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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size: 130%; margin: 6px 0px 6px 0px; background: #ddf;"><a href="Logical_connective" title="Logical connective">Logical connectives</a></th></tr><tr><td class="sidebar-content">

<mover>

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

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</math></span><img src="./8eab858e54d8de87e36fc80a991b32e74201a600.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.001ex; height:3.343ex;" alt="{\displaystyle \neg A,-A,{\overline {A}},\sim A}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Logical_conjunction" title="Logical conjunction">AND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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 B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}">

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

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</math></span><img src="./c041e99940ccd418648ea18d200af37e2b3548d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.68ex; height:2.509ex;" alt="{\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Sheffer_stroke" title="Sheffer stroke">NAND</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}">

<mover>

</mover>

</mrow>

<mover>

<mrow>

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</mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}</annotation>

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</math></span><img src="./b05374b45c2316947f052c6a46ca0f1d9381ed0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.98ex; height:3.509ex;" alt="{\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Logical_disjunction" title="Logical disjunction">OR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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,A+B,A\mid B,A\parallel B}">

</mstyle>

</mrow>

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

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</math></span><img src="./a262d8ab1dd1738c2b888661fe847101b624992d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.943ex; height:2.843ex;" alt="{\displaystyle A\lor B,A+B,A\mid B,A\parallel B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Logical_NOR" title="Logical NOR">NOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}">

<mover>

</mover>

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<mover>

<mrow>

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

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</math></span><img src="./331ccd940d0039678505e971d3e13a63fca14354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.663ex; height:3.343ex;" alt="{\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="XNOR_gate" title="XNOR gate">XNOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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\odot B,{\overline {A{\overline {\lor }}B}}}">

<mover>

<mrow>

<mover>

</mover>

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</mover>

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

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</math></span><img src="./7e5a7f5c2cebe8c2903dea347e6ce9223cc47e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.669ex; height:3.843ex;" alt="{\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └ <a href="Logical_biconditional" title="Logical biconditional">equivalent</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}">

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

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</math></span><img src="./73fd8a2bddea3e7553e1905a4b2b8944269d5430.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.916ex; height:2.509ex;" alt="{\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Exclusive_or" title="Exclusive or">XOR</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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{\underline {\lor }}B,A\oplus B}">

</munder>

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</mstyle>

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

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</math></span><img src="./d48ea5022d9d865ea81c6f954cf73429be684009.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.562ex; margin-bottom: -0.776ex; width:12.441ex; height:3.176ex;" alt="{\displaystyle A{\underline {\lor }}B,A\oplus B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> └ nonequivalent</td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}</annotation>

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</math></span><img src="./e31480781c46a0001e81f596615bc56e20d8aaa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.917ex; height:2.676ex;" alt="{\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Material_conditional" title="Material conditional">implies</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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\Rightarrow B,A\supset B,A\rightarrow B}">

</mstyle>

</mrow>

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

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</math></span><img src="./da2d4ee4d40286755cb17f11743dcece3224fa90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.916ex; height:2.509ex;" alt="{\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Material_nonimplication" title="Material nonimplication">nonimplication</a> (<a href="NIMPLY_gate" title="NIMPLY gate">NIMPLY</a>)</td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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\not \Rightarrow B,A\not \supset B,A\nrightarrow B}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}</annotation>

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</math></span><img src="./4d66f3ed3dc468f35292dfe91a75d59b3b5d4915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.917ex; height:2.676ex;" alt="{\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Converse_(logic)" title="Converse (logic)">converse</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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\Leftarrow B,A\subset B,A\leftarrow B}">

</mstyle>

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

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</math></span><img src="./128eb93aed65dd2e3aa1a4aaef4171a44f9a6718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.916ex; height:2.509ex;" alt="{\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B}" loading="lazy"></span></td></tr><tr style="vertical-align:top"><td style="text-align:left;"> <a href="Converse_nonimplication" title="Converse nonimplication">converse nonimplication</a></td><td style="text-align:right;font-size:125%;line-height:0.8em;vertical-align:middle;white-space:nowrap;font-family:serif;"> <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\not \Leftarrow B,A\not \subset B,A\nleftarrow B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}</annotation>

</semantics>

</math></span><img src="./651dce7a12fa2331a8c610ee47b32982552a01f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.917ex; height:2.676ex;" alt="{\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}" loading="lazy"></span></td></tr></tbody></table></td>

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Related concepts</th></tr><tr><td class="sidebar-content">

<div class="hlist" style="line-height:1.3em;"><ul><li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li><li><a href="First-order_logic" title="First-order logic">Predicate logic</a></li><li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li><li><a href="Truth_function" title="Truth function">Truth function</a></li><li><a href="Boolean_function" title="Boolean function">Boolean function</a></li><li><a href="Functional_completeness" title="Functional completeness">Functional completeness</a></li><li><a href="Scope_(logic)" title="Scope (logic)">Scope (logic)</a></li></ul></div></td>

</tr><tr><th class="sidebar-heading" style="background: #eef; text-align: center;">

Applications</th></tr><tr><td class="sidebar-content">

<div class="hlist"><ul><li><a href="Logic_gate" title="Logic gate">Digital logic</a></li><li><a href="Programming_language" title="Programming language">Programming languages</a></li><li><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li><li><a href="Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li></ul></div></td>

</tr><tr><td class="sidebar-below hlist" style="background: #eef; text-align: center;">

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

<p>A <b>truth table</b> is a <a href="Mathematical_table" title="Mathematical table">mathematical table</a> used in <a href="Logic" title="Logic">logic</a>—specifically in connection with <a href="Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a>, <a href="Boolean_function" title="Boolean function">Boolean functions</a>, and <a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">propositional calculus</a>—which sets out the functional values of logical <a href="Expression_(mathematics)" title="Expression (mathematics)">expressions</a> on each of their functional arguments, that is, for each <a href="Valuation_(logic)" title="Valuation (logic)">combination of values taken by their logical variables</a>.<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> In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, <a href="Validity_(logic)" title="Validity (logic)">logically valid</a>.

</p><p>A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, <a href="#Exclusive_disjunction">A</a> <a href="XOR" class="mw-redirect" title="XOR">XOR</a> <a href="#Exclusive_disjunction">B</a>). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values.

</p><p>A proposition's truth table is a graphical representation of its <a href="Truth_function" title="Truth function">truth function</a>. The truth function can be more useful for mathematical purposes, although the same information is encoded in both.

</p><p><a href="Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a> is generally credited with inventing and popularizing the truth table in his <i><a href="Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></i>, which was completed in 1918 and published in 1921.<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> Such a system was also independently proposed in 1921 by <a href="Emil_Leon_Post" title="Emil Leon Post">Emil Leon Post</a>.<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>

</p>

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

<p><a href="Irving_Anellis" title="Irving Anellis">Irving Anellis</a>'s research shows that <a href="C.S._Peirce" class="mw-redirect" title="C.S. Peirce">C.S. Peirce</a> appears to be the earliest logician (in 1883) to devise a truth table matrix.<sup id="cite_ref-Peirce_4-0" class="reference"><a href="#cite_note-Peirce-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>From the summary of Anellis's paper:<sup id="cite_ref-Peirce_4-1" class="reference"><a href="#cite_note-Peirce-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<blockquote><p> In 1997, John Shosky discovered, on the <a href="Verso" class="mw-redirect" title="Verso">verso</a> of a page of the typed transcript of <a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>'s 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883–84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in the <i><a href="American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a></i> in 1885 includes an example of an indirect truth table for the conditional. </p></blockquote>

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

<p>Truth tables can be used to prove many other <a href="Logical_equivalence" title="Logical equivalence">logical equivalences</a>. For example, consider the following truth table:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p\rightarrow q)\equiv (\neg p\vee q)}</annotation>

</semantics>

</math></span><img src="./3d2dccc16cc397016e346627b96f736caad4e464.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.942ex; height:2.843ex;" alt="{\displaystyle (p\rightarrow q)\equiv (\neg p\vee q)}" loading="lazy"></span>

</caption>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e2b198c79234d926cbee42c0f271d903ea55dc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./bcd38fdfa90c931b98234caec5f6d4060f92123c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.372ex; height:2.343ex;" alt="{\displaystyle \neg p\vee q}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\rightarrow q}</annotation>

</semantics>

</math></span><img src="./4e3f9e5de9aaead8d19411bb3ad0dc490a50ec69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\rightarrow q}" loading="lazy"></span>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<p>This demonstrates the fact 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 p\rightarrow q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\rightarrow q}</annotation>

</semantics>

</math></span><img src="./4e3f9e5de9aaead8d19411bb3ad0dc490a50ec69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\rightarrow q}" loading="lazy"></span> is <a href="Logically_equivalent" class="mw-redirect" title="Logically equivalent">logically equivalent</a> to <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 p\vee q}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./bcd38fdfa90c931b98234caec5f6d4060f92123c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.372ex; height:2.343ex;" alt="{\displaystyle \neg p\vee q}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Truth_table_for_logic_gates">Truth table for logic gates</h3></div>

<p>Here is a truth table that gives definitions of each of the 6 possible 2-input <a href="Logic_gate" title="Logic gate">logic gate</a> functions of two Boolean variables P and Q:

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c5690bb4822d8c821a00cfe3c6644b046a884af4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}" loading="lazy"></span></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./57892b87b74754882daffcf850dd8b445b0fc436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\vee Q}" loading="lazy"></span></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\uparrow Q}</annotation>

</semantics>

</math></span><img src="./19a23f528d4498c639ace27b934930a92eceb3db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.037ex; height:2.509ex;" alt="{\displaystyle P\uparrow Q}" loading="lazy"></span></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\downarrow Q}</annotation>

</semantics>

</math></span><img src="./e93d2631c8b8fe533749526b8c5516a367d4fa6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.037ex; height:2.509ex;" alt="{\displaystyle P\downarrow Q}" loading="lazy"></span></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\nleftrightarrow Q}</annotation>

</semantics>

</math></span><img src="./9ee067b1350c8e8b118e2402e87b07841570a66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\nleftrightarrow Q}" loading="lazy"></span></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./43e9c3ed4d9717db81fc3794218f377bea4f6eb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\leftrightarrow Q}" loading="lazy"></span>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td colspan="2">Name<br>(function)

</td>

<td><a href="Logical_conjunction" title="Logical conjunction">AND</a><br>(conjunction)

</td>

<td><a href="Logical_disjunction" title="Logical disjunction">OR</a><br>(disjunction)

</td>

<td><a href="Sheffer_stroke" title="Sheffer stroke">NAND</a><br>(non-conjunction)

</td>

<td><a href="Logical_NOR" title="Logical NOR">NOR</a><br>(non-disjunction)

</td>

<td><a href="Exclusive_or" title="Exclusive or">XOR</a><br>(non-equivalence)

</td>

<td><a href="Logical_biconditional" title="Logical biconditional">XNOR</a><br>(equivalence)

</td></tr>

<tr>

<p><i>where</i> <style data-mw-deduplicate="TemplateStyles:r981673959">

/* start https://en.wikipedia.org/ */

.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}

/* end https://en.wikipedia.org/ */

</style><span class="legend-color mw-no-invert" style="forced-color-adjust:none; background-color:#9EFF9E; color:black;"> T </span> <i>means</i> <b>true</b> <i>and</i> <span class="legend-color mw-no-invert" style="forced-color-adjust:none; background-color:#FFC7C7; color:black;"> F </span> <i>means</i> <b>false</b>

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Condensed_truth_tables_for_binary_operators">Condensed truth tables for binary operators</h3></div>

<p>For binary operators, a condensed form of truth table is also used, where the row headings and the column headings specify the operands and the table cells specify the result. For example, <a href="Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a> uses this condensed truth table notation:

</p>

<table>

</td>

<td>

<th data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">∧

</th>

<th>T

</th>

<th>F

</th></tr>

<tr>

<th>T

</th>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<th>F

</th>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

</td>

</td>

<td>

<th data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na">∨

</th>

<th>T

</th>

<th>F

</th></tr>

<tr>

<th>T

</th>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<th>F

</th>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

<p>This notation is useful especially if the operations are commutative, although one can additionally specify that the rows are the first operand and the columns are the second operand. This condensed notation is particularly useful in discussing multi-valued extensions of logic, as it significantly cuts down on combinatoric explosion of the number of rows otherwise needed. It also provides for quickly recognizable characteristic "shape" of the distribution of the values in the table which can assist the reader in grasping the rules more quickly.

</p>

<div class="mw-heading mw-heading3"><h3 id="Truth_tables_in_digital_logic">Truth tables in digital logic</h3></div>

<p>Truth tables are also used to specify the function of <a href="Lookup_table#Hardware_LUTs" title="Lookup table">hardware look-up tables (LUTs)</a> in <a href="Digital_circuit" class="mw-redirect" title="Digital circuit">digital logic circuitry</a>. For an n-input LUT, the truth table will have <span class="nowrap">⁠<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 2^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation>

</semantics>

</math></span><img src="./8226f30650ee4fe4e640c6d2798127e80e9c160d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}" loading="lazy"></span>⁠</span> values (or rows in the above tabular format), completely specifying a Boolean function for the LUT. By representing each Boolean value as a <a href="Bit" title="Bit">bit</a> in a <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary number</a>, truth table values can be efficiently encoded as <a href="Integer" title="Integer">integer</a> values in <a href="Electronic_design_automation" title="Electronic design automation">electronic design automation (EDA)</a> <a href="Software" title="Software">software</a>. For example, a 32-bit integer can encode the truth table for a LUT with up to 5 inputs.

</p><p>When using an integer representation of a truth table, the output value of the LUT can be obtained by calculating a bit index <i>k</i> based on the input values of the LUT, in which case the LUT's output value is the <i>k</i>th bit of the integer. For example, to evaluate the output value of a LUT given an <a href="Array_data_structure" class="mw-redirect" title="Array data structure">array</a> of <i>n</i> Boolean input values, the bit index of the truth table's output value can be computed as follows: if the <i>i</i>th input is true, let <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 V_{i}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./af98ff0a21a287cb51f7be55e5d5c06c2421f3c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle V_{i}=1}" loading="lazy"></span>, else let <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 V_{i}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1f623dac5aad31a952bd5f4c00a1fbb6693c90de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle V_{i}=0}" loading="lazy"></span>. Then the <i>k</i>th bit of the binary representation of the truth table is the LUT's output value, where

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=V_{0}\times 2^{0}+V_{1}\times 2^{1}+V_{2}\times 2^{2}+\dots +V_{n-1}\times 2^{n-1}.}</annotation>

</semantics>

</math></span></span>

</p><p>Truth tables are a simple and straightforward way to encode Boolean functions, however given the <a href="Exponential_growth" title="Exponential growth">exponential growth</a> in size as the number of inputs increase, they are not suitable for functions with a large number of inputs. Other representations which are more memory efficient are text equations and <a href="Binary_decision_diagram" title="Binary decision diagram">binary decision diagrams</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Applications_of_truth_tables_in_digital_electronics">Applications of truth tables in digital electronics</h3></div>

<p>In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without the use of <a href="Logic_gate" title="Logic gate">logic gates</a> or code. For example, a binary addition can be represented with the truth table:

</p>

<caption>Binary addition

</caption>

</th>

</th>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

<p>where A is the first operand, B is the second operand, C is the carry digit, and R is the result.

</p><p>This truth table is read left to right:

</p>

<ul><li>Value pair (A, B) equals value pair (C, R).</li>

<li>Or for this example, A plus B equal result R, with the Carry C.</li></ul>

<p>This table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.

</p><p>With respect to the result, this example may be arithmetically viewed as modulo 2 binary addition, and as logically equivalent to the exclusive-or (exclusive disjunction) binary logic operation.

</p><p>In this case it can be used for only very simple inputs and outputs, such as 1s and 0s. However, if the number of types of values one can have on the inputs increases, the size of the truth table will increase.

</p><p>For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values is 2<span class="nowrap"> × </span>2, or four. So the result is four possible outputs of C and R. If one were to use base 3, the size would increase to 3<span class="nowrap"> × </span>3, or nine possible outputs.

</p><p>The first "addition" example above is called a half-adder. A full-adder is when the carry from the previous operation is provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe a <a href="Full_adder" class="mw-redirect" title="Full adder">full adder</a>'s logic:

</p>

<pre>A B C* | C R

0 0 0 | 0 0

0 1 0 | 0 1

1 0 0 | 0 1

1 1 0 | 1 0

0 0 1 | 0 1

0 1 1 | 1 0

1 0 1 | 1 0

1 1 1 | 1 1

Same as previous, but..

C* = Carry from previous adder

</pre>

<div class="mw-heading mw-heading2"><h2 id="Methods_of_writing_truth_tables">Methods of writing truth tables</h2></div>

<p>Regarding the <i>guide columns<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></i> to the left of a table, which represent <a href="Propositional_variable" title="Propositional variable">propositional variables</a>, different authors have different recommendations about how to fill them in, although this is of no logical significance.<sup id="cite_ref-:13_6-0" class="reference"><a href="#cite_note-:13-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Alternating_method">Alternating method</h3></div>

<p>Lee Archie, a professor at <a href="Lander_University" title="Lander University">Lander University</a>, recommends this procedure, which is commonly followed in published truth-tables:

</p>

<ol><li>Write out the number of variables (corresponding to the number of statements) in alphabetical order.</li>

<li>The number of lines needed is 2<sup><i>n</i></sup> where n is the number of variables. (E. g., with three variables, 2<sup>3</sup> = 8).</li>

<li>Start in the right-hand column and alternate <strong>T</strong>'s and <strong>F</strong>'s until you run out of lines.</li>

<li>Then move left to the next column and alternate pairs of <strong>T</strong>'s and <strong>F</strong>'s until you run out of lines.</li>

<li>Then continue to the next left-hand column and double the numbers of <strong>T</strong>'s and <strong>F</strong>'s until completed.<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ol>

<p>This method results in truth-tables such as the following table for <span class="texhtml"><i>P</i> → (<i>Q</i> ∨ <i>R</i> → (<i>R</i> → ¬<i>P</i>))</span>, produced by <a href="Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a>:<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>

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\rightarrow (Q\vee R\rightarrow (R\rightarrow \neg P))}</annotation>

</semantics>

</math></span><img src="./ba0f18a10e998ff4fefc7ec50975afdb10609dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.451ex; height:2.843ex;" alt="{\displaystyle P\rightarrow (Q\vee R\rightarrow (R\rightarrow \neg P))}" loading="lazy"></span>

</th></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<div class="mw-heading mw-heading3"><h3 id="Combinatorial_method">Combinatorial method</h3></div>

<p><a href="Colin_Howson" title="Colin Howson">Colin Howson</a>, on the other hand, believes that "it is a good practical rule" to do the following:</p><blockquote><p>to start with all Ts, then all the ways (three) two Ts can be combined with one F, then all the ways (three) one T can be combined with two Fs, and then finish with all Fs. If a compound is built up from n distinct sentence letters, its truth table will have 2<sup>n</sup> rows, since there are two ways of assigning T or F to the first letter, and for each of these there will be two ways of assigning T or F to the second, and for each of these there will be two ways of assigning T or F to the third, and so on, giving 2.2.2. …, n times, which is equal to 2<sup>n</sup>.<sup id="cite_ref-:13_6-1" class="reference"><a href="#cite_note-:13-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></p></blockquote>

<p>This results in truth tables like this table "showing that <span class="texhtml">(<i>A</i>→<i>C</i>)∧(<i>B</i>→<i>C</i>)</span> and <span class="texhtml">(<i>A</i>∨<i>B</i>)→<i>C</i></span> are <a href="Truth_function" title="Truth function">truth-functionally</a> <a href="Logical_biconditional" title="Logical biconditional">equivalent</a>", modeled after a table produced by <a href="Colin_Howson" title="Colin Howson">Howson</a>:<sup id="cite_ref-:13_6-2" class="reference"><a href="#cite_note-:13-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./bc1d8f6dd2060d2a73740a960e783f31573b5f27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.469ex; height:2.843ex;" alt="{\displaystyle (A\rightarrow C)\land (B\rightarrow C)}" loading="lazy"></span>

</th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./41a593064755d381137062e4dcfa2fa7e1a7d744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.279ex; height:2.843ex;" alt="{\displaystyle (A\vee B)\rightarrow C}" loading="lazy"></span>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<div class="mw-heading mw-heading2"><h2 id="Size_of_truth_tables">Size of truth tables</h2></div>

<p>If there are <i>n</i> input variables then there are 2<sup><i>n</i></sup> possible combinations of their truth values. A given function may produce true or false for each combination so the number of different functions of <i>n</i> variables is the <a href="Double_exponential_function" title="Double exponential function">double exponential</a> 2<sup>2<sup><i>n</i></sup></sup>.

</p>

</th></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

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

<p>Truth tables for functions of three or more variables are rarely given.

</p>

<div class="mw-heading mw-heading2"><h2 id="Function_Tables">Function Tables</h2></div>

<p>It can be useful to have the output of a truth table expressed as a function of some variable values, instead of just a literal truth or false value. These may be called "function tables" to differentiate them from the more general "truth tables".<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> For example, one value, <span class="texhtml mvar" style="font-style:italic;">G</span>, may be used with an XOR gate to conditionally invert another value, <span class="texhtml mvar" style="font-style:italic;">X</span>. In other words, when <span class="texhtml mvar" style="font-style:italic;">G</span> is false, the output is <span class="texhtml mvar" style="font-style:italic;">X</span>, and when <span class="texhtml mvar" style="font-style:italic;">G</span> is true, the output is <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="{\textstyle \neg X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9cacdf7422d5b4bce988f98c01416a422d7b76d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\textstyle \neg X}" loading="lazy"></span>. The function table for this would look like:

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./116808eaf254fe1f3f51ba1fcbc769ba55afa4eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.421ex; height:2.176ex;" alt="{\displaystyle G\nleftrightarrow X}" loading="lazy"></span>

</th></tr>

<tr>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./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>

</td></tr>

<tr>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./da203aa7c9d097e5bf16f6f9cc482b3347ebe28e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.176ex;" alt="{\displaystyle \neg X}" loading="lazy"></span>

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

<p>Similarly, a 4-to-1 <a href="Multiplexer" title="Multiplexer">multiplexer</a> with select imputs <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 S_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ebe0ac45a38c4437bd2689a14ec434cd499e7e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}" loading="lazy"></span> and <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 S_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"></span>, data inputs <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">B</span>, <span class="texhtml mvar" style="font-style:italic;">C</span> and <span class="texhtml mvar" style="font-style:italic;">D</span>, and output <span class="texhtml mvar" style="font-style:italic;">Z</span> (as displayed in the image) would have this function table:

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}" loading="lazy"></span></th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ebe0ac45a38c4437bd2689a14ec434cd499e7e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{0}}" loading="lazy"></span></th>

/* start https://en.wikipedia.org/ */

.mw-parser-output .nobold{font-weight:normal}

/* end https://en.wikipedia.org/ */

</style><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">Z</span></span>

</th></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

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

<div class="mw-heading mw-heading2"><h2 id="Sentential_operator_truth_tables">Sentential operator truth tables</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Overview_table">Overview table</h3></div>

<p>Here is an extended truth table giving definitions of all sixteen possible truth functions of two Boolean variables <i><b>p</b></i> and <i><b>q</b></i>:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup>

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>

</th>

</th>

</mstyle>

</mrow>

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

</semantics>

</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></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\downarrow q}</annotation>

</semantics>

</math></span><img src="./180f47d11f2d53c298c9148795ad3894196d586a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\downarrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftarrow q}</annotation>

</semantics>

</math></span><img src="./e1765bb0aa9fba9ec1207e588a65e9161eedcc4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftarrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e2b198c79234d926cbee42c0f271d903ea55dc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nrightarrow q}</annotation>

</semantics>

</math></span><img src="./20a1d85a014a2fa2b7b7fdf06835faabb579331d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nrightarrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9ab9e26db63fa52698ddf6ef2b7dc355a954a48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.62ex; height:2.009ex;" alt="{\displaystyle \neg q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftrightarrow q}</annotation>

</semantics>

</math></span><img src="./f42d587b5c154e4ff8b71ce713c005b9aad49eee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftrightarrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\uparrow q}</annotation>

</semantics>

</math></span><img src="./8550d6ba7874bcb1df647048db24e90887fea4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\uparrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4904e68d8180ec4c12e20fabc15017b60b098b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\land q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./85f8399d94c0a05764a2961ddee15ee8ac2a8ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftrightarrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\rightarrow q}</annotation>

</semantics>

</math></span><img src="./4e3f9e5de9aaead8d19411bb3ad0dc490a50ec69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\rightarrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leftarrow q}</annotation>

</semantics>

</math></span><img src="./8f055ec54a9de5980a8c646a857e4220b2448465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftarrow q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c36427540990cefa44590b848be653f39cc1d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\vee q}" loading="lazy"></span></a></th>

</mstyle>

</mrow>

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

</semantics>

</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></a>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

</td></tr>

<tr>

</td>

</td></tr>

<tr>

<td colspan="2" style="background: #;"><abbr title="Associative">Assoc</abbr>

</td>

</td></tr>

<tr>

</td>

</mstyle>

</mrow>

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

</semantics>

</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></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\downarrow q}</annotation>

</semantics>

</math></span><img src="./180f47d11f2d53c298c9148795ad3894196d586a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\downarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nrightarrow q}</annotation>

</semantics>

</math></span><img src="./20a1d85a014a2fa2b7b7fdf06835faabb579331d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9ab9e26db63fa52698ddf6ef2b7dc355a954a48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.62ex; height:2.009ex;" alt="{\displaystyle \neg q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftarrow q}</annotation>

</semantics>

</math></span><img src="./e1765bb0aa9fba9ec1207e588a65e9161eedcc4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e2b198c79234d926cbee42c0f271d903ea55dc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftrightarrow q}</annotation>

</semantics>

</math></span><img src="./f42d587b5c154e4ff8b71ce713c005b9aad49eee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\uparrow q}</annotation>

</semantics>

</math></span><img src="./8550d6ba7874bcb1df647048db24e90887fea4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\uparrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4904e68d8180ec4c12e20fabc15017b60b098b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\land q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./85f8399d94c0a05764a2961ddee15ee8ac2a8ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leftarrow q}</annotation>

</semantics>

</math></span><img src="./8f055ec54a9de5980a8c646a857e4220b2448465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\rightarrow q}</annotation>

</semantics>

</math></span><img src="./4e3f9e5de9aaead8d19411bb3ad0dc490a50ec69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\rightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c36427540990cefa44590b848be653f39cc1d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\vee q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</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>

</td></tr>

<tr>

</td>

</mstyle>

</mrow>

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

</semantics>

</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></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c36427540990cefa44590b848be653f39cc1d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\vee q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leftarrow q}</annotation>

</semantics>

</math></span><img src="./8f055ec54a9de5980a8c646a857e4220b2448465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\rightarrow q}</annotation>

</semantics>

</math></span><img src="./4e3f9e5de9aaead8d19411bb3ad0dc490a50ec69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\rightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./85f8399d94c0a05764a2961ddee15ee8ac2a8ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4904e68d8180ec4c12e20fabc15017b60b098b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\land q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\uparrow q}</annotation>

</semantics>

</math></span><img src="./8550d6ba7874bcb1df647048db24e90887fea4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\uparrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftrightarrow q}</annotation>

</semantics>

</math></span><img src="./f42d587b5c154e4ff8b71ce713c005b9aad49eee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9ab9e26db63fa52698ddf6ef2b7dc355a954a48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.62ex; height:2.009ex;" alt="{\displaystyle \neg q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nrightarrow q}</annotation>

</semantics>

</math></span><img src="./20a1d85a014a2fa2b7b7fdf06835faabb579331d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e2b198c79234d926cbee42c0f271d903ea55dc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftarrow q}</annotation>

</semantics>

</math></span><img src="./e1765bb0aa9fba9ec1207e588a65e9161eedcc4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\downarrow q}</annotation>

</semantics>

</math></span><img src="./180f47d11f2d53c298c9148795ad3894196d586a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\downarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</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>

</td></tr>

<tr>

</td>

</mstyle>

</mrow>

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

</semantics>

</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></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\uparrow q}</annotation>

</semantics>

</math></span><img src="./8550d6ba7874bcb1df647048db24e90887fea4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\uparrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\rightarrow q}</annotation>

</semantics>

</math></span><img src="./4e3f9e5de9aaead8d19411bb3ad0dc490a50ec69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\rightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e2b198c79234d926cbee42c0f271d903ea55dc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leftarrow q}</annotation>

</semantics>

</math></span><img src="./8f055ec54a9de5980a8c646a857e4220b2448465.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9ab9e26db63fa52698ddf6ef2b7dc355a954a48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.62ex; height:2.009ex;" alt="{\displaystyle \neg q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./85f8399d94c0a05764a2961ddee15ee8ac2a8ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\leftrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\downarrow q}</annotation>

</semantics>

</math></span><img src="./180f47d11f2d53c298c9148795ad3894196d586a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.781ex; height:2.509ex;" alt="{\displaystyle p\downarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8c36427540990cefa44590b848be653f39cc1d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\vee q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftrightarrow q}</annotation>

</semantics>

</math></span><img src="./f42d587b5c154e4ff8b71ce713c005b9aad49eee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nleftarrow q}</annotation>

</semantics>

</math></span><img src="./e1765bb0aa9fba9ec1207e588a65e9161eedcc4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nleftarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\nrightarrow q}</annotation>

</semantics>

</math></span><img src="./20a1d85a014a2fa2b7b7fdf06835faabb579331d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.009ex;" alt="{\displaystyle p\nrightarrow q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./4904e68d8180ec4c12e20fabc15017b60b098b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\land q}" loading="lazy"></span></td>

</mstyle>

</mrow>

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

</semantics>

</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>

</td></tr>

<tr>

</td>

<td>

</td></tr>

<tr>

</td>

<td>

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

<p>where

</p>

<dd>F = false.</dd>

<dd>The <b>Com</b> row indicates whether an operator, <b>op</b>, is <a href="Commutative_property" title="Commutative property">commutative</a> – <span class="texhtml"><i>P</i> op <i>Q</i> = <i>Q</i> op <i>P</i></span>.</dd>

<dd>The <b>Assoc</b> row indicates whether an operator, <b>op</b>, is <a href="Associative_property" title="Associative property">associative</a> – <span class="texhtml">(<i>P</i> op <i>Q</i>) op <i>R</i> = <i>P</i> op (<i>Q</i> op <i>R</i>)</span>.</dd>

<dd>The <b>Adj</b> row shows the operator <b>op2</b> such that <span class="texhtml"><i>P</i> op <i>Q</i> = <i>Q</i> op2 <i>P</i></span>.</dd>

<dd>The <b>Neg</b> row shows the operator <b>op2</b> such that <span class="texhtml"><i>P</i> op <i>Q</i> = ¬(<i>P</i> op2 <i>Q</i>)</span>.</dd>

<dd>The <b>Dual</b> row shows the <a href="Duality_principle_(Boolean_algebra)" class="mw-redirect" title="Duality principle (Boolean algebra)">dual operation</a> obtained by interchanging T with F, and AND with OR.</dd>

<dd>The <b>L id</b> row shows the operator's <a href="Left_identity" class="mw-redirect" title="Left identity">left identities</a> if it has any values <span class="texhtml mvar" style="font-style:italic;">I</span> such that <span class="texhtml"><i>I</i> op <i>Q</i> = <i>Q</i></span>.</dd>

<dd>The <b>R id</b> row shows the operator's <a href="Right_identity" class="mw-redirect" title="Right identity">right identities</a> if it has any values <span class="texhtml mvar" style="font-style:italic;">I</span> such that <span class="texhtml"><i>P</i> op <i>I</i> = <i>P</i></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Wittgenstein_table">Wittgenstein table</h3></div>

<p>In proposition 5.101 of the <i><a href="Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></i>,<sup id="cite_ref-tlp5.101_11-0" class="reference"><a href="#cite_note-tlp5.101-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Wittgenstein</a> listed the table above as follows:

</p>

</th>

<th scope="col">Truthvalues

</th>

</th>

<th scope="col" colspan="2">Operator

</th>

<th scope="col">Operation name

</th>

<th scope="col">Tractatus<sup id="cite_ref-different_mapping_12-0" class="reference"><a href="#cite_note-different_mapping-12"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup>

</th></tr>

<tr>

<td><a href="Falsum" class="mw-redirect" title="Falsum">false</a></td>

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

<td>p and not p; and q and not q

</td></tr>

<tr>

<td><a href="Logical_NOR" title="Logical NOR">Logical NOR</a></td>

<td>neither <i>p</i> nor <i>q</i>

</td></tr>

<tr>

<td><a href="Converse_nonimplication" title="Converse nonimplication">Converse nonimplication</a></td>

</td></tr>

<tr>

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

</td></tr>

<tr>

</td>

<td><a href="Material_nonimplication" title="Material nonimplication">Material nonimplication</a></td>

</td></tr>

<tr>

<td>Negation</td>

</td></tr>

<tr>

<td><a href="Exclusive_disjunction" class="mw-redirect" title="Exclusive disjunction">Exclusive disjunction</a></td>

<td><i>p</i> or <i>q</i>, but not both

</td></tr>

<tr>

<td><a href="Logical_NAND" class="mw-redirect" title="Logical NAND">Logical NAND</a></td>

</td></tr>

<tr>

<td><a href="Logical_conjunction" title="Logical conjunction">Logical conjunction</a></td>

</td></tr>

<tr>

<td><a href="Logical_biconditional" title="Logical biconditional">Logical biconditional</a></td>

</td></tr>

<tr>

<td><a href="Projection_function" class="mw-redirect" title="Projection function">Projection function</a></td>

</td></tr>

<tr>

<td><a href="Material_conditional" title="Material conditional">Material implication</a></td>

</td></tr>

<tr>

<td>Projection function</td>

</td></tr>

<tr>

<td><a href="Converse_implication" class="mw-redirect" title="Converse implication">Converse implication</a></td>

</td></tr>

<tr>

<td><a href="Logical_disjunction" title="Logical disjunction">Logical disjunction</a></td>

</td></tr>

<tr>

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

<td>if p then p; and if q then q

</td></tr></tbody></table></dd></dl>

<p>The truth table represented by each row is obtained by appending the sequence given in <b>Truthvalues</b><sub>row</sub> to the table<sup id="cite_ref-different_mapping_12-1" class="reference"><a href="#cite_note-different_mapping-12"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup>

</p>

</th>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

</th>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr></tbody></table></dd></dl>

<p>For example, the table

</p>

</th>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

</th>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

</th>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr></tbody></table></dd></dl>

<p>represents the truth table for <a href="Material_conditional" title="Material conditional">Material implication</a>. Logical operators can also be visualized using <a href="Venn_diagram#Overview" title="Venn diagram">Venn diagrams</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Nullary_operations">Nullary operations</h3></div>

<p>There are 2 nullary operations:

</p>

<ul><li>Always true</li>

<li>Never true, unary <i><a href="Falsum" class="mw-redirect" title="Falsum">falsum</a></i></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Logical_true">Logical true</h4></div>

<p>The output value is always true, because this operator has zero operands and therefore no input values

</p>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<div class="mw-heading mw-heading4"><h4 id="Logical_false">Logical false</h4></div>

<p>The output value is never true: that is, always false, because this operator has zero operands and therefore no input values

</p>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

<div class="mw-heading mw-heading3"><h3 id="Unary_operations">Unary operations</h3></div>

<p>There are 2 unary operations:

</p>

<ul><li>Unary <i>identity</i></li>

<li>Unary <i>negation</i></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Logical_identity">Logical identity</h4></div>

<p><a href="Identity_function" title="Identity function">Logical identity</a> is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on one <a href="Logical_value" class="mw-redirect" title="Logical value">logical value</a> p, for which the output value remains p.

</p><p>The truth table for the logical identity operator is as follows:

</p>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

<div class="mw-heading mw-heading4"><h4 id="Logical_negation">Logical negation</h4></div>

<p><a href="Logical_negation" class="mw-redirect" title="Logical negation">Logical negation</a> is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on one <a href="Logical_value" class="mw-redirect" title="Logical value">logical value</a>, typically the value of a <a href="Proposition" title="Proposition">proposition</a>, that produces a value of <i>true</i> if its operand is false and a value of <i>false</i> if its operand is true.

</p><p>The truth table for <b>NOT p</b> (also written as <b>¬p</b>, <b>Np</b>, <b>Fpq</b>, or <b>~p</b>) is as follows:

</p>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<div class="mw-heading mw-heading3"><h3 id="Binary_operations">Binary operations</h3></div>

<p>There are 16 possible <a href="Truth_function" title="Truth function">truth functions</a> of two <a href="Binary_variable" class="mw-redirect" title="Binary variable">binary variables</a>, each operator has its own name.

</p>

<div class="mw-heading mw-heading4"><h4 id="Logical_conjunction_(AND)">Logical conjunction (AND)</h4></div>

<p><a href="Logical_conjunction" title="Logical conjunction">Logical conjunction</a> is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if both of its operands are true.

</p><p>The truth table for <b>p AND q</b> (also written as <b>p ∧ q</b>, <b>Kpq</b>, <b>p & q</b>, or <b>p</b> <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 \cdot }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"></span> <b>q</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

<p>In ordinary language terms, if both <i>p</i> and <i>q</i> are true, then the conjunction <i>p</i> ∧ <i>q</i> is true. For all other assignments of logical values to <i>p</i> and to <i>q</i> the conjunction <i>p</i> ∧ <i>q</i> is false.

</p><p>It can also be said that if <i>p</i>, then <i>p</i> ∧ <i>q</i> is <i>q</i>, otherwise <i>p</i> ∧ <i>q</i> is <i>p</i>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Logical_disjunction_(OR)">Logical disjunction (OR)</h4></div>

<p><a href="Logical_disjunction" title="Logical disjunction">Logical disjunction</a> is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if at least one of its operands is true.

</p><p>The truth table for <b>p OR q</b> (also written as <b>p ∨ q</b>, <b>Apq</b>, <b>p || q</b>, or <b>p + q</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

<p>Stated in English, if <i>p</i>, then <i>p</i> ∨ <i>q</i> is <i>p</i>, otherwise <i>p</i> ∨ <i>q</i> is <i>q</i>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Logical_implication">Logical implication</h4></div>

<p>Logical implication and the <a href="Material_conditional" title="Material conditional">material conditional</a> are both associated with an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="Proposition" title="Proposition">propositions</a>, which produces a value of <i>false</i> if the first operand is true and the second operand is false, and a value of <i>true</i> otherwise.

</p><p>The truth table associated with the logical implication <b>p implies q</b> (symbolized as <b>p ⇒ q</b>, or more rarely <b>Cpq</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<p>The truth table associated with the material conditional <b>if p then q</b> (symbolized as <b>p → q</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<p><b>p ⇒ q</b> and <b>p → q</b> are equivalent to <b>¬p ∨ q</b>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Logical_equality">Logical equality</h4></div>

<p><a href="Logical_equality" title="Logical equality">Logical equality</a> (also known as <a href="Biconditional" class="mw-redirect" title="Biconditional">biconditional</a> or <a href="Exclusive_nor" class="mw-redirect" title="Exclusive nor">exclusive nor</a>) is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if both operands are false or both operands are true.

</p><p>The truth table for <b>p XNOR q</b> (also written as <b>p ↔ q</b>, <b>Epq</b>, <b>p = q</b>, or <b>p ≡ q</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<p>So p EQ q is true if p and q have the same <a href="Truth_value" title="Truth value">truth value</a> (both true or both false), and false if they have different truth values.

</p>

<div class="mw-heading mw-heading4"><h4 id="Exclusive_disjunction">Exclusive disjunction</h4></div>

<p><a href="Exclusive_disjunction" class="mw-redirect" title="Exclusive disjunction">Exclusive disjunction</a> is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if one but not both of its operands is true.

</p><p>The truth table for <b>p XOR q</b> (also written as <b>Jpq</b>, or <b>p ⊕ q</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

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

<p>For two propositions, <b>XOR</b> can also be written as (p ∧ ¬q) ∨ (¬p ∧ q).

</p>

<div class="mw-heading mw-heading4"><h4 id="Logical_NAND">Logical NAND</h4></div>

<p>The <a href="Logical_NAND" class="mw-redirect" title="Logical NAND">logical NAND</a> is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="Proposition" title="Proposition">propositions</a>, that produces a value of <i>false</i> if both of its operands are true. In other words, it produces a value of <i>true</i> if at least one of its operands is false.

</p><p>The truth table for <b>p NAND q</b> (also written as <b>p ↑ q</b>, <b>Dpq</b>, or <b>p | q</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<p>It is frequently useful to express a logical operation as a <a href="Compound_operation_(computing)" class="mw-redirect" title="Compound operation (computing)">compound operation</a>, that is, as an operation that is built up or composed from other operations. Many such compositions are possible, depending on the operations that are taken as basic or "primitive" and the operations that are taken as composite or "derivative".

</p><p>In the case of logical NAND, it is clearly expressible as a compound of NOT and AND.

</p><p>The negation of a conjunction: ¬(<i>p</i> ∧ <i>q</i>), and the disjunction of negations: (¬<i>p</i>) ∨ (¬<i>q</i>) can be tabulated as follows:

</p>

</th>

</th>

</th>

<th style="width:15%">¬(<i>p</i> ∧ <i>q</i>)

</th>

</th>

</th>

<th style="width:15%">(¬<i>p</i>) ∨ (¬<i>q</i>)

</th></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<div class="mw-heading mw-heading4"><h4 id="Logical_NOR">Logical NOR</h4></div>

<p>The <a href="Logical_NOR" title="Logical NOR">logical NOR</a> is an <a href="Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on two <a href="Logical_value" class="mw-redirect" title="Logical value">logical values</a>, typically the values of two <a href="Proposition" title="Proposition">propositions</a>, that produces a value of <i>true</i> if both of its operands are false. In other words, it produces a value of <i>false</i> if at least one of its operands is true. ↓ is also known as the <a href="Peirce_arrow" class="mw-redirect" title="Peirce arrow">Peirce arrow</a> after its inventor, <a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>, and is a <a href="Sole_sufficient_operator" class="mw-redirect" title="Sole sufficient operator">Sole sufficient operator</a>.

</p><p>The truth table for <b>p NOR q</b> (also written as <b>p ↓ q</b>, or <b>Xpq</b>) is as follows:

</p>

</th>

</th>

</th></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<p>The negation of a disjunction ¬(<i>p</i> ∨ <i>q</i>), and the conjunction of negations (¬<i>p</i>) ∧ (¬<i>q</i>) can be tabulated as follows:

</p>

</th>

</th>

</th>

<th style="width:10%">¬(<i>p</i> ∨ <i>q</i>)

</th>

</th>

</th>

<th style="width:10%">(¬<i>p</i>) ∧ (¬<i>q</i>)

</th></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

<td style="background: #9EFF9E; color:black; vertical-align: middle; text-align: center;" class="table-success">T

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

<p>Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional arguments <i>p</i> and <i>q</i>, produces the identical patterns of functional values for ¬(<i>p</i> ∧ <i>q</i>) as for (¬<i>p</i>) ∨ (¬<i>q</i>), and for ¬(<i>p</i> ∨ <i>q</i>) as for (¬<i>p</i>) ∧ (¬<i>q</i>). Thus the first and second expressions in each pair are logically equivalent, and may be substituted for each other in all contexts that pertain solely to their logical values.

</p><p>This equivalence is one of <a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a>.

</p>

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<ul><li><a href="Boolean_domain" title="Boolean domain">Boolean domain</a></li>

<li><a href="Boolean-valued_function" title="Boolean-valued function">Boolean-valued function</a></li>

<li><a href="Publicad" class="mw-redirect" title="Publicad">Espresso heuristic logic minimizer</a></li>

<li><a href="Excitation_table" title="Excitation table">Excitation table</a></li>

<li><a href="State-transition_table" title="State-transition table">State-transition table</a></li>

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

<li><a href="Functional_completeness" title="Functional completeness">Functional completeness</a></li>

<li><a href="Karnaugh_maps" class="mw-redirect" title="Karnaugh maps">Karnaugh maps</a></li>

<li><a href="Logic_gate" title="Logic gate">Logic gate</a></li>

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

<li><a href="Logical_graph" class="mw-redirect" title="Logical graph">Logical graph</a></li>

<li><a href="Mathematical_table" title="Mathematical table">Mathematical table</a></li>

<li><a href="Method_of_analytic_tableaux" title="Method of analytic tableaux">Method of analytic tableaux</a></li>

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

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

<li><a href="Decision_table" title="Decision table">Decision table</a></li></ul>

</div>

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

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Information about notation may be found in (<a href="#CITEREFBocheński1959">Bocheński 1959</a>), (<a href="#CITEREFEnderton2001">Enderton 2001</a>), and (<a href="#CITEREFQuine1982">Quine 1982</a>).</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">The operators here with equal left and right identities (XOR, AND, XNOR, and OR) are also <a href="Monoid#Commutative_monoid" title="Monoid">commutative monoids</a> because they are also <a href="Associative_property" title="Associative property">associative</a>. While this distinction may be irrelevant in a simple discussion of logic, it can be quite important in more advanced mathematics. For example, in <a href="Category_theory" title="Category theory">category theory</a> an <a href="Enriched_category" title="Enriched category">enriched category</a> is described as a base <a href="Category_(mathematics)" title="Category (mathematics)">category</a> enriched over a monoid, and any of these operators can be used for enrichment.</span>

</li>

<li id="cite_note-different_mapping-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-different_mapping_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-different_mapping_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Wittgenstein used a different mapping. In proposition 5.101 of the Tractatus one has to append <b>Truthvalues</b><sub>row</sub> to the table

</th>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr>

<tr>

</th>

<td style="background: #FFC7C7; color:black; vertical-align: middle; text-align: center;" class="table-failure">F

</td></tr></tbody></table></dd></dl>

<p>This explains why <b>Tractatus</b><sub>row</sub> in the table given here does not point to the same <b>Truthvalues</b><sub>row</sub> as in the Tractatus.

</p>

</span></li>

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

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

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFEnderton2001">Enderton 2001</a></span>

</li>

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</style><cite id="CITEREFvon_Wright1955" class="citation journal cs1"><a href="Georg_Henrik_von_Wright" title="Georg Henrik von Wright">von Wright, Georg Henrik</a> (1955). "Ludwig Wittgenstein, A Biographical Sketch". <i>The Philosophical Review</i>. <b>64</b> (4): 527–545 (p. 532, note 9). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2182631">10.2307/2182631</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2182631">2182631</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFPost1921" class="citation journal cs1"><a href="Emil_Post" class="mw-redirect" title="Emil Post">Post, Emil</a> (July 1921). "Introduction to a general theory of elementary propositions". <i>American Journal of Mathematics</i>. <b>43</b> (3): <span class="nowrap">163–</span>185. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2370324">10.2307/2370324</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fuiuo.ark%3A%2F13960%2Ft9j450f7q">2027/uiuo.ark:/13960/t9j450f7q</a></span>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2370324">2370324</a>.</cite></span>

</li>

<li id="cite_note-Peirce-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Peirce_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Peirce_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFAnellis2012" class="citation journal cs1"><a href="Irving_Anellis" title="Irving Anellis">Anellis, Irving H.</a> (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table". <i>History and Philosophy of Logic</i>. <b>33</b>: <span class="nowrap">87–</span>97. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01445340.2011.621702">10.1080/01445340.2011.621702</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:170654885">170654885</a>.</cite></span>

</li>

<li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://philosophy.lander.edu/logic/table.html">"How to Construct a Truth Table"</a>. <i>philosophy.lander.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-04-05</span></span>.</cite></span>

</li>

<li id="cite_note-:13-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:13_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:13_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:13_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFHowson1997" class="citation book cs1">Howson, Colin (1997). <i>Logic with trees: an introduction to symbolic logic</i>. London; New York: Routledge. p. 10. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-415-13342-5</bdi>.</cite></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="CITEREFKleene2013" class="citation book cs1"><a href="Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene, Stephen Cole</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4GzCAgAAQBAJ&pg=PA11"><i>Mathematical Logic</i></a>. Dover Books on Mathematics. Courier Corporation. p. 11. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486317076</bdi>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFManoCiletti2018" class="citation book cs1">Mano, M. Morris; Ciletti, Michael (2018-07-13). <i>Digital Design, Global Edition</i> (6th ed.). Pearson Education, Limited. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781292231167</bdi>.</cite></span>

</li>

<li id="cite_note-tlp5.101-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-tlp5.101_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWittgenstein1922" class="citation book cs1"><a href="Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Wittgenstein, Ludwig</a> (1922). <a rel="nofollow" class="external text" href="https://www.gutenberg.org/files/5740/5740-pdf.pdf"><i>Tractatus Logico-Philosophicus</i></a> <span class="cs1-format">(PDF)</span>. Proposition 5.101.</cite></span>

</li>

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

<div class="mw-heading mw-heading3"><h3 id="Works_cited">Works cited</h3></div>

<ul><li><cite id="CITEREFBocheński1959" class="citation book cs1"><a href="J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Bocheński, Józef Maria</a> (1959). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YX8iCQAAQBAJ&pg=PP5"><i>A Précis of Mathematical Logic</i></a>. Translated by Bird, Otto. D. Reidel. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-017-0592-9">10.1007/978-94-017-0592-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-017-0592-9</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li>

<li><cite id="CITEREFEnderton2001" class="citation book cs1"><a href="Herbert_Enderton" title="Herbert Enderton">Enderton, H.</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dVncCl_EtUkC&pg=PR7"><i>A Mathematical Introduction to Logic</i></a> (2nd ed.). Harcourt Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-238452-0</bdi>.</cite></li>

<li><cite id="CITEREFQuine1982" class="citation book cs1"><a href="W.V._Quine" class="mw-redirect" title="W.V. Quine">Quine, W.V.</a> (1982). <i>Methods of Logic</i> (4th ed.). Harvard University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-674-57175-4</bdi>.</cite></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">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Truth_tables" class="extiw external" title="commons:Category:Truth tables">Truth tables</a></span>.</div></div>

</div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Truth_table">"Truth table"</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><a rel="nofollow" class="external text" href="http://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html">Truth Tables, Tautologies, and Logical Equivalence</a></li>

<li><a rel="nofollow" class="external text" href="http://www.allaboutcircuits.com/vol_4/chpt_7/9.html">Converting truth tables into Boolean expressions</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Classical_logic61" 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="Classical_logic61" style="font-size:114%;margin:0 4em"><a href="Classical_logic" title="Classical logic">Classical 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="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="Logical_connective" title="Logical connective">Connective</a></li>

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

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

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

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

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

<li><a href="Problem_of_multiple_generality" title="Problem of multiple generality">Problem of multiple generality</a></li>

<li><a href="Associative_property" title="Associative property">Associativity</a></li>

<li><a href="Distributive_property" title="Distributive property">Distribution</a></li>

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

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

</div></td><td class="noviewer navbox-image" rowspan="6" 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%">Classical logics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Principles</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="Commutativity_of_conjunction" title="Commutativity of conjunction">Commutativity of conjunction</a></li>

<li><a href="Law_of_excluded_middle" title="Law of excluded middle">Excluded middle</a></li>

<li><a href="Principle_of_bivalence" title="Principle of bivalence">Bivalence</a></li>

<li><a href="Law_of_noncontradiction" title="Law of noncontradiction">Noncontradiction</a></li>

<li><a href="Monotonicity_of_entailment" title="Monotonicity of entailment">Monotonicity of entailment</a></li>

<li><a href="Principle_of_explosion" title="Principle of explosion">Explosion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Rules</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="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a></li>

<li><a href="Material_implication_(rule_of_inference)" title="Material implication (rule of inference)">Material implication</a></li>

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

<li><a href="Modus_ponens" title="Modus ponens">modus ponens</a></li>

<li><a href="Modus_tollens" title="Modus tollens">modus tollens</a></li>

<li><a href="Modus_ponendo_tollens" title="Modus ponendo tollens">modus ponendo tollens</a></li>

<li><a href="Constructive_dilemma" title="Constructive dilemma">Constructive dilemma</a></li>

<li><a href="Destructive_dilemma" title="Destructive dilemma">Destructive dilemma</a></li>

<li><a href="Disjunctive_syllogism" title="Disjunctive syllogism">Disjunctive syllogism</a></li>

<li><a href="Hypothetical_syllogism" title="Hypothetical syllogism">Hypothetical syllogism</a></li>

<li><a href="Absorption_(logic)" title="Absorption (logic)">Absorption</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Introduction</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="Negation_introduction" title="Negation introduction">Negation</a></li>

<li><a href="Double_negation_introduction" class="mw-redirect" title="Double negation introduction">Double negation</a></li>

<li><a href="Existential_generalization" title="Existential generalization">Existential</a></li>

<li><a href="Universal_generalization" title="Universal generalization">Universal</a></li>

<li><a href="Biconditional_introduction" title="Biconditional introduction">Biconditional</a></li>

<li><a href="Conjunction_introduction" title="Conjunction introduction">Conjunction</a></li>

<li><a href="Disjunction_introduction" title="Disjunction introduction">Disjunction</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Elimination</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="Double_negation_elimination" class="mw-redirect" title="Double negation elimination">Double negation</a></li>

<li><a href="Existential_instantiation" title="Existential instantiation">Existential</a></li>

<li><a href="Universal_instantiation" title="Universal instantiation">Universal</a></li>

<li><a href="Biconditional_elimination" title="Biconditional elimination">Biconditional</a></li>

<li><a href="Conjunction_elimination" title="Conjunction elimination">Conjunction</a></li>

<li><a href="Disjunction_elimination" title="Disjunction elimination">Disjunction</a></li></ul>

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</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="Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a></li>

<li><a href="George_Boole" title="George Boole">George Boole</a></li>

<li><a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li>

<li><a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li>

<li><a href="Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a></li>

<li><a href="Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a></li>

<li><a href="Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li>

<li><a href="Hugh_MacColl" title="Hugh MacColl">Hugh MacColl</a></li>

<li><a href="Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a></li>

<li><a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a></li>

<li><a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li>

<li><a href="Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a></li>

<li><a href="Henry_M._Sheffer" title="Henry M. Sheffer">Henry M. Sheffer</a></li>

<li><a href="Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a></li>

<li><a href="Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a></li>

<li><a href="Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a></li>

<li><a href="Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</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="Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></li>

<li><a href="Function_and_Concept" title="Function and Concept">Function and Concept</a></li>

<li><a href="The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></li>

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

<li><a href="Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</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="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>

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

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

<li><a href="Porphyrian_tree" title="Porphyrian tree">Porphyrian tree</a></li>

<li><a href="Karnaugh_map" title="Karnaugh map">Karnaugh map</a></li>

<li><a href="Binary_decision_diagram" title="Binary decision diagram">Binary decision diagram</a></li>

<li><a href="Propositional_directed_acyclic_graph" title="Propositional directed acyclic graph">Propositional directed acyclic graph</a></li>

<li><a href="Sentential_decision_diagram" title="Sentential decision diagram">Sentential decision diagram</a></li>

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

<li><a href="Method_of_analytic_tableaux" title="Method of analytic tableaux">Method of analytic tableaux</a></li></ul>

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