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<title>Contradiction</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Contradiction</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Contradiction_(disambiguation)" class="mw-disambig" title="Contradiction (disambiguation)">Contradiction (disambiguation)</a>.</div>

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<p>In <a href="Traditional_logic" class="mw-redirect" title="Traditional logic">traditional logic</a>, a <b>contradiction</b> involves a <a href="Proposition" title="Proposition">proposition</a> conflicting either with itself or established <a href="Fact" title="Fact">fact</a>. It is often used as a tool to detect <a href="Disingenuous" class="mw-redirect" title="Disingenuous">disingenuous</a> beliefs and <a href="Bias" title="Bias">bias</a>. Illustrating a general tendency in applied logic, <a href="Aristotle" title="Aristotle">Aristotle</a>'s <a href="Law_of_noncontradiction" title="Law of noncontradiction">law of noncontradiction</a> states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>In modern <a href="Formal_logic" class="mw-redirect" title="Formal logic">formal logic</a> and <a href="Type_theory" title="Type theory">type theory</a>, the term is mainly used instead for a <i>single</i> proposition, often denoted by the <a href="Falsum" class="mw-redirect" title="Falsum">falsum</a> symbol <a href="Bottom_type" title="Bottom type"><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 \bot }">

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

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</math></span><img src="./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></a>; a proposition is a contradiction if <a href="False_(logic)" title="False (logic)">false</a> can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This can be generalized to a collection of propositions, which is then said to "contain" a contradiction.

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>By creation of a <a href="Paradox" title="Paradox">paradox</a>, <a href="Plato" title="Plato">Plato</a>'s <i><a href="Euthydemus_(dialogue)" title="Euthydemus (dialogue)">Euthydemus</a></i> dialogue demonstrates the need for the notion of <i>contradiction</i>. In the ensuing dialogue, <a href="Dionysodorus_(sophist)" title="Dionysodorus (sophist)">Dionysodorus</a> denies the existence of "contradiction", all the while that <a href="Socrates" title="Socrates">Socrates</a> is contradicting him:

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</style><blockquote class="templatequote"><p>... I in my astonishment said: What do you mean Dionysodorus? I have often heard, and have been amazed to hear, this thesis of yours, which is maintained and employed by the disciples of Protagoras and others before them, and which to me appears to be quite wonderful, and suicidal as well as destructive, and I think that I am most likely to hear the truth about it from you. The dictum is that there is no such thing as a falsehood; a man must either say what is true or say nothing. Is not that your position?</p></blockquote>

<p>Indeed, Dionysodorus agrees that "there is no such thing as false opinion ... there is no such thing as ignorance", and demands of Socrates to "Refute me." Socrates responds "But how can I refute you, if, as you say, to tell a falsehood is impossible?".<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="In_formal_logic">In formal logic</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Note: The symbol <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 \bot }">

</mstyle>

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

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</math></span><img src="./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> (<a href="Falsum" class="mw-redirect" title="Falsum">falsum</a>) represents an arbitrary contradiction, with the dual <a href="Tee_(symbol)" title="Tee (symbol)">tee</a> symbol <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 \top }">

</mstyle>

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

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</math></span><img src="./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> used to denote an arbitrary tautology. Contradiction is sometimes symbolized by "O<i>pq</i>", and tautology by "V<i>pq</i>". The turnstile symbol, <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 \vdash }">

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

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</math></span><img src="./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> is often read as "yields" or "proves".</div>

<p>In classical logic, particularly in <a href="Propositional_logic" title="Propositional logic">propositional</a> and <a href="First-order_logic" title="First-order logic">first-order logic</a>, a proposition <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 \varphi }">

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> is a contradiction <a href="If_and_only_if" title="If and only if">if and only if</a> <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 \varphi \vdash \bot }">

</mstyle>

</mrow>

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

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</math></span><img src="./cb2a11d466849d8b206a4a6c934264fe16835bb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.039ex; height:2.676ex;" alt="{\displaystyle \varphi \vdash \bot }" loading="lazy"></span>. Since for contradictory <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 \varphi }">

</mstyle>

</mrow>

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> it is true that <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 \vdash \varphi \rightarrow \psi }">

</mstyle>

</mrow>

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

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</math></span><img src="./60875b3cd620fe51bd3d52cde99877beece9a333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.713ex; height:2.676ex;" alt="{\displaystyle \vdash \varphi \rightarrow \psi }" loading="lazy"></span> for all <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>

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</math></span><img src="./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> (because <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 \bot \vdash \psi }">

</mstyle>

</mrow>

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

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</math></span><img src="./64ffc2cef9ffd8f84c60c0c55f393d8e5f8df8b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.032ex; height:2.509ex;" alt="{\displaystyle \bot \vdash \psi }" loading="lazy"></span>), one may prove any proposition from a set of axioms which contains contradictions. This is called the "<a href="Principle_of_explosion" title="Principle of explosion">principle of explosion</a>", or "ex falso quodlibet" ("from falsity, anything follows").<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>In a <a href="Completeness_(logic)" title="Completeness (logic)">complete</a> logic, a formula is contradictory if and only if it is <a href="Unsatisfiable" class="mw-redirect" title="Unsatisfiable">unsatisfiable</a>.

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<div class="mw-heading mw-heading3"><h3 id="Proof_by_contradiction">Proof by contradiction</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Proof_by_contradiction" title="Proof by contradiction">Proof by contradiction</a></div>

<p>For a set of consistent premises <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 \Sigma }">

</mstyle>

</mrow>

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

</semantics>

</mstyle>

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span>, it is true in <a href="Classical_logic" title="Classical logic">classical logic</a> that <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 \Sigma \vdash \varphi }">

</mstyle>

</mrow>

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

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</math></span><img src="./93affa0a70251d15804ea808eb6b56e135585f50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.909ex; height:2.676ex;" alt="{\displaystyle \Sigma \vdash \varphi }" loading="lazy"></span> (i.e., <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 \Sigma }">

</mstyle>

</mrow>

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

</semantics>

</mstyle>

</mrow>

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span>) if and only if <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 \Sigma \cup \{\neg \varphi \}\vdash \bot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }</annotation>

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</math></span><img src="./feae6450e7beacc7aece5f6bc6f09a91a81d14f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.175ex; height:2.843ex;" alt="{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }" loading="lazy"></span> (i.e., <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 \Sigma }">

</mstyle>

</mrow>

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

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</mstyle>

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

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</math></span><img src="./b6c4b627f483f3efdd7c27e708f6f37c20c63502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.07ex; height:2.176ex;" alt="{\displaystyle \neg \varphi }" loading="lazy"></span> leads to a contradiction). Therefore, a <a href="Proof_(logic)" class="mw-redirect" title="Proof (logic)">proof</a> that <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 \Sigma \cup \{\neg \varphi \}\vdash \bot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }</annotation>

</semantics>

</math></span><img src="./feae6450e7beacc7aece5f6bc6f09a91a81d14f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.175ex; height:2.843ex;" alt="{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }" loading="lazy"></span> also proves that <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 \varphi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> is true under the premises <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 \Sigma }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"></span>. The use of this fact forms the basis of a <a href="Proof_techniques" class="mw-redirect" title="Proof techniques">proof technique</a> called <a href="Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a>, which mathematicians use extensively to establish the validity of a wide range of theorems. This applies only in a logic where the <a href="Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a> <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\vee \neg A}">

</mstyle>

</mrow>

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

</semantics>

</p><p>Using <a href="Minimal_logic" title="Minimal logic">minimal logic</a>, a logic with similar axioms to classical logic but without <i>ex falso quodlibet</i> and proof by contradiction, we can investigate the axiomatic strength and properties of various rules that treat contradiction by considering theorems of classical logic that are not theorems of minimal logic.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Each of these extensions leads to an <a href="Intermediate_logic" title="Intermediate logic">intermediate logic</a>:

</p>

<ol><li>Double-negation elimination (DNE) is the strongest principle, axiomatized <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 \neg A\implies A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./07f0de4b75feccf724289b9534f5ea9d4796044a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.973ex; height:2.176ex;" alt="{\displaystyle \neg \neg A\implies A}" loading="lazy"></span>, and when it is added to minimal logic yields classical logic.</li>

<li>Ex falso quodlibet (EFQ), axiomatized <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 \bot \implies A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0af677ecbf6b939e434be7063a506422653dc7a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.937ex; height:2.176ex;" alt="{\displaystyle \bot \implies A}" loading="lazy"></span>, licenses many consequences of negations, but typically does not help to infer propositions that do not involve absurdity from consistent propositions that do. When added to minimal logic, EFQ yields <a href="Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. EFQ is equivalent to <i>ex contradiction quodlibet</i>, axiomatized <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\land \neg A\implies B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2de31b9d2e4db9afffe15b8a42f0115d69432625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.769ex; height:2.176ex;" alt="{\displaystyle A\land \neg A\implies B}" loading="lazy"></span>, over minimal logic.</li>

<li><a href="Peirce's_law" title="Peirce's law">Peirce's rule</a> (PR) is an axiom <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\implies B)\implies A)\implies A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./bec1c55819a500c8b7116934538a37ccf446d466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.771ex; height:2.843ex;" alt="{\displaystyle ((A\implies B)\implies A)\implies A}" loading="lazy"></span> that captures proof by contradiction without explicitly referring to absurdity. Minimal logic + PR + EFQ yields classical logic.</li>

<li>The Gödel-Dummett (GD) axiom <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\implies B\vee B\implies A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ac78081b0f6e3ec4ca66b38f8e7dd8be18113aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.369ex; height:2.176ex;" alt="{\displaystyle A\implies B\vee B\implies A}" loading="lazy"></span>, whose most simple reading is that there is a linear order on truth values. Minimal logic + GD yields <a href="G%C3%B6del-Dummett_logic" class="mw-redirect" title="Gödel-Dummett logic">Gödel-Dummett logic</a>. Peirce's rule entails but is not entailed by GD over minimal logic.</li>

<li>Law of the excluded middle (LEM), axiomatised <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\vee \neg A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./323f5c2a8924ef707dcc24cee3bce1b148ec1367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.619ex; height:2.176ex;" alt="{\displaystyle A\vee \neg A}" loading="lazy"></span>, is the most often cited formulation of the <a href="Principle_of_bivalence" title="Principle of bivalence">principle of bivalence</a>, but in the absence of EFQ it does not yield full classical logic. Minimal logic + LEM + EFQ yields classical logic. PR entails but is not entailed by LEM in minimal logic. If the formula B in Peirce's rule is restricted to absurdity, giving the axiom schema <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\implies A)\implies A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba03ffc1838513d69ca130c83789c7f7403f1928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.361ex; height:2.843ex;" alt="{\displaystyle (\neg A\implies A)\implies A}" loading="lazy"></span>, the scheme is equivalent to LEM over minimal logic.</li>

<li>Weak law of the excluded middle (WLEM) is axiomatised <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\vee \neg \neg A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ef67eb5aaa11449121671846b3dba54794dde62b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.72ex; height:2.176ex;" alt="{\displaystyle \neg A\vee \neg \neg A}" loading="lazy"></span> and yields a system where disjunction behaves more like in classical logic than intuitionistic logic, i.e. the <a href="Disjunction_and_existence_properties" title="Disjunction and existence properties">disjunction and existence properties</a> don't hold, but where use of non-intuitionistic reasoning is marked by occurrences of double-negation in the conclusion. LEM entails but is not entailed by WLEM in minimal logic. WLEM is equivalent to the instance of <a href="De_Morgan's_law" class="mw-redirect" title="De Morgan's law">De Morgan's law</a> that distributes negation over conjunction: <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 B)\iff (\neg A)\vee (\neg B)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b6266cb8dd6a3528c68da19b03f0ba6e746f9e47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.155ex; height:2.843ex;" alt="{\displaystyle \neg (A\land B)\iff (\neg A)\vee (\neg B)}" loading="lazy"></span>.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Symbolic_representation">Symbolic representation</h3></div>

<p>In mathematics, the symbol used to represent a contradiction within a proof varies.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Some symbols that may be used to represent a contradiction include ↯, Opq, <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 \Rightarrow \Leftarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0910f38ab10fa4e15bb398d89588f6a2147e6d0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.648ex; height:1.843ex;" alt="{\displaystyle \Rightarrow \Leftarrow }" 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 \leftrightarrow \ \!\!\!\!\!\!\!}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e0fa602e48c1a1c87ca9ced0a053da531f9c6f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -2.323ex; width:3.162ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow \ \!\!\!\!\!\!\!}" loading="lazy"></span>/ , and ※; in any symbolism, a contradiction may be substituted for the truth value "<a href="False_(logic)" title="False (logic)">false</a>", as symbolized, for instance, by "0" (as is common in <a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a>). It is not uncommon to see <a href="Q.E.D." title="Q.E.D.">Q.E.D.</a>, or some of its variants, immediately after a contradiction symbol. In fact, this often occurs in a proof by contradiction to indicate that the original assumption was proved false—and hence that its negation must be true.

</p>

<div class="mw-heading mw-heading3"><h3 id="The_notion_of_contradiction_in_an_axiomatic_system_and_a_proof_of_its_consistency">The notion of contradiction in an axiomatic system and a proof of its consistency</h3></div>

<p>In general, a <a href="Consistency_proof" class="mw-redirect" title="Consistency proof">consistency proof</a> requires the following two things:

</p>

<ol><li>An <a href="Axiomatic_system" title="Axiomatic system">axiomatic system</a></li>

<li>A demonstration that it is <i>not</i> the case that both the formula <i>p</i> and its negation <i>~p</i> can be derived in the system.</li></ol>

<p>But by whatever method one goes about it, all consistency proofs would <i>seem</i> to necessitate the primitive notion of <i>contradiction.</i> Moreover, it <i>seems</i> as if this notion would simultaneously have to be "outside" the formal system in the definition of tautology.

</p><p>When <a href="Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a>, in his 1921 "Introduction to a General Theory of Elementary Propositions", extended his proof of the consistency of the <a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">propositional calculus</a> (i.e. the logic) beyond that of <i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> (PM), he observed that with respect to a <i>generalized</i> set of postulates (i.e. axioms), he would no longer be able to automatically invoke the notion of "contradiction"—such a notion might not be contained in the postulates:

</p>

<blockquote class="templatequote"><p>The prime requisite of a set of postulates is that it be consistent. Since the ordinary notion of consistency involves that of contradiction, which again involves negation, and since this function does not appear in general as a primitive in [the <i>generalized</i> set of postulates] a new definition must be given.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></p></blockquote>

<p>Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute Proof of Consistency", offered by <a href="Ernest_Nagel" title="Ernest Nagel">Ernest Nagel</a> and <a href="James_R._Newman" title="James R. Newman">James R. Newman</a> in their 1958 <i><a href="G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a>'s Proof</i>. They too observed a problem with respect to the notion of "contradiction" with its usual "truth values" of "truth" and "falsity". They observed that:

</p>

<blockquote class="templatequote"><p>The property of being a tautology has been defined in notions of truth and falsity. Yet these notions obviously involve a reference to something <i>outside</i> the formula calculus. Therefore, the procedure mentioned in the text in effect offers an <i>interpretation</i> of the calculus, by supplying a model for the system. This being so, the authors have not done what they promised, namely, "<b>to define a property of formulas in terms of purely structural features of the formulas themselves</b>". [Indeed] ... proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></p></blockquote>

<p>Given some "primitive formulas" such as PM's primitives S<sub>1</sub> V S<sub>2</sub> [inclusive OR] and ~S (negation), one is forced to define the axioms in terms of these primitive notions. In a thorough manner, Post demonstrates in PM, and defines (as do Nagel and Newman, see below) that the property of <i>tautologous</i> – as yet to be defined – is "inherited": if one begins with a set of tautologous axioms (postulates) and a <a href="Deduction_system" class="mw-redirect" title="Deduction system">deduction system</a> that contains <a href="Substitution_(logic)" title="Substitution (logic)">substitution</a> and <a href="Modus_ponens" title="Modus ponens">modus ponens</a>, then a <i>consistent</i> system will yield only tautologous formulas.

</p><p>On the topic of the definition of <i>tautologous</i>, Nagel and Newman create two <a href="Mutually_exclusive" class="mw-redirect" title="Mutually exclusive">mutually exclusive</a> and <a href="Collectively_exhaustive_events" title="Collectively exhaustive events">exhaustive</a> classes K<sub>1</sub> and K<sub>2</sub>, into which fall (the outcome of) the axioms when their variables (e.g. S<sub>1</sub> and S<sub>2</sub> are assigned from these classes). This also applies to the primitive formulas. For example: "A formula having the form S<sub>1</sub> V S<sub>2</sub> is placed into class K<sub>2</sub>, if both S<sub>1</sub> and S<sub>2</sub> are in K<sub>2</sub>; otherwise it is placed in K<sub>1</sub>", and "A formula having the form ~S is placed in K<sub>2</sub>, if S is in K<sub>1</sub>; otherwise it is placed in K<sub>1</sub>".<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p>Hence Nagel and Newman can now define the notion of <i><a href="Tautology_(logic)" title="Tautology (logic)">tautologous</a></i>: "a formula is a tautology if and only if it falls in the class K<sub>1</sub>, no matter in which of the two classes its elements are placed".<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> This way, the property of "being tautologous" is described—without reference to a model or an interpretation.

</p>

<blockquote class="templatequote"><p>For example, given a formula such as ~S<sub>1</sub> V S<sub>2</sub> and an assignment of K<sub>1</sub> to S<sub>1</sub> and K<sub>2</sub> to S<sub>2</sub> one can evaluate the formula and place its outcome in one or the other of the classes. The assignment of K<sub>1</sub> to S<sub>1</sub> places ~S<sub>1</sub> in K<sub>2</sub>, and now we can see that our assignment causes the formula to fall into class K<sub>2</sub>. Thus by definition our formula is not a tautology.</p></blockquote>

<p>Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulas derived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from either class K<sub>1</sub> or K<sub>2</sub>, the deduction violates the inheritance characteristic of tautology (i.e., the derivation must yield an evaluation of a formula that will fall into class K<sub>1</sub>). From this, Post was able to derive the following definition of inconsistency—<i>without the use of the notion of contradiction</i>:

</p>

<blockquote class="templatequote"><p>Definition. <i>A system will be said to be inconsistent if it yields the assertion of the unmodified variable p [S in the Newman and Nagel examples].</i></p></blockquote>

<p>In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes. An axiomatic system need not include the notion of "contradiction".<sup id="cite_ref-jstor1921_12-0" class="reference"><a href="#cite_note-jstor1921-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 177">: 177 </span></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Philosophy">Philosophy</h2></div>

<p>Adherents of the <a href="Epistemology" title="Epistemology">epistemological</a> theory of <a href="Coherentism" title="Coherentism">coherentism</a> typically claim that as a necessary condition of the justification of a <a href="Belief" title="Belief">belief</a>, that belief must form a part of a logically non-contradictory <a href="System" title="System">system</a> of beliefs. Some <a href="Dialetheism" title="Dialetheism">dialetheists</a>, including <a href="Graham_Priest" title="Graham Priest">Graham Priest</a>, have argued that coherence may not require consistency.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Pragmatic_contradictions">Pragmatic contradictions</h3></div>

<p>A pragmatic contradiction occurs when the very statement of the argument contradicts the claims it purports. An inconsistency arises, in this case, because the act of utterance, rather than the content of what is said, undermines its conclusion.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Dialectical_materialism">Dialectical materialism</h3></div>

<p>In <a href="Dialectical_materialism" title="Dialectical materialism">dialectical materialism</a>: Contradiction—as derived from <a href="Hegelianism" class="mw-redirect" title="Hegelianism">Hegelianism</a>—usually refers to an opposition inherently existing within one realm, one unified force or object. This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to <a href="Marxist_theory" class="mw-redirect" title="Marxist theory">Marxist theory</a>, such a contradiction can be found, for example, in the fact that:

</p>

<ul><li>(a) enormous wealth and productive powers coexist alongside:</li>

<li>(b) extreme poverty and misery;</li>

<li>(c) the existence of (a) being contrary to the existence of (b).</li></ul>

<p>Hegelian and Marxist theories stipulate that the <a href="Dialectic#Modern_philosophy" title="Dialectic">dialectic</a> nature of history will lead to the <a href="Aufheben" title="Aufheben">sublation</a>, or <a href="Thesis%2C_antithesis%2C_synthesis" class="mw-redirect" title="Thesis, antithesis, synthesis">synthesis</a>, of its contradictions. Marx therefore postulated that history would logically make <a href="Capitalism" title="Capitalism">capitalism</a> evolve into a <a href="Socialism" title="Socialism">socialist</a> society where the <a href="Means_of_production" title="Means of production">means of production</a> would equally serve the <a href="Proletariat" title="Proletariat">working and producing class</a> of society, thus resolving the prior contradiction between (a) and (b).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Outside_formal_logic">Outside formal logic</h2></div>

<p><a href="Colloquialism" title="Colloquialism">Colloquial usage</a> can label actions or statements as contradicting each other when due (or perceived as due) to <a href="Presupposition" title="Presupposition">presuppositions</a> which are contradictory in the logical sense.

</p><p><a href="Proof_by_contradiction" title="Proof by contradiction">Proof by contradiction</a> is used in <a href="Mathematics" title="Mathematics">mathematics</a> to construct <a href="Mathematical_proof" title="Mathematical proof">proofs</a>.

</p>

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<ul><li><a href="Argument_Clinic" title="Argument Clinic">"Argument Clinic"</a> – Monty Python sketch, in which one of the two disputants repeatedly uses only contradictions in his argument</li>

<li><a href="Auto-antonym" class="mw-redirect" title="Auto-antonym">Auto-antonym</a> – Word that has two opposing meanings<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Contrary_(logic)" class="mw-redirect" title="Contrary (logic)">Contrary (logic)</a> – Type of logic diagram<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Contronym" title="Contronym">Contronym</a> – Word that has two opposing meanings</li>

<li><a href="Dialetheism" title="Dialetheism">Dialetheism</a> – View that there are statements that are both true and false</li>

<li><a href="Double_standard" title="Double standard">Double standard</a> – Inconsistent application of principles</li>

<li><a href="Doublethink" title="Doublethink">Doublethink</a> – Simultaneously accepting two mutually contradictory beliefs as correct</li>

<li><a href="Paul_Graham_(programmer)#Graham's_hierarchy_of_disagreement" title="Paul Graham (programmer)">Graham's hierarchy of disagreement</a></li>

<li><a href="Irony" title="Irony">Irony</a> – Literary and rhetorical device or general attitude towards life</li>

<li><a href="Law_of_noncontradiction" title="Law of noncontradiction">Law of noncontradiction</a> – Logic theorem</li>

<li><a href="On_Contradiction" title="On Contradiction">On Contradiction</a> – 1937 essay by Mao Zedong</li>

<li><a href="Oxymoron" title="Oxymoron">Oxymoron</a> – Figure of speech</li>

<li><a href="Paraconsistent_logic" title="Paraconsistent logic">Paraconsistent logic</a> – Type of formal logic without explosion principle</li>

<li><a href="Paradox" title="Paradox">Paradox</a> – Logically self-contradictory statement</li>

<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a> – In logic, a statement which is always true</li>

<li><a href="TRIZ" title="TRIZ">TRIZ</a> – Problem-solving tools</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2></div>

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</style><cite id="CITEREFHorn2018" class="citation cs2">Horn, Laurence R. (2018), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/win2018/entries/contradiction/">"Contradiction"</a>, in Zalta, Edward N. (ed.), <i>The Stanford Encyclopedia of Philosophy</i> (Winter 2018 ed.), Metaphysics Research Lab, Stanford University<span class="reference-accessdate">, retrieved <span class="nowrap">2019-12-10</span></span></cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.thefreedictionary.com/Contradiction+(logic)">"Contradiction (logic)"</a>. <i>TheFreeDictionary.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-14</span></span>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.skillfulreasoning.com/propositional_logic/properties_of_propositions.html">"Tautologies, contradictions, and contingencies"</a>. <i>www.skillfulreasoning.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-14</span></span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Dialog <i>Euthydemus</i> from <i>The Dialogs of Plato translated by <a href="Benjamin_Jowett" title="Benjamin Jowett">Benjamin Jowett</a></i> appearing in: BK 7 <i>Plato</i>: <a href="Robert_Maynard_Hutchins" title="Robert Maynard Hutchins">Robert Maynard Hutchins</a>, editor in chief, 1952, <i><a href="Great_Books_of_the_Western_World" title="Great Books of the Western World">Great Books of the Western World</a></i>, <a href="Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a>, Inc., <a href="Chicago" title="Chicago">Chicago</a>.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095804354">"Ex falso quodlibet - Oxford Reference"</a>. <i>www.oxfordreference.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-10</span></span>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Diener and Maarten McKubre-Jordens, 2020. <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1606.08092">Classifying Material Implications over Minimal Logic</a>. <a href="Archive_for_Mathematical_Logic" title="Archive for Mathematical Logic">Archive for Mathematical Logic</a> 59 (7-8):905-924.</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFPakin2017" class="citation web cs1">Pakin, Scott (January 19, 2017). <a rel="nofollow" class="external text" href="http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf">"The Comprehensive LATEX Symbol List"</a> <span class="cs1-format">(PDF)</span>. <i>ctan.mirror.rafal.ca</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-10</span></span>.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Post 1921 "Introduction to a General Theory of Elementary Propositions" in van Heijenoort 1967:272.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">boldface italics added, Nagel and Newman:109-110.</span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Nagel and Newman:110-111</span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Nagel and Newman:111</span>

</li>

<li id="cite_note-jstor1921-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-jstor1921_12-0">^</a></b></span> <span class="reference-text">Emil L. Post <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2370324">(1921) Introduction to a General Theory of Elementary Propositions</a> <i>American Journal of Mathematics</i> <b>43</b> (3):163—185 (1921) The Johns Hopkins University Press </span>

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<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=bId-cRNUWdAC&q=contradiction&pg=PP1">In Contradiction: A Study of the Transconsistent</a> By Graham Priest</span>

</li>

<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFStoljar2006" class="citation book cs1">Stoljar, Daniel (2006). <i>Ignorance and Imagination</i>. Oxford University Press - U.S. p. 87. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-530658-9</bdi>.</cite></span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFSørensen2006" class="citation journal cs1">Sørensen, Michael Kuur (2006). <a rel="nofollow" class="external text" href="https://journals.aau.dk/index.php/ijis/article/download/181/121">"Capital and Labour: Can the Conflict Be Solved?"</a>. <i>The Interdisciplinary Journal of International Studies</i>. <b>4</b> (1): <span class="nowrap">29–</span>48<span class="reference-accessdate">. Retrieved <span class="nowrap">28 May</span> 2017</span>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><a href="J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Józef Maria Bocheński</a> 1960 <i>Précis of Mathematical Logic</i>, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.</li>

<li>Jean van Heijenoort 1967 <i>From Frege to Gödel: A Source Book in Mathematical Logic 1879-1931</i>, Harvard University Press, Cambridge, MA, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-674-32449-8</bdi> (pbk.)</li>

<li>Ernest Nagel and James R. Newman 1958 <i>Gödel's Proof</i>, New York University Press, Card Catalog Number: 58-5610.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/contradiction" class="extiw external" title="wiktionary:contradiction">contradiction</a></b></i> or <i><b><a href="https://en.wiktionary.org/wiki/although" class="extiw external" title="wiktionary:although">although</a></b></i> in Wiktionary, the free dictionary.</div></div>

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<div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Contradiction" class="extiw external" title="q:Special:Search/Contradiction">Contradiction</a></b></i>.</div></div>

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<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Contradiction_(inconsistency)">"Contradiction (inconsistency)"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Contradiction,_law_of">"Contradiction, law of"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite id="CITEREFHorn" class="citation encyclopaedia cs1"><a href="Laurence_R._Horn" title="Laurence R. Horn">Horn, Laurence R.</a> <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/contradiction/">"Contradiction"</a>. In <a href="Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite></li></ul>

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</style><div id="Logic605" style="font-size:114%;margin:0 4em"><a href="Logic" title="Logic">Logic</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_logic" title="History of logic">History</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Major fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Logic_in_computer_science" title="Logic in computer science">Computer science</a></li>

<li><a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical</a></li>

<li><a href="Informal_logic" title="Informal logic">Informal</a>

<ul><li><a href="Critical_thinking" title="Critical thinking">Critical thinking</a></li>

<li><a href="Reason" title="Reason">Reason</a></li></ul></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Mathematical</a></li>

<li><a href="Non-classical_logic" title="Non-classical logic">Non-classical</a></li>

<li><a href="Philosophical_logic" title="Philosophical logic">Philosophical</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Argumentation_theory" title="Argumentation theory">Argumentation</a></li>

<li><a href="Metalogic" title="Metalogic">Metalogic</a></li>

<li><a href="Metamathematics" title="Metamathematics">Metamathematics</a></li>

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Foundations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abductive_reasoning" title="Abductive reasoning">Abduction</a></li>

<li><a href="Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li>

<li><a href="Antecedent_(logic)" title="Antecedent (logic)">Antecedent</a></li>

<li><a href="Consequent" title="Consequent">Consequent</a></li>

<li>

<ul><li><a href="Paradox" title="Paradox">Paradox</a></li>

<li><a href="Antinomy" title="Antinomy">Antinomy</a></li></ul></li>

<li><a href="Deductive_reasoning" title="Deductive reasoning">Deduction</a></li>

<li><a href="Deductive_closure" title="Deductive closure">Deductive closure</a></li>

<li><a href="Definition" title="Definition">Definition</a></li>

<li><a href="Description" title="Description">Description</a></li>

<li><a href="Dichotomy" title="Dichotomy">Dichotomy</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Entailment</a>

<ul><li><a href="Linguistic_entailment" title="Linguistic entailment">Linguistic</a></li></ul></li>

<li><a href="Inductive_reasoning" title="Inductive reasoning">Induction</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Necessity_and_sufficiency" title="Necessity and sufficiency">Necessity and sufficiency</a></li>

<li><a href="Premise" title="Premise">Premise</a></li>

<li><a href="Probability" title="Probability">Probability</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Reference" title="Reference">Reference</a></li>

<li><a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">Statement</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Truth" title="Truth">Truth</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Index_of_logic_articles" class="mw-redirect" title="Index of logic articles">Topics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li>

<li><a href="List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">Boolean algebra</a></li>

<li><a href="List_of_set_theory_topics" title="List of set theory topics">Set theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_logicians" title="List of logicians">Logicians</a></li>

<li><a href="List_of_rules_of_inference" title="List of rules of inference">Rules of inference</a></li>

<li><a href="List_of_paradoxes" title="List of paradoxes">Paradoxes</a></li>

<li><a href="List_of_fallacies" title="List of fallacies">Fallacies</a></li>

<li><a href="List_of_logic_symbols" title="List of logic symbols">Logic symbols</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li>

<li><span class="noviewer" typeof="mw:File"><span title="Outline"></span></span> <a href="Outline_of_logic" title="Outline of logic">Outline</a></li>

<li><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3APhilosophy" title="Portal:Philosophy">Portal</a></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> WikiProject</li>

<li><a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Special:Recentchangeslinked&target=Template:Logic&hidebots=0">changes</a></li></ul>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic344" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Axiom" title="Axiom">Axiom</a>

<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>

<li><a href="First-order_logic" title="First-order logic">First-order logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li>

<li><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (list)<br> and <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li>

<li><a href="Tarski's_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li>

<li><a href="Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li>

<li>Cantor's <a href="Cantor's_theorem" title="Cantor's theorem">theorem,</a> <a href="Cantor's_paradox" title="Cantor's paradox">paradox</a> and <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li>

<li><a href="Compactness_theorem" title="Compactness theorem">Compactness</a></li>

<li><a href="Halting_problem" title="Halting problem">Halting problem</a></li>

<li><a href="Lindstr%C3%B6m's_theorem" title="Lindström's theorem">Lindström's</a></li>

<li><a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li>

<li><a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional95" scope="row" class="navbox-group" style="width:1%"><a href="Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical logic</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li>

<li><a href="Consistency" title="Consistency">Consistency</a>

<ul><li><a href="Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li>

<li><a href="Argument" title="Argument">Argument</a></li>

<li><a href="Soundness" title="Soundness">Soundness</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li>

<li><a href="Syllogism" title="Syllogism">Syllogism</a></li>

<li><a href="Square_of_opposition" title="Square of opposition">Square of opposition</a></li>

<li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li>

<li><a href="Boolean_function" title="Boolean function">Boolean functions</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connectives</a></li>

<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>

<li><a href="Propositional_formula" title="Propositional formula">Propositional formula</a></li>

<li><a href="Truth_table" title="Truth table">Truth tables</a></li>

<li><a href="Many-valued_logic" title="Many-valued logic">Many-valued logic</a>

<li><a href="Finite-valued_logic" title="Finite-valued logic">finite</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="First-order_logic" title="First-order logic">First-order</a>

<li><a href="Second-order_logic" title="Second-order logic">Second-order</a>

<ul><li><a href="Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li>

<li><a href="Higher-order_logic" title="Higher-order logic">Higher-order</a></li>

<li><a href="Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li>

<li><a href="Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li>

<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>

<li>(<a href="Urelement" title="Urelement">Ur-</a>)<a href="Element_(mathematics)" title="Element (mathematics)">Element</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>

<li><a href="Extensionality" title="Extensionality">Extensionality</a></li>

<li><a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li>

<li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a>

<ul><li><a href="Equivalence_relation" title="Equivalence relation">equivalence</a></li>

<li><a href="Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li>

<li>Set operations:

<ul><li><a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li>

<li><a href="Union_(set_theory)" title="Union (set theory)">union</a></li>

<li><a href="Complement_(set_theory)" title="Complement (set theory)">complement</a></li>

<li><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></li>

<li><a href="Power_set" title="Power set">power set</a></li>

<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Countable_set" title="Countable set">Countable</a></li>

<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>

<li><a href="Empty_set" title="Empty set">Empty</a></li>

<li><a href="Inhabited_set" title="Inhabited set">Inhabited</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a></li>

<li><a href="Infinite_set" title="Infinite set">Infinite</a></li>

<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>

<li><a href="Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li>

<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>

<li><a href="Universal_set" title="Universal set">Universal</a></li>

<li><a href="Universe_(mathematics)" title="Universe (mathematics)">Universe</a>

<ul><li><a href="Constructible_universe" title="Constructible universe">constructible</a></li>

<li><a href="Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li>

<li><a href="Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="Map_(mathematics)" title="Map (mathematics)">Map</a>

<ul><li><a href="Domain_of_a_function" title="Domain of a function">domain</a></li>

<li><a href="Codomain" title="Codomain">codomain</a></li>

<li><a href="Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li>

<li><a href="Injective_function" title="Injective function">In</a>/<a href="Surjective_function" title="Surjective function">Sur</a>/<a href="Bijection" title="Bijection">Bi</a>-jection</li>

<li><a href="Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li>

<li><a href="Isomorphism" title="Isomorphism">Isomorphism</a></li>

<li><a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li>

<li><a href="Enumeration" title="Enumeration">Enumeration</a></li>

<li><a href="Large_cardinal" title="Large cardinal">Large cardinal</a>

<ul><li><a href="Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li>

<li><a href="Aleph_number" title="Aleph number">Aleph number</a></li>

<li><a href="Operation_(mathematics)" title="Operation (mathematics)">Operation</a>

<ul><li><a href="Binary_operation" title="Binary operation">binary</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a>

<ul><li><a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li>

<li><a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li>

<li><a href="General_set_theory" title="General set theory">General</a></li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>

<li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li>

<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>

<li><a href="New_Foundations" title="New Foundations">New Foundations</a></li>

<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li>

<li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li>

<li><a href="Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li>

<li><a href="Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Formal_system" title="Formal system">Formal systems</a> (<a href="List_of_formal_systems" title="List of formal systems"><span style="font-size: 85%;">list</span></a>),<br><a href="Formal_language" title="Formal language">language</a> and <a href="Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li>

<li><a href="Arity" title="Arity">Arity</a></li>

<li><a href="Automata_theory" title="Automata theory">Automata</a></li>

<li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a></li>

<li><a href="Expression_(mathematics)" title="Expression (mathematics)">Expression</a>

<ul><li><a href="Ground_expression" title="Ground expression">ground</a></li></ul></li>

<li><a href="Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a>

<ul><li><a href="Extension_by_definitions" class="mw-redirect" title="Extension by definitions">by definition</a></li>

<li><a href="Conservative_extension" title="Conservative extension">conservative</a></li></ul></li>

<li><a href="Finitary_relation" title="Finitary relation">Relation</a></li>

<li><a href="Formation_rule" title="Formation rule">Formation rule</a></li>

<li><a href="Formal_grammar" title="Formal grammar">Grammar</a></li>

<li><a href="Well-formed_formula" title="Well-formed formula">Formula</a>

<ul><li><a href="Atomic_formula" title="Atomic formula">atomic</a></li>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li>

<li><a href="Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li>

<li><a href="Formal_language" title="Formal language">Language</a></li>

<li><a href="Metalanguage" title="Metalanguage">Metalanguage</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connective</a>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a>

<ul><li><a href="Functional_predicate" title="Functional predicate">functional</a></li>

<li><a href="Predicate_variable" title="Predicate variable">variable</a></li>

<li><a href="Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a>

<ul><li><a href="Atomic_sentence" title="Atomic sentence">atomic</a></li>

<li><a href="Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li>

<li><a href="Signature_(logic)" title="Signature (logic)">Signature</a></li>

<li><a href="String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Symbol_(formal)" title="Symbol (formal)">Symbol</a>

<ul><li><a href="Uninterpreted_function" title="Uninterpreted function">function</a></li>

<li><a href="Logical_constant" title="Logical constant">logical/constant</a></li>

<li><a href="Non-logical_symbol" title="Non-logical symbol">non-logical</a></li>

<li><a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="Axiomatic_system" title="Axiomatic system">axiomatic<br>systems</a> <span style="font-size: 85%;">(<a href="List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>of <a href="True_arithmetic" title="True arithmetic">arithmetic</a>:

<ul><li><a href="Peano_axioms" title="Peano axioms">Peano</a></li>

<li><a href="Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li>

<li><a href="Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li>

<li><a href="Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li>

<li><a href="Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li>

<li><a href="Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li>

<li>of the <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a>

<ul><li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li>

<li>of <a href="Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a>

<ul><li><a href="Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li>

<li><a href="Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li>

<li>of <a href="Foundations_of_geometry" title="Foundations of geometry">geometry</a>:

<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a>:

<ul><li><a href="Euclid's_Elements" title="Euclid's Elements"><i>Elements</i></a></li>

<li><a href="Hilbert's_axioms" title="Hilbert's axioms">Hilbert's</a></li>

<li><a href="Tarski's_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li>

<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul>

<ul><li><i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Natural_deduction" title="Natural deduction">Natural deduction</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Rule_of_inference" title="Rule of inference">Rule of inference</a></li>

<li><a href="Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Formal_system" title="Formal system">Systems</a>

<ul><li><a href="Axiomatic_system" title="Axiomatic system">axiomatic</a></li>

<li><a href="Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li>

<li><a href="Hilbert_system" title="Hilbert system">Hilbert</a>

<li><a href="Complete_theory" title="Complete theory">Complete theory</a></li>

<li><a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li>

<li><a href="Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li>

<li><a href="Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li>

<li><a href="Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li>

<li><a href="Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a>

<ul><li><a href="Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li>

<li><a href="Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a>

<ul><li><a href="Elementary_equivalence" title="Elementary equivalence">equivalence</a></li>

<li><a href="Finite_model_theory" title="Finite model theory">finite</a></li>

<li><a href="Saturated_model" title="Saturated model">saturated</a></li>

<li><a href="Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li>

<li><a href="Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li>

<li><a href="Non-standard_model" title="Non-standard model">Non-standard model</a>

<ul><li><a href="Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li>

<li><a href="Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a>

<ul><li><a href="Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li>

<li><a href="Categorical_theory" title="Categorical theory">Categorical theory</a></li>

<li><a href="Model_complete_theory" title="Model complete theory">Model complete theory</a></li>

<li><a href="Satisfiability" title="Satisfiability">Satisfiability</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li>

<li><a href="Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a>

<ul><li><a href="Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li>

<li><a href="Tarski's_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li>

<li><a href="Kripke's_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li>

<li><a href="T-schema" title="T-schema">T-schema</a></li>

<li><a href="Transfer_principle" title="Transfer principle">Transfer principle</a></li>

<li><a href="Truth_predicate" title="Truth predicate">Truth predicate</a></li>

<li><a href="Truth_value" title="Truth value">Truth value</a></li>

<li><a href="Ultraproduct" title="Ultraproduct">Ultraproduct</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Church_encoding" title="Church encoding">Church encoding</a></li>

<li><a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li>

<li><a href="Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li>

<li><a href="Computable_function" title="Computable function">Computable function</a></li>

<li><a href="Computable_set" title="Computable set">Computable set</a></li>

<li><a href="Decision_problem" title="Decision problem">Decision problem</a>

<ul><li><a href="Decidability_(logic)" title="Decidability (logic)">decidable</a></li>

<li><a href="Undecidable_problem" title="Undecidable problem">undecidable</a></li>

<li><a href="P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li>

<li><a href="Recursion" title="Recursion">Recursion</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li>

<li><a href="Turing_machine" title="Turing machine">Turing machine</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_logic" title="Abstract logic">Abstract logic</a></li>

<li><a href="Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Category_theory" title="Category theory">Category theory</a></li>

<li><a href="Concrete_category" title="Concrete category">Concrete</a>/<a href="Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li>

<li><a href="Category_of_sets" title="Category of sets">Category of sets</a></li>

<li><a href="History_of_logic" title="History of logic">History of logic</a></li>

<li><a href="History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a>

<ul><li><a href="Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li>

<li><a href="Logicism" title="Logicism">Logicism</a></li>

<li><a href="Mathematical_object" title="Mathematical object">Mathematical object</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li>

<li><a href="Supertask" title="Supertask">Supertask</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="‌Logical_truth_⊤61" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="‌Logical_truth_⊤61" style="font-size:114%;margin:0 4em">‌<a href="Logical_truth" title="Logical truth">Logical truth</a> ⊤</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Functional:</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Truth_value" title="Truth value">Truth value</a></li>

<li><a href="Truth_function" title="Truth function">Truth function</a></li>

<li>⊨ <a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="3" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formal:</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li>

<li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Negation </th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><b>⊥</b> <a href="False_(logic)" title="False (logic)">False</a></li>

<li><a href="Consistency" title="Consistency">Inconsistency</a></li></ul>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Common_logical_symbols352" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Common_logical_symbols352" style="font-size:114%;margin:0 4em">Common <a href="List_of_logic_symbols" title="List of logic symbols">logical symbols</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0;background:transparent;color:inherit;"><div style="padding:0px"><table class="navbox-columns-table" style="border-spacing: 0px; text-align:left;width:100%;"><tbody><tr style="vertical-align:top"><td class="navbox-list" style="padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>