English+GermanWikipedia - /page/entry.mu

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<title>Formelsammlung Logik</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Formelsammlung Logik</span></h1>

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</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Dies ist eine <b>Formelsammlung</b> zum mathematischen Teilgebiet der <a href="Mathematische_Logik" title="Mathematische Logik">Logik</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Aussagenlogik"><a href="Aussagenlogik" title="Aussagenlogik">Aussagenlogik</a></h2></div>

<p>Logische Werte:

</p>

<li>falsch (false) 0</li></ul>

<p>Erweiterte Logik:

</p>

<ul><li>unbestimmt (<a href="Don%E2%80%99t-Care" title="Don’t-Care">Don’t-Care</a>) X</li></ul>

<p><a href="Aussage_(Logik)" title="Aussage (Logik)">Aussagen</a> können durch <a href="Logischer_Operator" title="Logischer Operator">logische Operatoren</a>, auch <a href="Junktor" title="Junktor">Junktoren</a> genannt, verknüpft werden. Die üblichen Junktoren sind:

</p>

<th>Name

</th>

<th>Symbol

</th>

<th>sprachliche Umschreibung

</th>

<th>Operation

</th>

<th>Definition

</th></tr>

<tr>

<td>Negator</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa78fd02085d39aa58c9e47a6d4033ce41e02fad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }" loading="lazy"></span></td>

<td>nicht</td>

<td>Negation</td>

<td>Die Negation eines logischen Werts ist genau dann wahr, wenn der Wert falsch ist.

</td></tr>

<tr>

<td>Konjunktor</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \land }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6823e5a222eb3ca49672818ac3d13ec607052c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }" loading="lazy"></span></td>

<td>Konjunktion</td>

<td>Die Konjunktion von zwei Werten ist genau dann wahr, wenn beide Werte wahr sind.

</td></tr>

<tr>

<td>Disjunktor</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lor }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab47f6b1f589aedcf14638df1d63049d233d851a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }" loading="lazy"></span></td>

<td>Disjunktion</td>

<td>Die Disjunktion von zwei Werten ist genau dann wahr, wenn mindestens ein Wert wahr ist.

</td></tr></tbody></table>

<p>Um die Symbole des Konjunktors und des Disjunktors leicht auseinanderhalten zu können, gibt es die Eselsbrücke mit den drei O: „Oder ist Oben Offen.“ Alternativ merkt man sich "<b>A</b>nd" (Englisch) für und, sowie "<b>v</b>el" (Latein) für oder.

</p>

<div class="mw-heading mw-heading3"><h3 id="Verknüpfungen_zweier_Aussagen"><span id="Verkn.C3.BCpfungen_zweier_Aussagen"></span>Verknüpfungen zweier Aussagen</h3></div>

<th rowspan="3" colspan="2">Name

</th>

<th rowspan="3">sprachliche Umschreibung

</th>

<th colspan="2">äquivalente Darstellungen

</th>

<th colspan="4"><a href="Wahrheitstabelle" title="Wahrheitstabelle">Wahrheitstabelle</a>

</th>

<th rowspan="3"><a href="Logikgatter" title="Logikgatter">Logikgatter</a>

</th></tr>

<tr>

<th rowspan="2">durch Negator, Konjunktor und Disjunktor

</th>

<th rowspan="2">durch andere Junktoren

</th>

<th colspan="2">A=1

</th>

<th colspan="2">A=0

</th></tr>

<tr>

<th>B=1

</th>

<th>B=0

</th>

<th>B=1

</th>

<th>B=0

</th></tr>

<tr>

<td colspan="2"><a href="Konjunktion_(Logik)" title="Konjunktion (Logik)">Konjunktion</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74954195333a8593163b93a9688695b8dc74da55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (B\to \neg A)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78b923ed48440972ed5625e50723b7d11e961252.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.031ex; height:2.843ex;" alt="{\displaystyle \neg (B\to \neg A)}" loading="lazy"></span>

</td>

</td></tr>

<tr>

<td colspan="2">Exklusion, <a href="Gegensatz#Der_kontradiktorische,_konträre,_subkonträre_und_subalterne_Gegensatz_(logisches_Quadrat)" title="Gegensatz">konträrer Gegensatz</a></td>

<td>nicht zugleich A und B</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (A\land B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91ba373711ac586009d3c4d0117525fbe04a1a4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.449ex; height:2.843ex;" alt="{\displaystyle \neg (A\land B)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\lor \neg B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg A\lor \neg B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1020d4923bd093b4d10a73c88d3db0b3211b4ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.19ex; height:2.176ex;" alt="{\displaystyle \neg A\lor \neg B}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mid B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1400973074b4691cc0638a68118716a2b218fce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.444ex; height:2.843ex;" alt="{\displaystyle A\mid B}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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 \neg B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\to \neg B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c640bcf9ec73c0218bc586c8757953a86f7cef08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.672ex; height:2.176ex;" alt="{\displaystyle A\to \neg B}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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 \neg A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\to \neg A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2733729e9e5a0ffec0e3be176c33945b0821f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.672ex; height:2.176ex;" alt="{\displaystyle B\to \neg A}" loading="lazy"></span></td>

</td></tr>

<tr>

<td colspan="2"><a href="Disjunktion" title="Disjunktion">Disjunktion</a></td>

<td>A oder B (oder beide)</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\lor B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b9c9c90857c12727201dd9e47a4e7c8658fdbc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\lor B}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg A\to B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46b0af8bc3d53e1e686ffc609aec72cd6f38bc7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.672ex; height:2.176ex;" alt="{\displaystyle \neg A\to B}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\to A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg B\to A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0846f5afa338a75e2f847db1f9603cff4bbc3b2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.672ex; height:2.176ex;" alt="{\displaystyle \neg B\to A}" loading="lazy"></span></td>

</td></tr>

<tr>

<td colspan="2">Nihilition, Rejektion</td>

<td>weder A noch B</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (A\lor B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4be2de29b87215d778d98ab2d26c4e166056ca0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.449ex; height:2.843ex;" alt="{\displaystyle \neg (A\lor B)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\land \neg B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg A\land \neg B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5008fff96d0c52ff4bfd0361d07b622fd23a8f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.19ex; height:2.176ex;" alt="{\displaystyle \neg A\land \neg B}" loading="lazy"></span></td>

</td></tr>

<tr>

<td colspan="2"><a href="Kontravalenz" title="Kontravalenz">Kontravalenz</a>, <a href="Gegensatz#Der_kontradiktorische,_konträre,_subkonträre_und_subalterne_Gegensatz_(logisches_Quadrat)" title="Gegensatz">kontradiktorischer Gegensatz</a></td>

<td>entweder A oder B</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A\land \neg B)\lor (\neg A\land B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb15e111a02acdae5d4aae8dd48998920a6d107d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.481ex; height:2.843ex;" alt="{\displaystyle (A\land \neg B)\lor (\neg A\land B)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\lor B)\land (\neg A\lor \neg B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A\lor B)\land (\neg A\lor \neg B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf45e799118780fd0e8257f58c832b385a8a09e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.481ex; height:2.843ex;" alt="{\displaystyle (A\lor B)\land (\neg A\lor \neg B)}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\veebar B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c6b95dc15cf07d2c1ce14624b7790ec1336e5eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.96ex; height:2.176ex;" alt="{\displaystyle A\veebar B}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\leftrightarrow B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (A\leftrightarrow B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/094ec19a54ae6fa2c4d13499807f57cb089362b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.481ex; height:2.843ex;" alt="{\displaystyle \neg (A\leftrightarrow B)}" loading="lazy"></span></td>

</td></tr>

<tr>

<td colspan="2"><a href="Bikonditional" title="Bikonditional">Bikonditional</a>, Bisubjunktion, materiale Äquivalenz</td>

<td>B dann und nur dann, wenn A; genau dann B, wenn A</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A\land B)\lor (\neg A\land \neg B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79bc1955d36134c749809af76eb79c623e6e12fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.481ex; height:2.843ex;" alt="{\displaystyle (A\land B)\lor (\neg A\land \neg B)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\lor \neg B)\land (\neg A\lor B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A\lor \neg B)\land (\neg A\lor B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65f844afea2d650ac897b9ba61b1f7f4b53ebb1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.481ex; height:2.843ex;" alt="{\displaystyle (A\lor \neg B)\land (\neg A\lor B)}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A\leftrightarrow B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/377a507f43f614500c9c326f3135fbb062cd5fc5.svg" 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)}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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)\land (B\to A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A\to B)\land (B\to A)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21459944695dadd2c3234fee78fa67e2f2752be2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.444ex; height:2.843ex;" alt="{\displaystyle (A\to B)\land (B\to A)}" loading="lazy"></span></td>

</td></tr>

<tr>

<td rowspan="2">Konditional, <a href="Subjunktion" title="Subjunktion">Subjunktion</a>, materiale Implikation</td>

<td>Implikation</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg A\lor B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c13321f163366a0912c15b09d401b33e29a1ad46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle \neg A\lor B}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\to \neg A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg B\to \neg A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fed8b28f42f63937db9b96078fbbcf92a8fc1cd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.222ex; height:2.176ex;" alt="{\displaystyle \neg B\to \neg A}" loading="lazy"></span></td>

</td></tr>

<tr>

<td>Replikation</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg B\lor A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad65aa82f31b9db072e634b154c6dd0f9b8f6327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle \neg B\lor A}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\to A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ef7d663f25816b2edf065af9a473eebebeea56a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle B\to A}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><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\to \neg B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg A\to \neg B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9df94953960eb04fd856b2ed494a3b391f4933dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.222ex; height:2.176ex;" alt="{\displaystyle \neg A\to \neg B}" loading="lazy"></span></td>

<td>1

</td></tr>

<tr>

<td rowspan="2">Inhibition</td>

<td>Postsektion</td>

<td>A und nicht B</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\land \neg B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64a16a6d22f82742c644c45dd32d0b2fa68ced9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle A\land \neg B}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (A\to B)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/069bd7cd9b81bd51a836de458917f7e370b39890.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.481ex; height:2.843ex;" alt="{\displaystyle \neg (A\to B)}" loading="lazy"></span></td>

</td></tr>

<tr>

<td>Präsektion</td>

<td>B und nicht A</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\land \neg A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afa5a12ee5e33d74b0c954b5b2dabd66c88cc0de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.64ex; height:2.176ex;" alt="{\displaystyle B\land \neg A}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (B\to A)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4010c8b4d296e34202683e31c7c1fb7b14f08872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.481ex; height:2.843ex;" alt="{\displaystyle \neg (B\to A)}" loading="lazy"></span></td>

<td>0

</td></tr></tbody></table>

<p>Insgesamt sind 16 Funktionen mit zwei Input-Variablen möglich. Zu den genannten zehn Funktionen kommen noch die eher uninteressanten Funktionen konstant_0, konstant_1, <span class="mwe-math-element mwe-math-element-inline"><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,\neg A,B,\neg B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A,\neg A,B,\neg B}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a69d2ad765ecaf5da381ef8afb86e59313d13ea5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.217ex; height:2.509ex;" alt="{\displaystyle A,\neg A,B,\neg B}" loading="lazy"></span> dazu.

</p>

<div class="mw-heading mw-heading3"><h3 id="Logische_Grundgesetze">Logische Grundgesetze</h3></div>

<td><a href="Gesetz_der_doppelten_Negation" title="Gesetz der doppelten Negation">Gesetz der doppelten Negation</a>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\leftrightarrow \neg (\neg x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf7b5d744d28ff4c5babde0ec71da109f6ffb334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.183ex; height:2.843ex;" alt="{\displaystyle x\leftrightarrow \neg (\neg x)}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Kommutativgesetz#Aussagenlogik" title="Kommutativgesetz">Kommutativgesetze</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\land y\leftrightarrow y\land x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/239150a51b84f587fe80e8a8b5a26f6e9550d267.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.75ex; height:2.343ex;" alt="{\displaystyle x\land y\leftrightarrow y\land x}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\lor y\leftrightarrow y\lor x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e9ba43af9b68a86189a1b4d8bfdcbd4e13c00d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.75ex; height:2.343ex;" alt="{\displaystyle x\lor y\leftrightarrow y\lor x}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Assoziativgesetz" title="Assoziativgesetz">Assoziativgesetze</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\land (y\land z)\leftrightarrow (x\land y)\land z}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cedf58bdaf52342eb0b1fbb8884dc773f9412ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.71ex; height:2.843ex;" alt="{\displaystyle x\land (y\land z)\leftrightarrow (x\land y)\land z}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x\lor y)\lor z\leftrightarrow x\lor (y\lor z)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cbce797cd248e40ac10470ebaa08852310a63c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.71ex; height:2.843ex;" alt="{\displaystyle (x\lor y)\lor z\leftrightarrow x\lor (y\lor z)}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Distributivgesetz" title="Distributivgesetz">Distributivgesetze</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\land (y\lor z)\leftrightarrow (x\land y)\lor (x\land z)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec22b12b93bf37d874b744e9c18e1f5558cb0d94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.431ex; height:2.843ex;" alt="{\displaystyle x\land (y\lor z)\leftrightarrow (x\land y)\lor (x\land z)}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\lor (y\land z)\leftrightarrow (x\lor y)\land (x\lor z)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1285b2125e3bd19350188fadb7b708697ab4513f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.431ex; height:2.843ex;" alt="{\displaystyle x\lor (y\land z)\leftrightarrow (x\lor y)\land (x\lor z)}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Idempotenz" title="Idempotenz">Idempotenz</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\land x\leftrightarrow x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfc9a7f12f15409804fa6f6ceb68c549376760ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.186ex; height:2.009ex;" alt="{\displaystyle x\land x\leftrightarrow x}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\lor x\leftrightarrow x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b50683496cd059b3569493e8c22716eea2d3d9e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.186ex; height:2.009ex;" alt="{\displaystyle x\lor x\leftrightarrow x}" loading="lazy"></span>

</td></tr>

<tr>

<td>Gesetze der <a href="Negation#Logik" title="Negation">Negation</a> (Tautologie / Kontradiktion)</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\lor \neg x\leftrightarrow 1}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/902237832b9c4355d721b286231b152d4ec5c17c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.569ex; height:2.176ex;" alt="{\displaystyle x\lor \neg x\leftrightarrow 1}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\land \neg x\leftrightarrow 0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f60bd611bbf9bd66b26856dac05b0530e1c8a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.569ex; height:2.176ex;" alt="{\displaystyle x\land \neg x\leftrightarrow 0}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Absorptionsgesetz_(Logik)" class="mw-redirect" title="Absorptionsgesetz (Logik)">Absorptionsgesetze</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\land (x\lor y)\leftrightarrow x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/106901a67e6ba90eb900fddd5ed2d7f302b96ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.733ex; height:2.843ex;" alt="{\displaystyle x\land (x\lor y)\leftrightarrow x}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\lor (x\land y)\leftrightarrow x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f330752f1163bb45a650ea33258c108d1d310315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.733ex; height:2.843ex;" alt="{\displaystyle x\lor (x\land y)\leftrightarrow x}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Neutrales_Element" title="Neutrales Element">Neutralität</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\lor 0\leftrightarrow x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cccd737472c80a61b9d761358fb4a19fc24bdbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.018ex; height:2.176ex;" alt="{\displaystyle x\lor 0\leftrightarrow x}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\land 1\leftrightarrow x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bedfe298fbdd55ba5e91aa4bf68aed3ea6e1bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.018ex; height:2.176ex;" alt="{\displaystyle x\land 1\leftrightarrow x}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="De_Morgansche_Gesetze" class="mw-redirect" title="De Morgansche Gesetze">De Morgansche Gesetze</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (x\land y)\leftrightarrow \neg x\lor \neg y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eea06a6436aba2f1fa7879d55e9e076366d8c50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.21ex; height:2.843ex;" alt="{\displaystyle \neg (x\land y)\leftrightarrow \neg x\lor \neg y}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (x\lor y)\leftrightarrow \neg x\land \neg y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f5b150a762a072db14b9238e7d5ffab9e9177ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.21ex; height:2.843ex;" alt="{\displaystyle \neg (x\lor y)\leftrightarrow \neg x\land \neg y}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Schlussregeln"><a href="Schlussregel" title="Schlussregel">Schlussregeln</a></h3></div>

<td><a href="Modus_ponens" title="Modus ponens">Modus ponens</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{a\rightarrow b,a\}\vdash b}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35df6f28d05ffbd36e6f8a4f8ebf0355117d9f2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.138ex; height:2.843ex;" alt="{\displaystyle \{a\rightarrow b,a\}\vdash b}" loading="lazy"></span>

</td></tr>

<tr>

<td><a href="Modus_tollens" title="Modus tollens">Modus tollens</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{a\rightarrow b,\neg b\}\vdash \neg a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38b2c7c97fde6566044ccf1af93f0669fcd05511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.239ex; height:2.843ex;" alt="{\displaystyle \{a\rightarrow b,\neg b\}\vdash \neg a}" loading="lazy"></span>

</td></tr>

<tr>

<td>Hypothetischer <a href="Syllogismus" title="Syllogismus">Syllogismus</a></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{a\rightarrow b,b\rightarrow c\}\vdash a\rightarrow c}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1a26a2d69f1c7b962d5ba8ece5424549fb8856d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.38ex; height:2.843ex;" alt="{\displaystyle \{a\rightarrow b,b\rightarrow c\}\vdash a\rightarrow c}" loading="lazy"></span>

</td></tr>

<tr>

<td>Disjunktiver Syllogismus</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{a\lor b,\neg a\}\vdash b}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5db1acf12097c3e6711bbb01917badcb104d3f2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.657ex; height:2.843ex;" alt="{\displaystyle \{a\lor b,\neg a\}\vdash b}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Prädikatenlogik"><span id="Pr.C3.A4dikatenlogik"></span>Prädikatenlogik</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Quantoren"><a href="Quantor" title="Quantor">Quantoren</a></h3></div>

<p>p ist Platzhalter für eine prädikatenlogische Aussageform.

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall _{x}p\leftrightarrow \neg (\exists _{x}\neg p)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/388f8b6841c5167917ab433882de068dad4ace42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.793ex; height:2.843ex;" alt="{\displaystyle \forall _{x}p\leftrightarrow \neg (\exists _{x}\neg p)}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exists _{x}p\leftrightarrow \neg (\forall _{x}\neg p)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/285b755397b4fc74d0a2e6ab3591662f7001103e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.793ex; height:2.843ex;" alt="{\displaystyle \exists _{x}p\leftrightarrow \neg (\forall _{x}\neg p)}" loading="lazy"></span>

</td></tr>

<tr>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg \forall _{x}p\leftrightarrow (\exists _{x}\neg p)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/015ca41a696cb382eb87b37306221db6df45e918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.793ex; height:2.843ex;" alt="{\displaystyle \neg \forall _{x}p\leftrightarrow (\exists _{x}\neg p)}" loading="lazy"></span>

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg \exists _{x}p\leftrightarrow (\forall _{x}\neg p)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23c2a079444acea480fe81ad5953bdf038db0a14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.793ex; height:2.843ex;" alt="{\displaystyle \neg \exists _{x}p\leftrightarrow (\forall _{x}\neg p)}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Pränexform"><span id="Pr.C3.A4nexform"></span><a href="Pr%C3%A4nexform" title="Pränexform">Pränexform</a></h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> sind im Folgenden Platzhalter für prädikatenlogische Aussageformen. Die Umformungen in Zeilen 1, 2, 4 und 5 der Tabelle gelten nur, wenn x innerhalb von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> nicht frei vorkommt, d. h., wenn durch das Verschieben des Quantors keine Variablenbindung entsteht (bzw. aufgelöst wird), die zuvor nicht da war (bzw. da war).

</p><p>Unproblematisch ist das, wenn die Variablen in den Aussageformen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"></span> jeweils unterschiedlich benannt sind.

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\forall x\phi )\land \psi \leftrightarrow \forall x(\phi \land \psi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8479d03410232ddc37b5a4c48b79c642dd60a205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.439ex; height:2.843ex;" alt="{\displaystyle (\forall x\phi )\land \psi \leftrightarrow \forall x(\phi \land \psi )}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\forall x\phi )\lor \psi \leftrightarrow \forall x(\phi \lor \psi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e12494b11001d98c6040fd3d0b0444f8c6fd5151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.439ex; height:2.843ex;" alt="{\displaystyle (\forall x\phi )\lor \psi \leftrightarrow \forall x(\phi \lor \psi )}" loading="lazy"></span>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\exists x\phi )\land \psi \leftrightarrow \exists x(\phi \land \psi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b01c497c174062c46f149c23f24d4bcbb6b3452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.439ex; height:2.843ex;" alt="{\displaystyle (\exists x\phi )\land \psi \leftrightarrow \exists x(\phi \land \psi )}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\exists x\phi )\lor \psi \leftrightarrow \exists x(\phi \lor \psi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a30828e043df5720f08de1e7163c9279225ae45f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.439ex; height:2.843ex;" alt="{\displaystyle (\exists x\phi )\lor \psi \leftrightarrow \exists x(\phi \lor \psi )}" loading="lazy"></span>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lnot \exists x\phi \leftrightarrow \forall x\lnot \phi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d81bdad51757cf8d61840b72b267fddc7ea76036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.73ex; height:2.509ex;" alt="{\displaystyle \lnot \exists x\phi \leftrightarrow \forall x\lnot \phi }" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lnot \forall x\phi \leftrightarrow \exists x\lnot \phi }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b54c6e3a5e7a33620c5cfb9563b4c01e9befaba6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.73ex; height:2.509ex;" alt="{\displaystyle \lnot \forall x\phi \leftrightarrow \exists x\lnot \phi }" loading="lazy"></span>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((\forall x\phi )\rightarrow \psi )\leftrightarrow \exists x(\phi \rightarrow \psi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3219d4f45b9bc141f747260f67e2084b8dfdec9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.312ex; height:2.843ex;" alt="{\displaystyle ((\forall x\phi )\rightarrow \psi )\leftrightarrow \exists x(\phi \rightarrow \psi )}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ((\exists x\phi )\rightarrow \psi )\leftrightarrow \forall x(\phi \rightarrow \psi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4aeef7bf8c635d5c3b1659f85e5ca5af4af84a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.312ex; height:2.843ex;" alt="{\displaystyle ((\exists x\phi )\rightarrow \psi )\leftrightarrow \forall x(\phi \rightarrow \psi )}" loading="lazy"></span>

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\psi \rightarrow (\exists x\phi ))\leftrightarrow \exists x(\psi \rightarrow \phi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2d865a4e07418586d3ea9b6f5be57a62f4d828e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.312ex; height:2.843ex;" alt="{\displaystyle (\psi \rightarrow (\exists x\phi ))\leftrightarrow \exists x(\psi \rightarrow \phi )}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\psi \rightarrow (\forall x\phi ))\leftrightarrow \forall x(\psi \rightarrow \phi )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a224aee9d8d15cb487fb6629bff7c3ff6b511e7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.312ex; height:2.843ex;" alt="{\displaystyle (\psi \rightarrow (\forall x\phi ))\leftrightarrow \forall x(\psi \rightarrow \phi )}" loading="lazy"></span>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Minimale_Schlussregeln">Minimale Schlussregeln</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Quasiordnung">Quasiordnung</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0c0d30cf8cb7dba179e317fcde9583d842e80f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }" loading="lazy"></span> ist im Folgenden eine Quasiordnung zwischen Aussagen.

</p><p><span class="mwe-math-element mwe-math-element-inline"><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{array}{c}{~}\\\hline A\vdash A\end{array}}\qquad {\begin{array}{c}A\vdash B\qquad B\vdash C\\\hline A\vdash C\end{array}}}">

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{~}\\\hline A\vdash A\end{array}}\qquad {\begin{array}{c}A\vdash B\qquad B\vdash C\\\hline A\vdash C\end{array}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70467f4979ebabcbc6f472d42898245ae6f23719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.165ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}{~}\\\hline A\vdash A\end{array}}\qquad {\begin{array}{c}A\vdash B\qquad B\vdash C\\\hline A\vdash C\end{array}}}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Konjunktion">Konjunktion</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \top }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf12e436fef2365e76fcb1034a51179d8328bb33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \top }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \land }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6823e5a222eb3ca49672818ac3d13ec607052c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }" loading="lazy"></span> werden durch folgende Regeln definiert.

</p><p><span class="mwe-math-element mwe-math-element-inline"><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{array}{c}{~}\\\hline A\vdash \top \end{array}}\qquad {\begin{array}{c}A\vdash B\qquad A\vdash C\\\hline A\vdash B\land C\end{array}}{\uparrow }{\downarrow }}">

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{~}\\\hline A\vdash \top \end{array}}\qquad {\begin{array}{c}A\vdash B\qquad A\vdash C\\\hline A\vdash B\land C\end{array}}{\uparrow }{\downarrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00230ceab62c6e4355540e96185fd1d41bae05e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.534ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}{~}\\\hline A\vdash \top \end{array}}\qquad {\begin{array}{c}A\vdash B\qquad A\vdash C\\\hline A\vdash B\land C\end{array}}{\uparrow }{\downarrow }}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Disjunktion">Disjunktion</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bot }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f282c7bc331cc3bfcf1c57f1452cc23c022f58de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \bot }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lor }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab47f6b1f589aedcf14638df1d63049d233d851a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }" loading="lazy"></span> werden durch folgende Regeln definiert.

</p><p><span class="mwe-math-element mwe-math-element-inline"><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{array}{c}{~}\\\hline \bot \vdash A\end{array}}\qquad {\begin{array}{c}A\vdash C\qquad B\vdash C\\\hline A\lor B\vdash C\end{array}}{\uparrow }{\downarrow }}">

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{~}\\\hline \bot \vdash A\end{array}}\qquad {\begin{array}{c}A\vdash C\qquad B\vdash C\\\hline A\lor B\vdash C\end{array}}{\uparrow }{\downarrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0cbed6cf2f6440f30ef1bfdaaabab31342ba18a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.557ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}{~}\\\hline \bot \vdash A\end{array}}\qquad {\begin{array}{c}A\vdash C\qquad B\vdash C\\\hline A\lor B\vdash C\end{array}}{\uparrow }{\downarrow }}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Heyting-Implikation_und_-Negation">Heyting-Implikation und -Negation</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \to }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span> wird durch die Regel

</p><p><span class="mwe-math-element mwe-math-element-inline"><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{array}{lcr}A\land B&\vdash &C\\\hline A&\vdash &B\to C\end{array}}{\uparrow }{\downarrow }}">

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcr}A\land B&\vdash &C\\\hline A&\vdash &B\to C\end{array}}{\uparrow }{\downarrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f37ef655557939fa30c6d74e855f6400ff6c4ec5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.234ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{lcr}A\land B&\vdash &C\\\hline A&\vdash &B\to C\end{array}}{\uparrow }{\downarrow }}" loading="lazy"></span>

</p><p>definiert, und <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lnot }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/099107443792f5fec9bebe39b919a690db7198c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \lnot }" loading="lazy"></span> per

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lnot A:=A\to \bot }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f36ef3c75c69d0a159badfb25220b896e3bb27f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.204ex; height:2.176ex;" alt="{\displaystyle \lnot A:=A\to \bot }" loading="lazy"></span>.

</p><p>Es gelten

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\land \lnot A\vdash \bot }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d9269b5cdbd8b29b07d5070a6cf115ca5ba6b61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.138ex; height:2.176ex;" alt="{\displaystyle A\land \lnot A\vdash \bot }" loading="lazy"></span>,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lnot \top \vdash \bot }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efd0cfe15799049bc31e5d7c14eb608e9fcc2d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.877ex; height:2.176ex;" alt="{\displaystyle \lnot \top \vdash \bot }" loading="lazy"></span> und</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \top \vdash \lnot \bot }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09f8f563a06f5587954e94e09b79f3a946bafa68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.877ex; height:2.176ex;" alt="{\displaystyle \top \vdash \lnot \bot }" loading="lazy"></span>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Ko-Heyting-Implikation_und_-Negation">Ko-Heyting-Implikation und -Negation</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \to }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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 }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lnot }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/099107443792f5fec9bebe39b919a690db7198c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \lnot }" loading="lazy"></span> sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \setminus }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \setminus }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d0e20e45087a97f0448fc3d4bc27b060084830f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \setminus }" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sim }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }" loading="lazy"></span>.

</p><p><span class="mwe-math-element mwe-math-element-inline"><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{array}{lcr}A\setminus B&\vdash &C\\\hline A&\vdash &B\lor C\end{array}}{\uparrow }{\downarrow }}">

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcr}A\setminus B&\vdash &C\\\hline A&\vdash &B\lor C\end{array}}{\uparrow }{\downarrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b8c284b25e37febd4ec5b1ead8f35512657e1a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.815ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{lcr}A\setminus B&\vdash &C\\\hline A&\vdash &B\lor C\end{array}}{\uparrow }{\downarrow }}" loading="lazy"></span>,

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sim }A:=\top \setminus A}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sim }A:=\top \setminus A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44570ac04a86b4a2804a23586d61426831f9feba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.042ex; height:2.843ex;" alt="{\displaystyle {\sim }A:=\top \setminus A}" loading="lazy"></span>.

</p><p>Es gelten

</p>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \top \vdash A\lor {\sim }A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51f239a8792986f9b673fb8bbd9ac90eaaf227bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.396ex; height:2.176ex;" alt="{\displaystyle \top \vdash A\lor {\sim }A}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle {\sim }\top \vdash \bot }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09b30bbe233ed29490c58aa445bb323978446cb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.135ex; height:2.176ex;" alt="{\displaystyle {\sim }\top \vdash \bot }" loading="lazy"></span> und</li>

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<annotation encoding="application/x-tex">{\displaystyle \top \vdash {\sim }\bot }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/abea046a11f533ce3f233c44085945177c200293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.135ex; height:2.176ex;" alt="{\displaystyle \top \vdash {\sim }\bot }" loading="lazy"></span>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Beziehung_zwischen_den_Negationen">Beziehung zwischen den Negationen</h3></div>

<p>Es gilt immer <span class="mwe-math-element mwe-math-element-inline"><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 {\sim }A}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lnot A\vdash {\sim }A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa4ef6bdade2618e557a05c2b7f3302e82519fd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.555ex; height:2.176ex;" alt="{\displaystyle \lnot A\vdash {\sim }A}" loading="lazy"></span>. Gilt auch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sim }A\vdash \lnot A}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sim }A\vdash \lnot A}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7b9bd2de64f460de5677f0e0e48fb8a00ac9260.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.555ex; height:2.176ex;" alt="{\displaystyle {\sim }A\vdash \lnot A}" loading="lazy"></span>, erhält man klassische Logik.

</p>

<div class="mw-heading mw-heading3"><h3 id="Quantoren_2">Quantoren</h3></div>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07b9ff205beb51e7899846aeae788ae5e5546a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" loading="lazy"></span> eine Abbildung. Eine beliebige Aussage <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" 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}" loading="lazy"></span> über Elemente von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> kann per <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> in eine Aussage über <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>-Elemente transformiert werden; Notation: <span class="mwe-math-element mwe-math-element-inline"><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\circ f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\circ f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55470f2d2b2d8e4c7b829bb86b4d98b361176e36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.216ex; height:2.509ex;" alt="{\displaystyle A\circ f}" loading="lazy"></span>.

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\mathord {-}}\circ f)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77443300faddf252c48a93072ded25c9f2729a1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.091ex; height:2.843ex;" alt="{\displaystyle ({\mathord {-}}\circ f)}" loading="lazy"></span> ist ein Funktor. Sein Rechts- und Linksadjungierter ist der All- bzw. Existenzquantor, d. h.,

</p><p><span class="mwe-math-element mwe-math-element-inline"><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{array}{lcr}A\circ f&\vdash _{X}&B\\\hline A&\vdash _{Y}&\forall _{f}B\end{array}}{\uparrow }{\downarrow }\qquad {\begin{array}{rcl}C&\vdash _{X}&A\circ f\\\hline \exists _{f}C&\vdash _{Y}&A\end{array}}{\uparrow }{\downarrow }}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lcr}A\circ f&\vdash _{X}&B\\\hline A&\vdash _{Y}&\forall _{f}B\end{array}}{\uparrow }{\downarrow }\qquad {\begin{array}{rcl}C&\vdash _{X}&A\circ f\\\hline \exists _{f}C&\vdash _{Y}&A\end{array}}{\uparrow }{\downarrow }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fcfe49b9c9bc1bb211fe2649ec8b93627fbf98a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.731ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{lcr}A\circ f&\vdash _{X}&B\\\hline A&\vdash _{Y}&\forall _{f}B\end{array}}{\uparrow }{\downarrow }\qquad {\begin{array}{rcl}C&\vdash _{X}&A\circ f\\\hline \exists _{f}C&\vdash _{Y}&A\end{array}}{\uparrow }{\downarrow }}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Liste_mathematischer_Symbole" title="Liste mathematischer Symbole">Liste mathematischer Symbole</a></li></ul></div><div><div class="zim-footer">

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