä tahansa totuusjakaumalla. </p><p><b>Esimerkki 7</b><br /> Lasketaan propositiolauseen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1}))\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1}))\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a83e10b6ef5328b27937136b8a22e6b15d15df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:25.039ex; height:2.843ex;" alt="{\displaystyle \neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1}))\,\!}"></span> totuusarvo. Olkoon totuusjakauma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f34217ded608407636238760709b92635f19dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,\!}"></span> kuten esimerkissä 5.<br /> Määritelmän 4 nojalla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(p_{0})=1,v(p_{1})=1,v(p_{2})=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(p_{0})=1,v(p_{1})=1,v(p_{2})=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/285d31adfa1cc59399d117c0dd43830a99131732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:30.719ex; height:2.843ex;" alt="{\displaystyle v(p_{0})=1,v(p_{1})=1,v(p_{2})=0\,\!}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(p_{3})=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(p_{3})=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c004486b05b34aa5356a312a356696d6c44dec67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.809ex; height:2.843ex;" alt="{\displaystyle v(p_{3})=0\,\!}"></span>.<br /> Määritelmän 6 toisen rivin ja kuudennen sarakkeen mukaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v((p_{0}\wedge p_{2}))=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v((p_{0}\wedge p_{2}))=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba918d65e1b06344fc61b99cf07a556f0c4ab23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:16.424ex; height:2.843ex;" alt="{\displaystyle v((p_{0}\wedge p_{2}))=0\,\!}"></span>.<br /> Kolmannen rivin ja kuudennen sarakkeen mukaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v((p_{3}\vee p_{1}))=1\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v((p_{3}\vee p_{1}))=1\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc05661a04df2fe69392cd79ed07d9da4c30df6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:16.424ex; height:2.843ex;" alt="{\displaystyle v((p_{3}\vee p_{1}))=1\,\!}"></span>.<br /> Kolmannen rivin ja seitsemännen sarakkeen mukaan siis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1})))=1\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1})))=1\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/911dc022a44f163e07f3ff9048e0e0dbc84821ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:30.687ex; height:2.843ex;" alt="{\displaystyle v(((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1})))=1\,\!}"></span>.<br /> Edelleen kolmannen sarakkeen mukaan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(\neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1})))=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(\neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1})))=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ada3b5d0b1779bc1524ca71923b17c7b085f918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:32.237ex; height:2.843ex;" alt="{\displaystyle v(\neg ((p_{0}\wedge p_{2})\to (p_{3}\vee p_{1})))=0\,\!}"></span>. </p><p>Totuustaulua voidaan käyttää myös apuvälineenä propositiolauseen totuusarvon laskemiseksi. Seuraavassa esimerkissä Arvo-riville on merkitty kunkin elementin totuusarvo ja Laskujärjestys-riville järjestys, jossa ne on merkitty. </p><p><b>Esimerkki 8</b> Propositiolauseen totuusarvo totuustaululla </p> <table border="0"> <tbody><tr> <td align="center"> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74835fe1a24df69036387829691ae9bc38736870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:2.7ex; height:2.009ex;" alt="{\displaystyle p_{0}\,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02da95873380cf3f6fcb2da34b3f42c60ead31ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:2.7ex; height:2.009ex;" alt="{\displaystyle p_{1}\,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9689210ce4d82389453bb3df3281dee081ea8e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:2.7ex; height:2.009ex;" alt="{\displaystyle p_{2}\,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{3}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{3}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02b5b2878ad260f8791817e50ee002991244a0e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; margin-right: -0.387ex; width:2.7ex; height:2.009ex;" alt="{\displaystyle p_{3}\,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25554743b2c355ba9d7e0dd3036bb17bb594d3c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; margin-right: -0.387ex; width:1.937ex; height:1.176ex;" alt="{\displaystyle \neg \,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((p_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56110c755e3b3f730ccbc5b884deef30ee20acc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:4.42ex; height:2.843ex;" alt="{\displaystyle ((p_{0}\,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c683a847f466ee483a637e7c77b8ae6e2266ad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.937ex; height:2.009ex;" alt="{\displaystyle \wedge \,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{2})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{2})\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c947db58cff21524b9581ec69273ae20fb05c648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; margin-right: -0.387ex; width:3.605ex; height:2.843ex;" alt="{\displaystyle p_{2})\,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9747a940e84e039c8f7e8a622ab3b70e99cba9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.711ex; height:1.843ex;" alt="{\displaystyle \to \,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{3}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p_{3}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0bdc89912ba3309d9b575c251059ddb7934069d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:3.515ex; height:2.843ex;" alt="{\displaystyle (p_{3}\,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vee \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ce25ddcea7ab813cd57406366222641f23cdd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.937ex; height:2.009ex;" alt="{\displaystyle \vee \,\!}"></span> </td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}))\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}))\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276322a9041b50219f9e712b534b42aaf23afa55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; margin-right: -0.387ex; width:4.509ex; height:2.843ex;" alt="{\displaystyle p_{1}))\,\!}"></span> </td></tr> <tr> <td align="right">Arvo </td> <td align="center">1 </td> <td align="center">1 </td> <td align="center">0 </td> <td align="center">0 </td> <td align="center">0 </td> <td align="center">1 </td> <td align="center">0 </td> <td align="center">0 </td> <td align="center">1 </td> <td align="center">0 </td> <td align="center">1 </td> <td align="center">1 </td></tr> <tr> <td align="right">Laskujärjestys </td> <td align="center">1 </td> <td align="center">1 </td> <td align="center">1 </td> <td align="center">1 </td> <td align="center">5 </td> <td align="center">2 </td> <td align="center">3 </td> <td align="center">2 </td> <td align="center">4 </td> <td align="center">2 </td> <td align="center">3 </td> <td align="center">2 </td></tr></tbody></table> <p>Propositiolauseen totuus riippuu (tietenkin sen rakenteen ohella) ainoastaan propositiosymbolien totuusarvoista. Koska jokaisella oikein muodostetulla propositiolauseella on yksiselitteinen rakennepuu, voidaan propositiolauseen totuusarvo yksiselitteisesti laskea sen sisältämien propositiosymbolien totuusarvojen perusteella, kuten esimerkeistä 7 ja 8 nähdään. Jos propositiosymbolin totuusarvo tiedetään, niin se voidaan korvata esityksessä totuusarvollaan. Myös propositiosymbolia kompleksisempi propositiolauseen osa voidaan korvata totuusarvollaan. </p> <div class="mw-heading mw-heading2"><h2 id="Tautologia_ja_looginen_seuraus">Tautologia ja looginen seuraus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=4" title="Muokkaa osiota Tautologia ja looginen seuraus" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=4" title="Muokkaa osion lähdekoodia: Tautologia ja looginen seuraus"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="" class="metadata plainlinks ambox ambox-content"> <tbody><tr> <td class="ambox-image"><div style="width:52px;text-align:center"> <span typeof="mw:File"><a href="/wiki/Tiedosto:Noto_Emoji_Oreo_1f3d7.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Noto_Emoji_Oreo_1f3d7.svg/40px-Noto_Emoji_Oreo_1f3d7.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Noto_Emoji_Oreo_1f3d7.svg/60px-Noto_Emoji_Oreo_1f3d7.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Noto_Emoji_Oreo_1f3d7.svg/80px-Noto_Emoji_Oreo_1f3d7.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div> </td> <td class="ambox-text"><b>Tämä artikkeli tai osio on keskeneräinen.</b><br /> <small>Voit auttaa Wikipediaa <span class="plainlinks"><a class="external text" href="https://fi.wikipedia.org/w/index.php?title=Propositiologiikka&action=edit">laajentamalla sivua</a></span>. Lisää tietoa saattaa olla <a href="/wiki/Keskustelu:Propositiologiikka" title="Keskustelu:Propositiologiikka">keskustelusivulla</a>. <br /></small> </td></tr></tbody></table> <p><i>Klassisessa</i> <i>propositiologiikassa</i> pätevät seuraavat lait: </p> <ul><li><i>Principium exclusi terti</i> (lat. kielletyn kolmannen laki), jonka mukaan jokainen lause on aina tosi tai epätosi.</li></ul> <ul><li><i>Principium exclusi contradictionis</i> (lat. kielletyn ristiriidan laki), jonka mukaan mikään lause ei voi olla sekä tosi että epätosi.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Propositiologiikan_täydellisyys-_ja_eheyslause"><span id="Propositiologiikan_t.C3.A4ydellisyys-_ja_eheyslause"></span>Propositiologiikan täydellisyys- ja eheyslause</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=5" title="Muokkaa osiota Propositiologiikan täydellisyys- ja eheyslause" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=5" title="Muokkaa osion lähdekoodia: Propositiologiikan täydellisyys- ja eheyslause"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Propositiologiikan <a href="/wiki/T%C3%A4ydellisyyslause" title="Täydellisyyslause"><b>täydellisyyslauseen</b></a> mukaan jos lause <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on tautologia, niin lauseella <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on <a href="/wiki/Luonnollinen_p%C3%A4%C3%A4ttely" title="Luonnollinen päättely">luonnollinen päättely</a>. </p><p>Propositiologiikan <b>eheyslauseen</b> mukaan jos lauseella <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on luonnollinen päättely oletuksista <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1},B_{2},\dots ,B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1},B_{2},\dots ,B_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2794a4c016fd32def39c2f2fd2bfaffc0ce505fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.831ex; height:2.509ex;" alt="{\displaystyle B_{1},B_{2},\dots ,B_{n}}"></span>; niin jos totuusjakaumalla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> pätee <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(B_{1})=v(B_{2})=\ldots =v(B_{n})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>…</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(B_{1})=v(B_{2})=\ldots =v(B_{n})=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ed18dc395ecd1796ca96b1d846a5b35b65e6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.709ex; height:2.843ex;" alt="{\displaystyle v(B_{1})=v(B_{2})=\ldots =v(B_{n})=1}"></span>, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c5c4828d4486a6156432e4499e408bced6ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=1}"></span>. </p><p>Kun täydellisyyslause ja eheyslause yhdistettään, niin saadaan tuloksena seuraava lause: propositiolause <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on tautologia jos ja vain jos propositiolauseella <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on luonnollinen päättely. </p><p>Eheyslauseesta siis seuraa, että jos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>:lla on luonnollinen päättely, niin lause <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on tautologia. Todistetaan tämä eheyslauseen avulla. </p> <div class="mw-heading mw-heading4"><h4 id="Todistus">Todistus</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=6" title="Muokkaa osiota Todistus" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=6" title="Muokkaa osion lähdekoodia: Todistus"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Väite: Jos lauseella <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on luonnollinen päättely, niin lause <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on tautologia. </p><p>Tehdään vastaoletus: lauseella <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on luonnollinen päättely mutta lause <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> ei ole tautologia. Tällöin on olemassa totuusjakauma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> siten, että <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806916a984904d5011ba81ef7d01790fb6defa4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=0}"></span>, joten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(\lnot A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(\lnot A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e75de36c00d055c4b3b9e0c731a99a74fcd95b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.491ex; height:2.843ex;" alt="{\displaystyle v(\lnot A)=1}"></span>. Koska </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\{\lnot A\}&\vdash A\land \lnot A,\\\{A\land \lnot A\}&\vdash \lnot \lnot A\quad {\text{ja}}\\\{\lnot \lnot A\}&\vdash A,\quad {\text{niin}}\\\{\lnot A\}&\vdash A.\end{aligned}}}"> <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> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mi></mi> <mo>⊢</mo> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mi>A</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mi></mi> <mo>⊢</mo> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>ja</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mi></mi> <mo>⊢</mo> <mi>A</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>niin</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> </mtd> <mtd> <mi></mi> <mo>⊢</mo> <mi>A</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\{\lnot A\}&\vdash A\land \lnot A,\\\{A\land \lnot A\}&\vdash \lnot \lnot A\quad {\text{ja}}\\\{\lnot \lnot A\}&\vdash A,\quad {\text{niin}}\\\{\lnot A\}&\vdash A.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd963534b4e44034ea73752d6286098edb64d1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:22.447ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\{\lnot A\}&\vdash A\land \lnot A,\\\{A\land \lnot A\}&\vdash \lnot \lnot A\quad {\text{ja}}\\\{\lnot \lnot A\}&\vdash A,\quad {\text{niin}}\\\{\lnot A\}&\vdash A.\end{aligned}}}"></span></dd></dl> <p>Edellä sovellettiin luonnolisen päättelyn tunnettuja sääntöjä konjunktion tuonti, negaation tuonti ja negaation eliminointi. </p><p>Toisaalta eheyslauseen nojalla koska <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lnot A\}\vdash A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊢</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lnot A\}\vdash A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0488035d58a811bad1594b8e8ddd60f484537f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.072ex; height:2.843ex;" alt="{\displaystyle \{\lnot A\}\vdash A}"></span> ja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(\lnot A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(\lnot A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e75de36c00d055c4b3b9e0c731a99a74fcd95b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.491ex; height:2.843ex;" alt="{\displaystyle v(\lnot A)=1}"></span>, niin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c5c4828d4486a6156432e4499e408bced6ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.843ex;" alt="{\displaystyle v(A)=1}"></span>. Ollaan päädytty ristiriitaan, joten alkuperäinen väite pätee. </p><p>Täten lause <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> on tautologia. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455831d58fa08f311b934d324adcff89a868b4e4" 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 \square }"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Konnektiivit_ja_loogiset_portit">Konnektiivit ja loogiset portit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=7" title="Muokkaa osiota Konnektiivit ja loogiset portit" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=7" title="Muokkaa osion lähdekoodia: Konnektiivit ja loogiset portit"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Elektroniikka" title="Elektroniikka">Elektroniikassa</a> tärkeimpiä loogisia konnektiiveja vastaavat tietyt <a href="/wiki/Looginen_portti" title="Looginen portti">loogiset portit</a> seuraavasti: </p> <table border="1" class="prettytable"> <tbody><tr> <th>Konnektiivi </th> <th>Yhdistetty lause </th> <th>Looginen portti </th></tr> <tr> <td>negaatio </td> <td>"ei A" </td> <td><a href="/wiki/NOT-portti" title="NOT-portti">NOT-portti</a> </td></tr> <tr> <td><a href="/wiki/Konjunktio_(logiikka)" title="Konjunktio (logiikka)">konjunktio</a> </td> <td>"A ja B" </td> <td><a href="/wiki/AND-portti" title="AND-portti">AND-portti</a> </td></tr> <tr> <td><a href="/wiki/Disjunktio" title="Disjunktio">disjunktio</a> </td> <td>"A tai B tai molemmat" </td> <td><a href="/wiki/OR-portti" title="OR-portti">OR-portti</a> </td></tr> <tr> <td><a href="/wiki/Eksklusiivinen_disjunktio" title="Eksklusiivinen disjunktio">eksklusiivinen disjunktio</a> </td> <td>"A tai B, mutta ei molemmat" </td> <td><a href="/wiki/XOR-portti" title="XOR-portti">XOR-portti</a> </td></tr> <tr> <td><a href="/wiki/Shefferin_viiva" title="Shefferin viiva">Shefferin viiva</a> </td> <td>"ei A tai ei B" </td> <td><a href="/wiki/NAND-portti" title="NAND-portti">NAND-portti</a> </td></tr> <tr> <td><a href="/wiki/Peircen_nuoli" title="Peircen nuoli">Peircen nuoli</a> </td> <td>"ei A eikä B" </td> <td><a href="/wiki/NOR-portti" title="NOR-portti">NOR-portti</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Katso_myös"><span id="Katso_my.C3.B6s"></span>Katso myös</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=8" title="Muokkaa osiota Katso myös" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=8" title="Muokkaa osion lähdekoodia: Katso myös"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Predikaattilogiikka" title="Predikaattilogiikka">Predikaattilogiikka</a></li> <li><a href="/wiki/Symbolinen_logiikka" title="Symbolinen logiikka">Symbolinen logiikka</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=9" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=9" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="viitteet-malline" class="viitteet-malline" style="list-style-type:decimal;"><ol class="references"> <li id="cite_note-a-1"><span class="mw-cite-backlink"><a href="#cite_ref-a_1-0">↑</a></span> <span class="reference-text"><span class="kirjaviite" title="Kirjaviite">Thompson, Jan & Martinsson, Thomas: <i>Matematiikan käsikirja</i>, s. 235–236.  Helsinki:  Tammi, 1994.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/951-31-0471-0" title="Toiminnot:Kirjalähteet/951-31-0471-0">ISBN 951-31-0471-0</a> </span></span> </li> </ol> </div> <div class="mw-heading mw-heading2"><h2 id="Kirjallisuutta">Kirjallisuutta</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=10" title="Muokkaa osiota Kirjallisuutta" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=10" title="Muokkaa osion lähdekoodia: Kirjallisuutta"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="kirjaviite" title="Kirjaviite">Thompson, Jan & Martinsson, Thomas: <i><a href="/wiki/Matematiikan_k%C3%A4sikirja" title="Matematiikan käsikirja">Matematiikan käsikirja</a></i>.  Helsinki:  Tammi, 1994.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/951-31-0471-0" title="Toiminnot:Kirjalähteet/951-31-0471-0">ISBN 951-31-0471-0</a> </span></li> <li><span class="kirjaviite" title="Kirjaviite">Miettinen, Seppo K.: <i>Logiikka: Perusteet</i>.  Helsinki:  <a href="/wiki/Gaudeamus" title="Gaudeamus">Gaudeamus</a>, 2002.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/951-662-865-6" title="Toiminnot:Kirjalähteet/951-662-865-6">ISBN 951-662-865-6</a> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Aiheesta_muualla">Aiheesta muualla</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Propositiologiikka&veaction=edit&section=11" title="Muokkaa osiota Aiheesta muualla" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Propositiologiikka&action=edit&section=11" title="Muokkaa osion lähdekoodia: Aiheesta muualla"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r22431496">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/Tiedosto:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commonsissa</a> on kuvia tai muita tiedostoja aiheesta <b><a href="https://commons.wikimedia.org/wiki/Category:Propositional_logic" class="extiw" title="commons:Category:Propositional logic">Propositiologiikka</a></b>.</div></div> </div> <ul><li><span class="verkkoviite" title="Verkkoviite">Klement, Kevin C.: <a rel="nofollow" class="external text" href="http://www.iep.utm.edu/prop-log/">Propositional Logic</a> <i>The Internet Encyclopedia of Philosophy</i>. <span style="font-size: 0.95em; 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mw-list-item"><a href="https://af.wikipedia.org/wiki/Proposisionele_logika" title="Proposisionele logika — afrikaans" lang="af" hreflang="af" data-title="Proposisionele logika" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%D8%A7%D9%84%D9%82%D8%B6%D8%A7%D9%8A%D8%A7" title="حساب القضايا — arabia" lang="ar" hreflang="ar" data-title="حساب القضايا" data-language-autonym="العربية" data-language-local-name="arabia" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/L%C3%B3xica_proposicional" title="Lóxica proposicional — asturia" lang="ast" hreflang="ast" data-title="Lóxica proposicional" data-language-autonym="Asturianu" data-language-local-name="asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kalkulus_proposisional" title="Kalkulus proposisional — indonesia" lang="id" hreflang="id" data-title="Kalkulus proposisional" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D1%96%D0%BA%D0%B0_%D0%B2%D1%8B%D0%BA%D0%B0%D0%B7%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F%D1%9E" title="Логіка выказванняў — valkovenäjä" lang="be" hreflang="be" data-title="Логіка выказванняў" data-language-autonym="Беларуская" data-language-local-name="valkovenäjä" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%97%D1%8C%D0%BB%D1%96%D1%87%D1%8D%D0%BD%D1%8C%D0%BD%D0%B5_%D0%B2%D1%8B%D0%BA%D0%B0%D0%B7%D0%B2%D0%B0%D0%BD%D1%8C%D0%BD%D1%8F%D1%9E" title="Зьлічэньне выказваньняў — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Зьлічэньне выказваньняў" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BF%D0%BE%D0%B7%D0%B8%D1%86%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Пропозиционална логика — bulgaria" lang="bg" hreflang="bg" data-title="Пропозиционална логика" data-language-autonym="Български" data-language-local-name="bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/L%C3%B2gica_proposicional" title="Lògica proposicional — katalaani" lang="ca" hreflang="ca" data-title="Lògica proposicional" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%D0%B0%D0%BD%C4%83%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D1%88%D1%83%D1%82%D0%BB%D0%B0%D0%B2%C4%95" title="Каланăлăхсен шутлавĕ — tšuvassi" lang="cv" hreflang="cv" data-title="Каланăлăхсен шутлавĕ" data-language-autonym="Чӑвашла" data-language-local-name="tšuvassi" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/V%C3%BDrokov%C3%A1_logika" title="Výroková logika — tšekki" lang="cs" hreflang="cs" data-title="Výroková logika" data-language-autonym="Čeština" data-language-local-name="tšekki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhesymeg_osodiadol" title="Rhesymeg osodiadol — kymri" lang="cy" hreflang="cy" data-title="Rhesymeg osodiadol" data-language-autonym="Cymraeg" data-language-local-name="kymri" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Aussagenlogik" title="Aussagenlogik — saksa" lang="de" hreflang="de" data-title="Aussagenlogik" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lauseloogika" title="Lauseloogika — viro" lang="et" hreflang="et" data-title="Lauseloogika" data-language-autonym="Eesti" data-language-local-name="viro" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CF%81%CE%BF%CF%84%CE%B1%CF%83%CE%B9%CE%B1%CE%BA%CF%8C%CF%82_%CE%BB%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Προτασιακός λογισμός — kreikka" lang="el" hreflang="el" data-title="Προτασιακός λογισμός" data-language-autonym="Ελληνικά" data-language-local-name="kreikka" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Propositional_calculus" title="Propositional calculus — englanti" lang="en" hreflang="en" data-title="Propositional calculus" data-language-autonym="English" data-language-local-name="englanti" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%B3gica_proposicional" title="Lógica proposicional — espanja" lang="es" hreflang="es" data-title="Lógica proposicional" data-language-autonym="Español" data-language-local-name="espanja" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logika_proposizional" title="Logika proposizional — baski" lang="eu" hreflang="eu" data-title="Logika proposizional" data-language-autonym="Euskara" data-language-local-name="baski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%DA%AF%D8%B2%D8%A7%D8%B1%D9%87%E2%80%8C%D8%A7%DB%8C" title="حساب گزاره‌ای — persia" lang="fa" hreflang="fa" data-title="حساب گزاره‌ای" data-language-autonym="فارسی" data-language-local-name="persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Calcul_des_propositions" title="Calcul des propositions — ranska" lang="fr" hreflang="fr" data-title="Calcul des propositions" data-language-autonym="Français" data-language-local-name="ranska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%B3xica_proposicional" title="Lóxica proposicional — galicia" lang="gl" hreflang="gl" data-title="Lóxica proposicional" data-language-autonym="Galego" data-language-local-name="galicia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%AA%85%EC%A0%9C_%EB%85%BC%EB%A6%AC" title="명제 논리 — korea" lang="ko" hreflang="ko" data-title="명제 논리" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%BD%D5%B8%D6%82%D5%B5%D5%A9%D5%B6%D5%A5%D6%80%D5%AB_%D5%BF%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ասույթների տրամաբանություն — armenia" lang="hy" hreflang="hy" data-title="Ասույթների տրամաբանություն" data-language-autonym="Հայերեն" data-language-local-name="armenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%9C%E0%A5%8D%E0%A4%9E%E0%A4%AA%E0%A5%8D%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%95%E0%A4%B2%E0%A4%A8" title="प्रतिज्ञप्तिक कलन — hindi" lang="hi" hreflang="hi" data-title="प्रतिज्ञप्तिक कलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logica_proposizionale" title="Logica proposizionale — italia" lang="it" hreflang="it" data-title="Logica proposizionale" data-language-autonym="Italiano" data-language-local-name="italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%97%D7%A9%D7%99%D7%91_%D7%94%D7%A4%D7%A1%D7%95%D7%A7%D7%99%D7%9D" title="תחשיב הפסוקים — heprea" lang="he" hreflang="he" data-title="תחשיב הפסוקים" data-language-autonym="עברית" data-language-local-name="heprea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D1%81%D2%AF%D0%B9%D0%BB%D3%A9%D3%A9" title="Логикалык сүйлөө — kirgiisi" lang="ky" hreflang="ky" data-title="Логикалык сүйлөө" data-language-autonym="Кыргызча" data-language-local-name="kirgiisi" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logica_propositionalis" title="Logica propositionalis — latina" lang="la" hreflang="la" data-title="Logica propositionalis" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Teigini%C5%B3_logika" title="Teiginių logika — liettua" lang="lt" hreflang="lt" data-title="Teiginių logika" data-language-autonym="Lietuvių" data-language-local-name="liettua" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%8Dt%C3%A9letlogika" title="Ítéletlogika — unkari" lang="hu" hreflang="hu" data-title="Ítéletlogika" data-language-autonym="Magyar" data-language-local-name="unkari" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Propositielogica" title="Propositielogica — hollanti" lang="nl" hreflang="nl" data-title="Propositielogica" data-language-autonym="Nederlands" data-language-local-name="hollanti" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%91%BD%E9%A1%8C%E8%AB%96%E7%90%86" title="命題論理 — japani" lang="ja" hreflang="ja" data-title="命題論理" data-language-autonym="日本語" data-language-local-name="japani" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/%C3%9Ctjsaagenloogik" title="Ütjsaagenloogik — pohjoisfriisi" lang="frr" hreflang="frr" data-title="Ütjsaagenloogik" data-language-autonym="Nordfriisk" data-language-local-name="pohjoisfriisi" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Setningslogikk" title="Setningslogikk — norjan bokmål" lang="nb" hreflang="nb" data-title="Setningslogikk" data-language-autonym="Norsk bokmål" data-language-local-name="norjan bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Utsegnslogikk" title="Utsegnslogikk — norjan nynorsk" lang="nn" hreflang="nn" data-title="Utsegnslogikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norjan nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%88%D9%8A_%D8%AD%D8%B3%D8%A7%D8%A8_(%D9%85%D9%86%D8%B7%D9%82)" title="قضیوي حساب (منطق) — paštu" lang="ps" hreflang="ps" data-title="قضیوي حساب (منطق)" data-language-autonym="پښتو" data-language-local-name="paštu" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rachunek_zda%C5%84" title="Rachunek zdań — puola" lang="pl" hreflang="pl" data-title="Rachunek zdań" data-language-autonym="Polski" data-language-local-name="puola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/L%C3%B3gica_proposicional" title="Lógica proposicional — portugali" lang="pt" hreflang="pt" data-title="Lógica proposicional" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0_%D0%B2%D1%8B%D1%81%D0%BA%D0%B0%D0%B7%D1%8B%D0%B2%D0%B0%D0%BD%D0%B8%D0%B9" title="Логика высказываний — venäjä" lang="ru" hreflang="ru" data-title="Логика высказываний" data-language-autonym="Русский" data-language-local-name="venäjä" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Propositional_logic" title="Propositional logic — Simple English" lang="en-simple" hreflang="en-simple" data-title="Propositional logic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/V%C3%BDrokov%C3%A1_logika" title="Výroková logika — slovakki" lang="sk" hreflang="sk" data-title="Výroková logika" data-language-autonym="Slovenčina" data-language-local-name="slovakki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Propozicijska_logika" title="Propozicijska logika — sloveeni" lang="sl" hreflang="sl" data-title="Propozicijska logika" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D1%81%D0%BA%D0%B0%D0%B7%D0%BD%D0%B8_%D1%80%D0%B0%D1%87%D1%83%D0%BD" title="Исказни рачун — serbia" lang="sr" hreflang="sr" data-title="Исказни рачун" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Satslogik" title="Satslogik — ruotsi" lang="sv" hreflang="sv" data-title="Satslogik" data-language-autonym="Svenska" data-language-local-name="ruotsi" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%84%E0%B8%A5%E0%B8%84%E0%B8%B9%E0%B8%A5%E0%B8%B1%E0%B8%AA%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B8%9E%E0%B8%88%E0%B8%99%E0%B9%8C" title="แคลคูลัสเชิงประพจน์ — thai" lang="th" hreflang="th" data-title="แคลคูลัสเชิงประพจน์" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BB%87nh_%C4%91%E1%BB%81_to%C3%A1n_h%E1%BB%8Dc" title="Mệnh đề toán học — vietnam" lang="vi" hreflang="vi" data-title="Mệnh đề toán học" data-language-autonym="Tiếng Việt" data-language-local-name="vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96nermeler_mant%C4%B1%C4%9F%C4%B1" title="Önermeler mantığı — turkki" lang="tr" hreflang="tr" data-title="Önermeler mantığı" data-language-autonym="Türkçe" data-language-local-name="turkki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F_%D0%B2%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D0%BB%D0%B5%D0%BD%D1%8C" title="Числення висловлень — ukraina" lang="uk" hreflang="uk" data-title="Числення висловлень" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%91%BD%E9%A1%8C%E9%82%8F%E8%BC%AF" title="命題邏輯 — kantoninkiina" lang="yue" hreflang="yue" data-title="命題邏輯" data-language-autonym="粵語" data-language-local-name="kantoninkiina" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%91%BD%E9%A2%98%E9%80%BB%E8%BE%91" title="命题逻辑 — kiina" lang="zh" hreflang="zh" data-title="命题逻辑" data-language-autonym="中文" data-language-local-name="kiina" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q200694#sitelinks-wikipedia" title="Muokkaa 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