SherbyNet - /page/wikibook/entry.mu

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<title>Material conditional</title>

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

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</style><div role="note" class="hatnote navigation-not-searchable">"Logical conditional" redirects here. For other related meanings, see <a href="Conditional_statement_(disambiguation)" class="mw-redirect mw-disambig" title="Conditional statement (disambiguation)">Conditional statement</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Material_inference" title="Material inference">Material inference</a> or <a href="Material_implication_(rule_of_inference)" title="Material implication (rule of inference)">Material implication (rule of inference)</a>.</div>

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</style><table class="infobox"><caption class="infobox-title" style="background:navy; color:white;">Material conditional</caption><tbody><tr><th colspan="2" class="infobox-above">IMPLY</th></tr><tr><td colspan="2" class="infobox-image"><span typeof="mw:File"></span></td></tr><tr><th scope="row" class="infobox-label">Definition</th><td class="infobox-data"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to y}">

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

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</math></span><img src="./43570cdee5baeaebcf546338f77cf67c27a7344f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:2.176ex;" alt="{\displaystyle x\to y}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Truth_table" title="Truth table">Truth table</a></th><td class="infobox-data"><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 (1011)}">

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<annotation encoding="application/x-tex">{\displaystyle (1011)}</annotation>

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</math></span><img src="./fae607ae798279320876b3b82ebb9c728590a14f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.459ex; height:2.843ex;" alt="{\displaystyle (1011)}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Logic_gate" title="Logic gate">Logic gate</a></th><td class="infobox-data"><span typeof="mw:File"></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;">Normal forms</th></tr><tr><th scope="row" class="infobox-label"><a href="Disjunctive_normal_form" title="Disjunctive normal form">Disjunctive</a></th><td class="infobox-data"><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 {\overline {x}}+y}">

<mover>

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

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</math></span><img src="./1e48860176df3326bc54f0431ee6b72e8ba6d37f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.441ex; height:2.676ex;" alt="{\displaystyle {\overline {x}}+y}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Conjunctive_normal_form" title="Conjunctive normal form">Conjunctive</a></th><td class="infobox-data"><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 {\overline {x}}+y}">

<mover>

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

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</math></span><img src="./1e48860176df3326bc54f0431ee6b72e8ba6d37f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.441ex; height:2.676ex;" alt="{\displaystyle {\overline {x}}+y}" loading="lazy"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="Zhegalkin_polynomial" title="Zhegalkin polynomial">Zhegalkin polynomial</a></th><td class="infobox-data"><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 1\oplus x\oplus xy}">

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

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</math></span><img src="./e5dcdddf5b82b6dd5d7e74a5fb728d6e371f53c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.658ex; height:2.509ex;" alt="{\displaystyle 1\oplus x\oplus xy}" loading="lazy"></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;"><a href="Post's_lattice" title="Post's lattice"><span style="color:white;">Post's lattices</span></a></th></tr><tr><th scope="row" class="infobox-label">0-preserving</th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label">1-preserving</th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label"><a href="Monotonic_function" title="Monotonic function">Monotone</a></th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label"><a href="Affine_transformation" title="Affine transformation">Affine</a></th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label">Self-dual</th><td class="infobox-data">no</td></tr><tr><td colspan="2" class="infobox-navbar"><style data-mw-deduplicate="TemplateStyles:r1129693374">

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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}">

</mstyle>

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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>

</mrow>

</mover>

</mrow>

</mstyle>

</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>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<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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</mstyle>

</mrow>

<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}">

</mstyle>

</mrow>

<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>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</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>

</semantics>

</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;"> </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>

</semantics>

</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>

</mrow>

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

</semantics>

</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>

</mrow>

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

</semantics>

</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>

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

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_table" title="Truth table">Truth table</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;">

<span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</td></tr><tr><td class="sidebar-navbar"></td></tr></tbody></table>

<p>The <b>material conditional</b> (also known as <b>material implication</b>) is a <a href="Binary_operation" title="Binary operation">binary operation</a> commonly used in <a href="Mathematical_logic" title="Mathematical logic">logic</a>. When the conditional symbol <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span> is <a href="Interpretation_(logic)" title="Interpretation (logic)">interpreted</a> as material implication, a formula <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\to Q}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d7cad5b2c2991ae1dbded560c5d875fbf49fe8ea.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\to Q}" loading="lazy"></span> is true unless <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}">

</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> is true 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 Q}">

</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> is false.

</p><p>Material implication is used in all the basic systems of <a href="Classical_logic" title="Classical logic">classical logic</a> as well as some <a href="Nonclassical_logic" class="mw-redirect" title="Nonclassical logic">nonclassical logics</a>. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many <a href="Programming_language" title="Programming language">programming languages</a>. However, many logics replace material implication with other operators such as the <a href="Strict_conditional" title="Strict conditional">strict conditional</a> and the <a href="Variably_strict_conditional" class="mw-redirect" title="Variably strict conditional">variably strict conditional</a>. Due to the <a href="Paradoxes_of_material_implication" title="Paradoxes of material implication">paradoxes of material implication</a> and related problems, material implication is not generally considered a viable analysis of <a href="Conditional_sentence" title="Conditional sentence">conditional sentences</a> in <a href="Natural_language" title="Natural language">natural language</a>.

</p>

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

<p>In logic and related fields, the material conditional is customarily notated with an infix operator <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 \to }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>.<sup id="cite_ref-FOOTNOTEHilbert1918_1-0" class="reference"><a href="#cite_note-FOOTNOTEHilbert1918-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The material conditional is also notated using the infixes <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 \supset }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./27bfe0828a2ed4c9c6b70987a85c02a1f005843c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \supset }" 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 \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span>.<sup id="cite_ref-FOOTNOTEMendelson2015_2-0" class="reference"><a href="#cite_note-FOOTNOTEMendelson2015-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the prefixed <a href="Polish_notation" title="Polish notation">Polish notation</a>, conditionals are notated as <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 Cpq}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./54d77892e3efc7c577e93c08774120a94c786148.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.005ex; height:2.509ex;" alt="{\displaystyle Cpq}" loading="lazy"></span>. In a conditional formula <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\to q}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8fccb3827df1efe7930f9d1febd15d2359971b93.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\to q}" loading="lazy"></span>, the subformula <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}">

</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> is referred to as the <i><a href="Antecedent_(logic)" title="Antecedent (logic)">antecedent</a></i> 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 q}">

</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> is termed the <i><a href="Consequent" title="Consequent">consequent</a></i> of the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula <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\to q)\to (r\to s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p\to q)\to (r\to s)}</annotation>

</semantics>

</math></span><img src="./a5dd89d4d4fbf0a537ff8ceb4ed7c56855ae00c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.839ex; height:2.843ex;" alt="{\displaystyle (p\to q)\to (r\to s)}" loading="lazy"></span>.

</p>

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

<p>In <i><a href="Arithmetices_principia%2C_nova_methodo_exposita" title="Arithmetices principia, nova methodo exposita">Arithmetices Principia: Nova Methodo Exposita</a></i> (1889), <a href="Giuseppe_Peano" title="Giuseppe Peano">Peano</a> expressed the proposition "If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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

</mstyle>

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

</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> with the symbol Ɔ, which is the opposite of C.<sup id="cite_ref-FOOTNOTEVan_Heijenoort1967_3-0" class="reference"><a href="#cite_note-FOOTNOTEVan_Heijenoort1967-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> He also expressed the proposition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\supset B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ee952838d8b3e67045072a8f2b71e7fc0467dea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}" loading="lazy"></span> as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

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

</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>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTENahas2022VI_5-0" class="reference"><a href="#cite_note-FOOTNOTENahas2022VI-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="David_Hilbert" title="David Hilbert">Hilbert</a> expressed the proposition "If <i>A</i>, then <i>B</i>" as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d5b8dd84619daff17b52a08b77d15db2b9ad6c2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}" loading="lazy"></span> in 1918.<sup id="cite_ref-FOOTNOTEHilbert1918_1-1" class="reference"><a href="#cite_note-FOOTNOTEHilbert1918-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Bertrand_Russell" title="Bertrand Russell">Russell</a> followed Peano in his <i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> (1910–1913), in which he expressed the proposition "If <i>A</i>, then <i>B</i>" as <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\supset B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ee952838d8b3e67045072a8f2b71e7fc0467dea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}" loading="lazy"></span>. Following Russell, <a href="Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen</a> expressed the proposition "If <i>A</i>, then <i>B</i>" as <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\supset B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ee952838d8b3e67045072a8f2b71e7fc0467dea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}" loading="lazy"></span>. <a href="Arend_Heyting" title="Arend Heyting">Heyting</a> expressed the proposition "If <i>A</i>, then <i>B</i>" as <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\supset B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ee952838d8b3e67045072a8f2b71e7fc0467dea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\supset B}" loading="lazy"></span> at first but later came to express it as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d5b8dd84619daff17b52a08b77d15db2b9ad6c2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}" loading="lazy"></span> with a right-pointing arrow. <a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a> expressed the proposition "If <i>A</i>, then <i>B</i>" as <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}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8e560143d45c97e6387c7c3aa90e9d7745002228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\Rightarrow B}" loading="lazy"></span> in 1954.<sup id="cite_ref-FOOTNOTEBourbaki195414_6-0" class="reference"><a href="#cite_note-FOOTNOTEBourbaki195414-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><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>

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

<div class="mw-heading mw-heading3"><h3 id="Truth_table">Truth table</h3></div>

<p>From a <a href="Classical_logic" title="Classical logic">classical</a> <a href="Semantics_of_logic" title="Semantics of logic">semantic perspective</a>, material implication is the <a href="Binary_operator" class="mw-redirect" title="Binary operator">binary</a> <a href="Truth_function" title="Truth function">truth functional</a> operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in the following <a href="Truth_table" title="Truth table">truth table</a>:

</p>

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

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</style><table class="wikitable sortable two-ary-truth-table"><tbody><tr><th><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

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

</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><th class="unsortable two-ary-truth-table-border"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d5b8dd84619daff17b52a08b77d15db2b9ad6c2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle A\to B}" loading="lazy"></span></th></tr><tr><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-true"><abbr title="true">T</abbr></td></tr><tr><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-true"><abbr title="true">T</abbr></td></tr><tr><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-false"><abbr title="false">F</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-false"><abbr title="false">F</abbr></td></tr><tr><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-true"><abbr title="true">T</abbr></td><td class="two-ary-truth-table-border two-ary-truth-table-true"><abbr title="true">T</abbr></td></tr></tbody></table>

<p>One can also consider the equivalence <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B\equiv \neg (A\land \neg B)\equiv \neg A\lor B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./25a5f968ca6f8f93668ded98e221578b83d3f643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.958ex; height:2.843ex;" alt="{\displaystyle A\to B\equiv \neg (A\land \neg B)\equiv \neg A\lor B}" loading="lazy"></span>.

</p><p>The conditionals <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\to B)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2874d32e52c03fcb9112bab28893069fed6fc3e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.93ex; height:2.843ex;" alt="{\displaystyle (A\to B)}" loading="lazy"></span> where the antecedent <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</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> is false, are called "<a href="Vacuous_truth" title="Vacuous truth">vacuous truths</a>".

Examples are ...

</p>

</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> false: <i>"If <a href="Marie_Curie" title="Marie Curie">Marie Curie</a> is a sister of <a href="Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a>, then Galileo Galilei is a brother of Marie Curie."</i></li>

</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> true: <i>"If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling."</i></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Analytic_tableaux">Analytic tableaux</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Method_of_analytic_tableaux" title="Method of analytic tableaux">Method of analytic tableaux</a></div>

<p>Formulas over the set of connectives <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 \{\to ,\bot \}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./55ebcc40157f1c4fb9352bcda77afdcfe135b11b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.491ex; height:2.843ex;" alt="{\displaystyle \{\to ,\bot \}}" loading="lazy"></span><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> are called <b>f-implicational</b>.<sup id="cite_ref-FOOTNOTEFrancoGoldsmithSchlipfSpeckenmeyer1999_9-0" class="reference"><a href="#cite_note-FOOTNOTEFrancoGoldsmithSchlipfSpeckenmeyer1999-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In <a href="Classical_logic" title="Classical logic">classical logic</a> the other connectives, such as <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 }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./fa78fd02085d39aa58c9e47a6d4033ce41e02fad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }" loading="lazy"></span> (<a href="Negation" title="Negation">negation</a>), <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d6823e5a222eb3ca49672818ac3d13ec607052c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }" loading="lazy"></span> (<a href="Logical_conjunction" title="Logical conjunction">conjunction</a>), <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ab47f6b1f589aedcf14638df1d63049d233d851a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }" loading="lazy"></span> (<a href="Disjunction" class="mw-redirect" title="Disjunction">disjunction</a>) 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 \leftrightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./046b918c43e05caf6624fe9b676c69ec9cd6b892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \leftrightarrow }" loading="lazy"></span> (<a href="If_and_only_if" title="If and only if">equivalence</a>), can be defined in terms of <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 \to }">

</mstyle>

</mrow>

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

</semantics>

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

</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 href="False_(logic)#False,_negation_and_contradiction" title="False (logic)">falsity</a>):<sup id="cite_ref-connective_needed_10-0" class="reference"><a href="#cite_note-connective_needed-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

<mtr>

<mtd>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\neg A&\quad {\overset {\text{def}}{=}}\quad A\to \bot \\A\land B&\quad {\overset {\text{def}}{=}}\quad (A\to (B\to \bot ))\to \bot \\A\lor B&\quad {\overset {\text{def}}{=}}\quad (A\to \bot )\to B\\A\leftrightarrow B&\quad {\overset {\text{def}}{=}}\quad \{(A\to B)\to [(B\to A)\to \bot ]\}\to \bot \\\end{aligned}}}</annotation>

</semantics>

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

</p><p>The validity of f-implicational formulas can be semantically established by the <a href="Method_of_analytic_tableaux" title="Method of analytic tableaux">method of analytic tableaux</a>. The logical rules are

</p>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\boldsymbol {\mathsf {T}}}(A\to B)}{{\boldsymbol {\mathsf {F}}}(A)\quad \mid \quad {\boldsymbol {\mathsf {T}}}(B)}}}</annotation>

</semantics>

</math></span><img src="./3931bded88c4a736444c1959a9dc8d0e8ce8b6b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.668ex; height:6.509ex;" alt="{\displaystyle {\frac {{\boldsymbol {\mathsf {T}}}(A\to B)}{{\boldsymbol {\mathsf {F}}}(A)\quad \mid \quad {\boldsymbol {\mathsf {T}}}(B)}}}" loading="lazy"></span></td>

<mfrac>

<mrow>

</mrow>

</mrow>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\boldsymbol {\mathsf {F}}}(A\to B)}{\begin{array}{c}{\boldsymbol {\mathsf {T}}}(A)\\{\boldsymbol {\mathsf {F}}}(B)\end{array}}}}</annotation>

</semantics>

</math></span><img src="./c32b07d35b4e10cdec3ddc6ad948fc53ffad9dca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:11.187ex; height:10.009ex;" alt="{\displaystyle {\frac {{\boldsymbol {\mathsf {F}}}(A\to B)}{\begin{array}{c}{\boldsymbol {\mathsf {T}}}(A)\\{\boldsymbol {\mathsf {F}}}(B)\end{array}}}}" loading="lazy"></span>

</td></tr>

<tr>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mathsf {T}}}(\bot )}</annotation>

</semantics>

</math></span><img src="./e1667f5ebe9a9e3ec9a739332683c298a6922219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.321ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\mathsf {T}}}(\bot )}" loading="lazy"></span> : Close the branch (contradiction)<br><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 {\boldsymbol {\mathsf {F}}}(\bot )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mathsf {F}}}(\bot )}</annotation>

</semantics>

</math></span><img src="./4a8cf3da097eedef97fd6014d6c310930635a395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\mathsf {F}}}(\bot )}" loading="lazy"></span> : Do nothing (since it just asserts no contradiction)

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

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</style>

<th class="cot-header-mainspace" style="background:#ffffff; font-size:87%; padding:0.2em 0.3em; text-align:center; color:#202122"><div style="font-size:115%;margin:0 4em"><span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Example: proof of <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\to \neg \neg p\quad }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./556d85352ac5063b02a9664e2d71de31f004eb1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.466ex; height:2.176ex;" alt="{\displaystyle p\to \neg \neg p\quad }" loading="lazy"></span>, by <a href="Method_of_analytic_tableaux" title="Method of analytic tableaux">method of analytic tableaux</a></span></div>

</th></tr>

<tr>

<pre> F[p → ((p → ⊥) → ⊥)]

|

T[p]

F[(p → ⊥) → ⊥]

|

T[p → ⊥]

F[⊥]

┌────────┴────────┐

F[p] T[⊥]

| |

CONTRADICTION CONTRADICTION

(T[p], F[p]) (⊥ is true)

</pre>

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

</div>

<th class="cot-header-mainspace" style="background:#ffffff; font-size:87%; padding:0.2em 0.3em; text-align:center; color:#202122"><div style="font-size:115%;margin:0 4em"><span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Example: proof of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg p\to p\quad }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9bf9b9ca9c5591c548dffa6de566a3bf60783606.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.376ex; height:2.176ex;" alt="{\displaystyle \neg \neg p\to p\quad }" loading="lazy"></span>, by method of analytic tableaux</span></div>

</th></tr>

<tr>

<pre> F[((p → ⊥) → ⊥) → p]

|

T[(p → ⊥) → ⊥]

F[p]

┌────────┴────────┐

F[p → ⊥] T[⊥]

| |

T[p] CONTRADICTION (⊥ is true)

F[⊥]

|

CONTRADICTION (T[p], F[p])

</pre>

<p><a href="Hilbert_system" title="Hilbert system">Hilbert-style proofs</a> can be found <a href="Implicational_propositional_calculus#An_alternative_axiomatization" title="Implicational propositional calculus">here</a> or <a href="Peirce's_law" title="Peirce's law">here</a>.

</p>

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

</div>

<th class="cot-header-mainspace" style="background:#ffffff; font-size:87%; padding:0.2em 0.3em; text-align:center; color:#202122"><div style="font-size:115%;margin:0 4em"><span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Example: proof of <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\to q)\to ((q\to r)\to (p\to r))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p\to q)\to ((q\to r)\to (p\to r))}</annotation>

</semantics>

</math></span><img src="./b7873ed29e4a1c3a3f6815c77d535544a732c718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.883ex; height:2.843ex;" alt="{\displaystyle (p\to q)\to ((q\to r)\to (p\to r))}" loading="lazy"></span>, by method of analytic tableaux</span></div>

</th></tr>

<tr>

<pre> 1. F[(p → q) → ((q → r) → (p → r))]

| // from 1

2. T[p → q]

3. F[(q → r) → (p → r)]

| // from 3

4. T[q → r]

5. F[p → r]

| // from 5

6. T[p]

7. F[r]

┌────────┴────────┐ // from 2

8a. F[p] 8b. T[q]

X ┌────────┴────────┐ // from 4

9a. F[q] 9b. T[r]

X X

</pre>

<p>A <a href="Hilbert_system" title="Hilbert system">Hilbert-style proof</a> can be found <a href="Implicational_propositional_calculus#The_Bernays–Tarski_axiom_system" title="Implicational propositional calculus">here</a>.

</p>

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

</div>

<div class="mw-heading mw-heading2"><h2 id="Syntactical_properties">Syntactical properties</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Natural_deduction" title="Natural deduction">Natural deduction</a></div>

<p>The semantic definition by truth tables does not permit the examination of structurally identical propositional forms in various <a href="Formal_system" title="Formal system">logical systems</a>, where different properties may be demonstrated. The language considered here is restricted to <b>f-implicational formulas</b>.

</p><p>Consider the following (candidate) <a href="Natural_deduction" title="Natural deduction">natural deduction</a> rules.

</p>

<td valign="top"><b>Implication Introduction</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 \to }">

</mstyle>

</mrow>

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

</semantics>

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

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

</mstyle>

</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> one can derive <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</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>, then one can conclude <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}">

</mstyle>

</mrow>

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

</semantics>

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

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\begin{array}{c}[A]\\\vdots \\B\end{array}}{A\to B}}}">

<mfrac>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\begin{array}{c}[A]\\\vdots \\B\end{array}}{A\to B}}}</annotation>

</semantics>

</math></span><img src="./429c3fda935111a3709b38770cd212db3ee7d58e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.957ex; height:13.509ex;" alt="{\displaystyle {\frac {\begin{array}{c}[A]\\\vdots \\B\end{array}}{A\to B}}}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }">

</mstyle>

</mrow>

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

</semantics>

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

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./79eaa334597b1861f1b08ca0c8fecb3858ebcb12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle [A]}" loading="lazy"></span> is an assumption that is discharged when applying the rule.

</p>

</td>

<td valign="top"><b>Implication Elimination</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 \to }">

</mstyle>

</mrow>

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

</semantics>

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

<p>This rule corresponds to <a href="Modus_ponens" title="Modus ponens">modus ponens</a>.

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

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./dca58721319a435f52b3f389504bfa3efe92399c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.023ex; height:5.343ex;" alt="{\displaystyle {\frac {A\to B\quad A}{B}}}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }">

</mstyle>

</mrow>

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

</semantics>

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

</p><p><br>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./01b3a9dd49f6e29ddccd8179fe6bdf9749a18f8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.023ex; height:5.343ex;" alt="{\displaystyle {\frac {A\quad A\to B}{B}}}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }">

</mstyle>

</mrow>

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

</semantics>

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

</p>

</td></tr>

<tr>

<td valign="top"><b><a href="Double_negation" title="Double negation">Double Negation Elimination</a></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 \neg \neg }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./686ca19f81cdf5295452c8fe216c451d17f5506b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:3.101ex; height:1.176ex;" alt="{\displaystyle \neg \neg }" loading="lazy"></span>E)

<p><br>

<mfrac>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\begin{array}{c}(A\to \bot )\to \bot \\\end{array}}{A}}}</annotation>

</semantics>

</math></span><img src="./87126e11d445d1f7bdff5ad51568f6d74af67e9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.209ex; height:5.843ex;" alt="{\displaystyle {\frac {\begin{array}{c}(A\to \bot )\to \bot \\\end{array}}{A}}}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./686ca19f81cdf5295452c8fe216c451d17f5506b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:3.101ex; height:1.176ex;" alt="{\displaystyle \neg \neg }" loading="lazy"></span>E)

</p>

</td>

<td valign="top"><b>Falsum Elimination</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 \bot }">

</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>E)

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

</mstyle>

</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>) one can derive any formula.<br>(ex falso quodlibet)

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

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9c269a3effe176c6cefe4a60837943b255ec2e99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.644ex; height:5.343ex;" alt="{\displaystyle {\frac {\bot }{A}}}" loading="lazy"></span> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }">

</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>E)

</p>

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

<ul><li><b><a href="Minimal_logic" title="Minimal logic">Minimal logic</a></b>: By limiting the <a href="Natural_deduction" title="Natural deduction">natural deduction</a> rules to <i>Implication Introduction</i> (<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 \to }">

</mstyle>

</mrow>

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

</semantics>

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E), one obtains (the implicational fragment of)<sup id="cite_ref-connective_needed_10-1" class="reference"><a href="#cite_note-connective_needed-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> minimal logic (as defined by <a href="Ingebrigt_Johansson" title="Ingebrigt Johansson">Johansson</a>).<sup id="cite_ref-FOOTNOTEJohansson1937_11-0" class="reference"><a href="#cite_note-FOOTNOTEJohansson1937-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul>

<th class="cot-header-mainspace" style="background:#ffffff; font-size:87%; padding:0.2em 0.3em; text-align:center; color:#202122"><div style="font-size:115%;margin:0 4em"><span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Proof of <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\to \neg \neg P\quad }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b5ac750814328dec6d7ecc37d170f585015f3ea7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.528ex; height:2.176ex;" alt="{\displaystyle P\to \neg \neg P\quad }" loading="lazy"></span>, within minimal logic</span></div>

</th></tr>

<tr>

<table>

</td>

<td>[ P ]

</td>

<td><span class="nowrap"> </span>// Assume

</td></tr>

<tr>

</td>

<td>[ P → ⊥ ]

</td>

<td><span class="nowrap"> </span>// Assume

</td></tr>

<tr>

</td>

<td>⊥

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E (1, 2)

</td></tr>

<tr>

</td>

<td>(P → ⊥) → ⊥)

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (2, 3), discharging 2

</td></tr>

<tr>

</td>

<td>P → ((P → ⊥) → ⊥)

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (1, 4), discharging 1

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

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

</div>

<ul><li><b><a href="Intuitionistic_logic" title="Intuitionistic logic">Intuitionistic logic</a></b>: By adding <i>Falsum Elimination</i> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }">

</mstyle>

</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>E) as a rule, one obtains (the implicational fragment of)<sup id="cite_ref-connective_needed_10-2" class="reference"><a href="#cite_note-connective_needed-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> intuitionistic logic.</li></ul>

<dl><dd>The statement <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\to \neg \neg P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./40e39162b78925788231e8debe44919114fc445e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.206ex; height:2.176ex;" alt="{\displaystyle P\to \neg \neg P}" loading="lazy"></span> is valid (already in minimal logic), unlike the reverse implication which would entail the <a href="Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a>.</dd></dl>

<ul><li><b><a href="Classical_logic" title="Classical logic">Classical logic</a></b>: If <i><a href="Double_negation" title="Double negation">Double Negation Elimination</a></i> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./686ca19f81cdf5295452c8fe216c451d17f5506b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:3.101ex; height:1.176ex;" alt="{\displaystyle \neg \neg }" loading="lazy"></span>E) is also permitted,<sup id="cite_ref-RAA_14-0" class="reference"><a href="#cite_note-RAA-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> the system defines (full!) classical logic.<sup id="cite_ref-FOOTNOTEPrawitz196521_12-1" class="reference"><a href="#cite_note-FOOTNOTEPrawitz196521-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEAyala-Rincónde_Moura201717–24_13-1" class="reference"><a href="#cite_note-FOOTNOTEAyala-Rincónde_Moura201717–24-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTETennant199048_15-0" class="reference"><a href="#cite_note-FOOTNOTETennant199048-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="A_selection_of_theorems_(classical_logic)">A selection of theorems (classical logic)</h2></div>

<p>In <a href="Classical_logic" title="Classical logic">classical logic</a> material implication validates the following:

</p>

<th class="cot-header-mainspace" style="background:#ffffff; font-size:87%; padding:0.2em 0.3em; text-align:center; color:#202122"><div style="font-size:115%;margin:0 4em"><span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Contraposition: <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 Q\to \neg P)\to (P\to Q)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0de4a784a0139080659f18cdab9351f6f01335a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.729ex; height:2.843ex;" alt="{\displaystyle (\neg Q\to \neg P)\to (P\to Q)}" loading="lazy"></span></span></div>

</th></tr>

<tr>

<table>

</td>

<td>[ (Q → ⊥) → (P → ⊥) ]

</td>

<td><span class="nowrap"> </span>// Assume (to discharge at 9)

</td></tr>

<tr>

</td>

<td>[ P ]

</td>

<td><span class="nowrap"> </span>// Assume (to discharge at 8)

</td></tr>

<tr>

</td>

<td>[ Q → ⊥ ]

</td>

<td><span class="nowrap"> </span>// Assume (to discharge at 6))

</td></tr>

<tr>

</td>

<td>P → ⊥

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E (1, 3)

</td></tr>

<tr>

</td>

<td>⊥

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E (2, 4)

</td></tr>

<tr>

</td>

<td>(Q → ⊥) → ⊥

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (3, 5) (discharging 3)

</td></tr>

<tr>

</td>

<td>Q

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./686ca19f81cdf5295452c8fe216c451d17f5506b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:3.101ex; height:1.176ex;" alt="{\displaystyle \neg \neg }" loading="lazy"></span>E (6)

</td></tr>

<tr>

</td>

<td>P → Q

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (2, 7) (discharging 2)

</td></tr>

<tr>

</td>

<td>((Q → ⊥) → (P → ⊥)) → (P → Q)

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (1, 8) (discharging 1)

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

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

</div>

<th class="cot-header-mainspace" style="background:#ffffff; font-size:87%; padding:0.2em 0.3em; text-align:center; color:#202122"><div style="font-size:115%;margin:0 4em"><span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;"><a href="Peirce's_law" title="Peirce's law">Peirce's law</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((P\to Q)\to P)\to P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./49578aceacd182b716c7a635cc964a2eef6b6bda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.535ex; height:2.843ex;" alt="{\displaystyle ((P\to Q)\to P)\to P}" loading="lazy"></span></span></div>

</th></tr>

<tr>

<table>

</td>

<td>[ (P → Q) → P ]

</td>

<td><span class="nowrap"> </span>// Assume (to discharge at 11)

</td></tr>

<tr>

</td>

<td>[ P → ⊥ ]

</td>

<td><span class="nowrap"> </span>// Assume (to discharge at 9)

</td></tr>

<tr>

</td>

<td>[ P ]

</td>

<td><span class="nowrap"> </span>// Assume (to discharge at 6)

</td></tr>

<tr>

</td>

<td>⊥

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E (2, 3)

</td></tr>

<tr>

</td>

<td>Q

</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>E (4)

</td></tr>

<tr>

</td>

<td>P → Q

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (3, 5) (discharging 3)

</td></tr>

<tr>

</td>

<td>P

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E (1, 6)

</td></tr>

<tr>

</td>

<td>⊥

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E (2, 7)

</td></tr>

<tr>

</td>

<td>(P → ⊥) → ⊥

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (2, 8) (discharging 2)

</td></tr>

<tr>

</td>

<td>P

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./686ca19f81cdf5295452c8fe216c451d17f5506b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:3.101ex; height:1.176ex;" alt="{\displaystyle \neg \neg }" loading="lazy"></span>E (9)

</td></tr>

<tr>

</td>

<td>((P → Q) → P) → P

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (1, 10) (discharging 1)

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

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

</div>

<th class="cot-header-mainspace" style="background:#ffffff; font-size:87%; padding:0.2em 0.3em; text-align:center; color:#202122"><div style="font-size:115%;margin:0 4em"><span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;"><a href="Vacuous_truth" title="Vacuous truth">Vacuous conditional</a> (IPC): <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\to (P\to Q)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./77db16115cb89a45f4831cff83098f2e3ce5b9e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.917ex; height:2.843ex;" alt="{\displaystyle \neg P\to (P\to Q)}" loading="lazy"></span></span></div>

</th></tr>

<tr>

<table>

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ab8801c12087e813015fb95a1cb13dfb515ddc1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.461ex; height:2.843ex;" alt="{\displaystyle [P\to \bot ]}" loading="lazy"></span>

</td>

<td><span class="nowrap"> </span>// Assume

</td></tr>

<tr>

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./25d78ad4ad13872df07ac9b02a2574250a0e54fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle [P]}" loading="lazy"></span>

</td>

<td><span class="nowrap"> </span>// Assume

</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 \to }</annotation>

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>E (1, 2)

</td></tr>

<tr>

</td>

</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>

</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>E (3)

</td></tr>

<tr>

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d7cad5b2c2991ae1dbded560c5d875fbf49fe8ea.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\to Q}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (2, 4) (discharging 2)

</td></tr>

<tr>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (P\to \bot )\to (P\to Q)}</annotation>

</semantics>

</math></span><img src="./256d220a2a7a9c0869fcb9aab66cd96c3c4fac3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.598ex; height:2.843ex;" alt="{\displaystyle (P\to \bot )\to (P\to Q)}" loading="lazy"></span>

</td>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1daab843254cfcb23a643070cf93f3badc4fbbbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }" loading="lazy"></span>I (1, 5) (discharging 1)

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

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

</div>

<ul><li><a href="Import-Export_(logic)" class="mw-redirect" title="Import-Export (logic)">Import-export</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to (Q\to R)\equiv (P\land Q)\to R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\to (Q\to R)\equiv (P\land Q)\to R}</annotation>

</semantics>

</math></span><img src="./c2ff649c0321573b5a051861317aa9bbfe013410.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.837ex; height:2.843ex;" alt="{\displaystyle P\to (Q\to R)\equiv (P\land Q)\to R}" loading="lazy"></span></li>

<li>Negated conditionals: <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\to Q)\equiv P\land \neg Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (P\to Q)\equiv P\land \neg Q}</annotation>

</semantics>

</math></span><img src="./c93a56ed24d4b679276ef835000b391ab98bc93e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.373ex; height:2.843ex;" alt="{\displaystyle \neg (P\to Q)\equiv P\land \neg Q}" loading="lazy"></span></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\to Q\equiv \neg P\lor Q}</annotation>

</semantics>

</math></span><img src="./e0ebbd885a87b001952561f6a20c615fe4e800f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.013ex; height:2.509ex;" alt="{\displaystyle P\to Q\equiv \neg P\lor Q}" loading="lazy"></span></li>

<li>Commutativity of antecedents: <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 {\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}}</annotation>

</semantics>

</math></span><img src="./7bc8f3d2e4f5a21f0282f82bf35b32f4b690f240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.129ex; height:3.176ex;" alt="{\displaystyle {\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}}" loading="lazy"></span></li>

<li><a href="Left_distributivity" class="mw-redirect" title="Left distributivity">Left distributivity</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big (}R\to (P\to Q){\big )}\equiv {\big (}(R\to P)\to (R\to Q){\big )}}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\big (}R\to (P\to Q){\big )}\equiv {\big (}(R\to P)\to (R\to Q){\big )}}</annotation>

</semantics>

</math></span><img src="./c18ffeb6bb414cd9fd2999761ce4d6624bd6553e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.316ex; height:3.176ex;" alt="{\displaystyle {\big (}R\to (P\to Q){\big )}\equiv {\big (}(R\to P)\to (R\to Q){\big )}}" loading="lazy"></span></li></ul>

<p>Similarly, on classical interpretations of the other connectives, material implication validates the following <a href="Logical_consequence#Semantic_consequence" title="Logical consequence">entailments</a>:

</p>

<ul><li>Antecedent strengthening: <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\to Q\models (P\land R)\to Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\to Q\models (P\land R)\to Q}</annotation>

</semantics>

</math></span><img src="./7f899c0a2618c91762d71f09832dbddc5a08d7ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.857ex; height:2.843ex;" alt="{\displaystyle P\to Q\models (P\land R)\to Q}" loading="lazy"></span></li>

<li><a href="Transitive_relation" title="Transitive relation">Transitivity</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\to Q)\land (Q\to R)\models P\to R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (P\to Q)\land (Q\to R)\models P\to R}</annotation>

</semantics>

</math></span><img src="./806fd3db14d25ed73d162cc95cbfaa416ad59b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.044ex; height:2.843ex;" alt="{\displaystyle (P\to Q)\land (Q\to R)\models P\to R}" loading="lazy"></span></li>

<li><a href="Simplification_of_disjunctive_antecedents" title="Simplification of disjunctive antecedents">Simplification of disjunctive antecedents</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\lor Q)\to R\models (P\to R)\land (Q\to R)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (P\lor Q)\to R\models (P\to R)\land (Q\to R)}</annotation>

</semantics>

</math></span><img src="./492c0745553efa1daa25b4355fceaab418ab66ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.2ex; height:2.843ex;" alt="{\displaystyle (P\lor Q)\to R\models (P\to R)\land (Q\to R)}" loading="lazy"></span></li></ul>

<p><a href="Tautology_(logic)" title="Tautology (logic)">Tautologies</a> involving material implication include:

</p>

<ul><li><a href="Reflexive_relation" title="Reflexive relation">Reflexivity</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models P\to P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3124ee503e74654ecd236c551d7572fc4d22693c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.765ex; height:2.843ex;" alt="{\displaystyle \models P\to P}" loading="lazy"></span></li>

<li><a href="Connex_relation" class="mw-redirect" title="Connex relation">Totality</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models (P\to Q)\lor (Q\to P)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \models (P\to Q)\lor (Q\to P)}</annotation>

</semantics>

</math></span><img src="./05840fba7a78368f9aa92ef42568cf162dac9a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.257ex; height:2.843ex;" alt="{\displaystyle \models (P\to Q)\lor (Q\to P)}" loading="lazy"></span></li>

<li><a href="Law_of_excluded_middle" title="Law of excluded middle">Conditional excluded middle</a>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models (P\to Q)\lor (P\to \neg Q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \models (P\to Q)\lor (P\to \neg Q)}</annotation>

</semantics>

</math></span><img src="./0e075de217f717cdea532f7ede48c3e168ce2db9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.807ex; height:2.843ex;" alt="{\displaystyle \models (P\to Q)\lor (P\to \neg Q)}" loading="lazy"></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Discrepancies_with_natural_language">Discrepancies with natural language</h2></div>

<p>Material implication does not closely match the usage of <a href="Conditional_sentence" title="Conditional sentence">conditional sentences</a> in <a href="Natural_language" title="Natural language">natural language</a>. For example, even though material conditionals with false antecedents are <a href="Vacuous_truth" title="Vacuous truth">vacuously true</a>, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris is in France". These classic problems have been called the <a href="Paradoxes_of_material_implication" title="Paradoxes of material implication">paradoxes of material implication</a>.<sup id="cite_ref-FOOTNOTEEdgington2008_16-0" class="reference"><a href="#cite_note-FOOTNOTEEdgington2008-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance, <a href="Counterfactual_conditional" title="Counterfactual conditional">counterfactual conditionals</a> would all be vacuously true on such an account, when in fact some are false.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p><p>In the mid-20th century, a number of researchers including <a href="Paul_Grice" title="Paul Grice">H. P. Grice</a> and <a href="Frank_Cameron_Jackson" title="Frank Cameron Jackson">Frank Jackson</a> proposed that <a href="Pragmatics" title="Pragmatics">pragmatic</a> principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals <a href="Denotation" title="Denotation">denote</a> material implication but end up conveying additional information when they interact with conversational norms such as <a href="Cooperative_principle#Grice's_maxims" title="Cooperative principle">Grice's maxims</a>.<sup id="cite_ref-FOOTNOTEEdgington2008_16-1" class="reference"><a href="#cite_note-FOOTNOTEEdgington2008-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEGillies2017_18-0" class="reference"><a href="#cite_note-FOOTNOTEGillies2017-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Recent work in <a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">formal semantics</a> and <a href="Philosophy_of_language" title="Philosophy of language">philosophy of language</a> has generally eschewed material implication as an analysis for natural-language conditionals.<sup id="cite_ref-FOOTNOTEGillies2017_18-1" class="reference"><a href="#cite_note-FOOTNOTEGillies2017-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In particular, such work has often rejected the assumption that natural-language conditionals are <a href="Truth_function" title="Truth function">truth functional</a> in the sense that the truth value of "If <i>P</i>, then <i>Q</i>" is determined solely by the truth values of <i>P</i> and <i>Q</i>.<sup id="cite_ref-FOOTNOTEEdgington2008_16-2" class="reference"><a href="#cite_note-FOOTNOTEEdgington2008-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as <a href="Modal_logic" title="Modal logic">modal logic</a>, <a href="Relevance_logic" title="Relevance logic">relevance logic</a>, <a href="Probability_theory" title="Probability theory">probability theory</a>, and <a href="Causal_graph" title="Causal graph">causal models</a>.<sup id="cite_ref-FOOTNOTEGillies2017_18-2" class="reference"><a href="#cite_note-FOOTNOTEGillies2017-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEEdgington2008_16-3" class="reference"><a href="#cite_note-FOOTNOTEEdgington2008-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEVon_Fintel2011_19-0" class="reference"><a href="#cite_note-FOOTNOTEVon_Fintel2011-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

</p><p>Similar discrepancies have been observed by psychologists studying conditional reasoning, for instance, by the notorious <a href="Wason_selection_task" title="Wason selection task">Wason selection task</a> study, where less than 10% of participants reasoned according to the material conditional. Some researchers have interpreted this result as a failure of the participants to conform to normative laws of reasoning, while others interpret the participants as reasoning normatively according to nonclassical laws.<sup id="cite_ref-FOOTNOTEOaksfordChater1994_20-0" class="reference"><a href="#cite_note-FOOTNOTEOaksfordChater1994-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEStenningvan_Lambalgen2004_21-0" class="reference"><a href="#cite_note-FOOTNOTEStenningvan_Lambalgen2004-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEVon_Sydow2006_22-0" class="reference"><a href="#cite_note-FOOTNOTEVon_Sydow2006-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p>

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

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

<li><a href="Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a></li>

<li><a href="Conditional_quantifier" title="Conditional quantifier">Conditional quantifier</a></li>

<li><a href="Implicational_propositional_calculus" title="Implicational propositional calculus">Implicational propositional calculus</a></li>

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

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

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

<li><a href="Peirce's_law" title="Peirce's law">Peirce's law</a></li>

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

<li><a href="Sole_sufficient_operator" class="mw-redirect" title="Sole sufficient operator">Sole sufficient operator</a></li></ul>

</div>

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

<ul><li><a href="Corresponding_conditional" title="Corresponding conditional">Corresponding conditional</a></li>

<li><a href="Counterfactual_conditional" title="Counterfactual conditional">Counterfactual conditional</a></li>

<li><a href="Indicative_conditional" title="Indicative conditional">Indicative conditional</a></li>

<li><a href="Strict_conditional" title="Strict conditional">Strict conditional</a></li></ul>

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

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<li id="cite_note-FOOTNOTEHilbert1918-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEHilbert1918_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEHilbert1918_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFHilbert1918">Hilbert 1918</a>.</span>

</li>

<li id="cite_note-FOOTNOTEMendelson2015-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMendelson2015_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMendelson2015">Mendelson 2015</a>.</span>

</li>

<li id="cite_note-FOOTNOTEVan_Heijenoort1967-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVan_Heijenoort1967_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVan_Heijenoort1967">Van Heijenoort 1967</a>.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂.</span>

</li>

<li id="cite_note-FOOTNOTENahas2022VI-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENahas2022VI_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNahas2022">Nahas 2022</a>, p. VI.</span>

</li>

<li id="cite_note-FOOTNOTEBourbaki195414-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBourbaki195414_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBourbaki1954">Bourbaki 1954</a>, p. 14.</span>

</li>

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

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</style><cite id="CITEREFMiller2020" class="citation web cs1">Miller, Jeff (2020). <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/">"Earliest Uses of Symbols for Set Theory and Logic"</a>. <i>Maths History (University of St Andrews)</i>. University of St Andrews<span class="reference-accessdate">. Retrieved <span class="nowrap">10 June</span> 2025</span>.</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">The <a href="Well-formed_formula" title="Well-formed formula">well-formed formulas</a> are:

<ol><li>Each <a href="Propositional_variable" title="Propositional variable">propositional variable</a> is a formula.</li>

</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>" is a formula.</li>

</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> 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 B}">

</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> are formulas, so 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="{\displaystyle (A\to B)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2874d32e52c03fcb9112bab28893069fed6fc3e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.93ex; height:2.843ex;" alt="{\displaystyle (A\to B)}" loading="lazy"></span>.</li>

<li>Nothing else is a formula.</li></ol>

</span></li>

<li id="cite_note-FOOTNOTEFrancoGoldsmithSchlipfSpeckenmeyer1999-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFrancoGoldsmithSchlipfSpeckenmeyer1999_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFrancoGoldsmithSchlipfSpeckenmeyer1999">Franco et al. 1999</a>.</span>

</li>

<li id="cite_note-connective_needed-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-connective_needed_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-connective_needed_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-connective_needed_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">f-implicational formulas cannot express all valid formulas in <a href="Minimal_logic" title="Minimal logic">minimal</a> (MPC) or <a href="Intuitionistic_logic" title="Intuitionistic logic">intuitionistic</a> (IPC) propositional logic — in particular, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ab47f6b1f589aedcf14638df1d63049d233d851a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }" loading="lazy"></span> (disjunction) cannot be defined within it. In contrast, <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 \{\to ,\lor ,\bot \}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0e3434a017d81e2f5a72f062e9829f2e8f9867b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.075ex; height:2.843ex;" alt="{\displaystyle \{\to ,\lor ,\bot \}}" loading="lazy"></span> is a complete basis for MPC / IPC: from these, all other connectives (e.g., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land ,\neg ,\leftrightarrow ,\bot }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2c576995e8a64417fdda58e1b804ecf05fbbb322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.334ex; height:2.509ex;" alt="{\displaystyle \land ,\neg ,\leftrightarrow ,\bot }" loading="lazy"></span>) can be defined.</span>

</li>

<li id="cite_note-FOOTNOTEJohansson1937-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJohansson1937_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJohansson1937">Johansson 1937</a>.</span>

</li>

<li id="cite_note-FOOTNOTEPrawitz196521-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEPrawitz196521_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEPrawitz196521_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFPrawitz1965">Prawitz 1965</a>, p. 21.</span>

</li>

<li id="cite_note-FOOTNOTEAyala-Rincónde_Moura201717–24-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEAyala-Rincónde_Moura201717–24_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAyala-Rincónde_Moura201717–24_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAyala-Rincónde_Moura2017">Ayala-Rincón & de Moura 2017</a>, pp. 17–24.</span>

</li>

<li id="cite_note-RAA-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-RAA_14-0">^</a></b></span> <span class="reference-text">Instead of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./686ca19f81cdf5295452c8fe216c451d17f5506b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:3.101ex; height:1.176ex;" alt="{\displaystyle \neg \neg }" loading="lazy"></span>E one can add <b><a href="Reductio_ad_absurdum" title="Reductio ad absurdum">reductio ad absurdum</a></b> as a rule to obtain (full) classical logic:<sup id="cite_ref-FOOTNOTEPrawitz196521_12-0" class="reference"><a href="#cite_note-FOOTNOTEPrawitz196521-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEAyala-Rincónde_Moura201717–24_13-0" class="reference"><a href="#cite_note-FOOTNOTEAyala-Rincónde_Moura201717–24-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

<mfrac>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\begin{array}{c}[A\to \bot ]\\\vdots \\\bot \end{array}}{A}}}</annotation>

</semantics>

</math></span><img src="./3aafc8561979ef204fe896f3eb3b0c2366fa4c7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.271ex; height:13.509ex;" alt="{\displaystyle {\frac {\begin{array}{c}[A\to \bot ]\\\vdots \\\bot \end{array}}{A}}}" loading="lazy"></span> (RAA)</dd></dl>

</span></li>

<li id="cite_note-FOOTNOTETennant199048-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETennant199048_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTennant1990">Tennant 1990</a>, p. 48.</span>

</li>

<li id="cite_note-FOOTNOTEEdgington2008-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEEdgington2008_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEEdgington2008_16-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEEdgington2008_16-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-FOOTNOTEEdgington2008_16-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFEdgington2008">Edgington 2008</a>.</span>

</li>

<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">For example, "If <a href="Janis_Joplin" title="Janis Joplin">Janis Joplin</a> were alive today, she would drive a <a href="Mercedes-Benz" title="Mercedes-Benz">Mercedes-Benz</a>", see <a href="#CITEREFStarr2019">Starr (2019)</a></span>

</li>

<li id="cite_note-FOOTNOTEGillies2017-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGillies2017_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGillies2017_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGillies2017_18-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGillies2017">Gillies 2017</a>.</span>

</li>

<li id="cite_note-FOOTNOTEVon_Fintel2011-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVon_Fintel2011_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVon_Fintel2011">Von Fintel 2011</a>.</span>

</li>

<li id="cite_note-FOOTNOTEOaksfordChater1994-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOaksfordChater1994_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOaksfordChater1994">Oaksford & Chater 1994</a>.</span>

</li>

<li id="cite_note-FOOTNOTEStenningvan_Lambalgen2004-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStenningvan_Lambalgen2004_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStenningvan_Lambalgen2004">Stenning & van Lambalgen 2004</a>.</span>

</li>

<li id="cite_note-FOOTNOTEVon_Sydow2006-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEVon_Sydow2006_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFVon_Sydow2006">Von Sydow 2006</a>.</span>

</li>

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

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

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<ul><li><cite id="CITEREFBourbaki1954" class="citation book cs1">Bourbaki, N. (1954). <i>Théorie des ensembles</i>. Paris: Hermann & Cie, Éditeurs. p. 14.</cite></li></ul>

<ul><li><cite id="CITEREFEdgington2008" class="citation encyclopaedia cs1">Edgington, Dorothy (2008). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/archives/win2008/entries/conditionals/">"Conditionals"</a>. In Edward N. Zalta (ed.). <i>The Stanford Encyclopedia of Philosophy</i> (Winter 2008 ed.).</cite></li></ul>

<ul><li><cite id="CITEREFVon_Fintel2011" class="citation encyclopaedia cs1">Von Fintel, Kai (2011). <a rel="nofollow" class="external text" href="http://mit.edu/fintel/fintel-2011-hsk-conditionals.pdf">"Conditionals"</a> <span class="cs1-format">(PDF)</span>. In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.). <i>Semantics: An international handbook of meaning</i>. de Gruyter Mouton. pp. <span class="nowrap">1515–</span>1538. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110255072.1515">10.1515/9783110255072.1515</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/1721.1%2F95781">1721.1/95781</a></span>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-11-018523-2</bdi>.</cite></li></ul>

<ul><li><cite id="CITEREFFrancoGoldsmithSchlipfSpeckenmeyer1999" class="citation journal cs1 cs1-prop-long-vol">Franco, John; Goldsmith, Judy; Schlipf, John; Speckenmeyer, Ewald; Swaminathan, R.P. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0166-218X%2899%2900038-4">"An algorithm for the class of pure implicational formulas"</a>. <i>Discrete Applied Mathematics</i>. <span class="nowrap">96–</span>97: <span class="nowrap">89–</span>106. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0166-218X%2899%2900038-4">10.1016/S0166-218X(99)00038-4</a></span>.</cite></li></ul>

<ul><li><cite id="CITEREFGillies2017" class="citation encyclopaedia cs1">Gillies, Thony (2017). <a rel="nofollow" class="external text" href="http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf">"Conditionals"</a> <span class="cs1-format">(PDF)</span>. In Hale, B.; Wright, C.; Miller, A. (eds.). <i>A Companion to the Philosophy of Language</i>. Wiley Blackwell. pp. <span class="nowrap">401–</span>436. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781118972090.ch17">10.1002/9781118972090.ch17</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118972090</bdi>.</cite></li></ul>

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<ul><li><cite id="CITEREFHilbert1918" class="citation book cs1">Hilbert, D. (1918). <i>Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.)</i>.</cite></li></ul>

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<ul><li><cite id="CITEREFNahas2022" class="citation web cs1">Nahas, Michael (25 Apr 2022). <a rel="nofollow" class="external text" href="https://github.com/mdnahas/Peano_Book/blob/46e27bdb5aed51c078ad99e5a78d134fd2a0c3ca/Peano.pdf">"English Translation of 'Arithmetices Principia, Nova Methodo Exposita'"</a> <span class="cs1-format">(PDF)</span>. GitHub<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-08-10</span></span>.</cite></li></ul>

<ul><li><cite id="CITEREFOaksfordChater1994" class="citation journal cs1">Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection". <i><a href="Psychological_Review" title="Psychological Review">Psychological Review</a></i>. <b>101</b> (4): <span class="nowrap">608–</span>631. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.174.4085">10.1.1.174.4085</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0033-295X.101.4.608">10.1037/0033-295X.101.4.608</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:2912209">2912209</a>.</cite></li></ul>

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<ul><li><cite id="CITEREFStarr2019" class="citation encyclopaedia cs1">Starr, Will (2019). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/fall2019/entries/counterfactuals">"Counterfactuals"</a>. In Zalta, Edward N. (ed.). <i>The Stanford Encyclopedia of Philosophy</i>.</cite></li></ul>

<ul><li><cite id="CITEREFStenningvan_Lambalgen2004" class="citation journal cs1">Stenning, K.; van Lambalgen, M. (2004). "A little logic goes a long way: basing experiment on semantic theory in the cognitive science of conditional reasoning". <i>Cognitive Science</i>. <b>28</b> (4): <span class="nowrap">481–</span>530. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.1854">10.1.1.13.1854</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cogsci.2004.02.002">10.1016/j.cogsci.2004.02.002</a>.</cite></li></ul>

<ul><li><cite id="CITEREFVon_Sydow2006" class="citation thesis cs1">Von Sydow, M. (2006). <a rel="nofollow" class="external text" href="https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-AC29-9"><i>Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules</i></a> (doctoralThesis). Göttingen: Göttingen University Press. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.53846%2Fgoediss-161">10.53846/goediss-161</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:246924881">246924881</a>.</cite></li></ul>

<ul><li><cite id="CITEREFTennant1990" class="citation book cs1">Tennant, Neil (1990) [1978]. <i>Natural Logic</i> (1st, repr. with corrections ed.). <a href="Edinburgh_University_Press" title="Edinburgh University Press">Edinburgh University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0852245793</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Brown, Frank Markham (2003), <i>Boolean Reasoning: The Logic of Boolean Equations</i>, 1st edition, <a href="Kluwer" class="mw-redirect" title="Kluwer">Kluwer</a> Academic Publishers, <a href="Norwell%2C_Massachusetts" title="Norwell, Massachusetts">Norwell</a>, MA. 2nd edition, <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="Mineola%2C_New_York" title="Mineola, New York">Mineola</a>, NY, 2003.</li>

<li><a href="Dorothy_Edgington" title="Dorothy Edgington">Edgington, Dorothy</a> (2001), "Conditionals", in Lou Goble (ed.), <i>The Blackwell Guide to Philosophical Logic</i>, <a href="Wiley-Blackwell" title="Wiley-Blackwell">Blackwell</a>.</li>

<li><a href="W._V._Quine" class="mw-redirect" title="W. V. Quine">Quine, W.V.</a> (1982), <i>Methods of Logic</i>, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, <a href="Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, <a href="Cambridge%2C_Massachusetts" title="Cambridge, Massachusetts">Cambridge</a>, MA.</li>

<li><a href="Robert_Stalnaker" title="Robert Stalnaker">Stalnaker, Robert</a>, "Indicative Conditionals", <i><a href="Philosophia_(journal)" title="Philosophia (journal)">Philosophia</a></i>, <b>5</b> (1975): 269–286.</li></ul>

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

<ul><li><cite id="CITEREFEdgington" class="citation encyclopaedia cs1">Edgington, Dorothy. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/conditionals/">"Conditionals"</a>. In <a href="Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Common_logical_connectives157" 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="Common_logical_connectives157" style="font-size:114%;margin:0 4em">Common <a href="Logical_connective" title="Logical connective">logical connectives</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a>/<a href="Logical_truth" title="Logical truth">True</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \top }">

</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></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"></span></div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sheffer_stroke" title="Sheffer stroke">Alternative denial</a> (<a href="NAND_gate" title="NAND gate">NAND gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\wedge }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2f076fb91b5d1276ab165ed8dfaa3cacab8d4cef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\wedge }}}" loading="lazy"></span></li>

<li><a href="Converse_(logic)" title="Converse (logic)">Converse implication</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./682eb97b10e06ba3d2dcc642ecd753d34dbb4ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftarrow }" loading="lazy"></span></li>

<li> (<a href="IMPLY_gate" title="IMPLY gate">IMPLY gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./469b737d167b9b28a74e27c7f5e35b5ea9256100.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }" loading="lazy"></span></li>

<li><a href="Logical_disjunction" title="Logical disjunction">Disjunction</a> (<a href="OR_gate" title="OR gate">OR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }">

</mstyle>

</mrow>

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

</semantics>

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

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Negation" title="Negation">Negation</a> (<a href="Inverter_(logic_gate)" title="Inverter (logic gate)">NOT gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }">

</mstyle>

</mrow>

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

</semantics>

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

<li><a href="Exclusive_or" title="Exclusive or">Exclusive or</a> (<a href="XOR_gate" title="XOR gate">XOR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8b16e2bdaefee9eed86d866e6eba3ac47c710f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }" loading="lazy"></span></li>

<li><a href="Logical_biconditional" title="Logical biconditional">Biconditional</a> (<a href="XNOR_gate" title="XNOR gate">XNOR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e89e009eb8a8839c82aa5c76c15e9f2d67006276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }" loading="lazy"></span></li>

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

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Logical_NOR" title="Logical NOR">Joint denial</a> (<a href="NOR_gate" title="NOR gate">NOR gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\vee }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a6f9bdf4cb18d1b79d370c396dc425e80f8340f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.665ex; height:2.843ex;" alt="{\displaystyle {\overline {\vee }}}" loading="lazy"></span></li>

<li><a href="Material_nonimplication" title="Material nonimplication">Nonimplication</a> (<a href="NIMPLY_gate" title="NIMPLY gate">NIMPLY gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nRightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3e05a42e88f019861cf404b6f982d8f729a4c4ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \nRightarrow }" loading="lazy"></span></li>

<li><a href="Converse_nonimplication" title="Converse nonimplication">Converse nonimplication</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nLeftarrow }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./746948304d4d6a4903cc7bf82bf687b6f284eb92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \nLeftarrow }" loading="lazy"></span></li>

<li><a href="Logical_conjunction" title="Logical conjunction">Conjunction</a> (<a href="AND_gate" title="AND gate">AND gate</a>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }">

</mstyle>

</mrow>

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

</semantics>

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

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contradiction" title="Contradiction">Contradiction</a>/<a href="False_(logic)" title="False (logic)">False</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }">

</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></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div><span class="nowrap"><span class="noviewer" typeof="mw:File"><span></span></span> </span><a href="Portal%3APhilosophy" title="Portal:Philosophy">Philosophy portal</a></div></td></tr></tbody></table></div>

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

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Horseshoe_(symbol)" title="Horseshoe (symbol)">⊃</a> </div> ,<br><a href="Subset" title="Subset">superset</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Turned_A" title="Turned A">∀</a> </div> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0; line-height:1.15em"><a href="Universal_quantification" title="Universal quantification">universal<br>quantification</a></div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Existential_quantification" title="Existential quantification">∃</a> </div> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0; line-height:1.15em"><a href="Existential_quantification" title="Existential quantification">existential<br>quantification</a></div>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Tee_(symbol)" title="Tee (symbol)">⊤</a> </div> <a href="True_(logic)" class="mw-redirect" title="True (logic)">true</a>,<br><a href="Tautology_(logic)" title="Tautology (logic)">tautology</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Up_tack" title="Up tack">⊥</a> </div> <a href="False_(logic)" title="False (logic)">false</a>,<br><a href="Contradiction" title="Contradiction">contradiction</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Turnstile_(symbol)" title="Turnstile (symbol)">⊢</a> </div> <a href="Turnstile_(symbol)" title="Turnstile (symbol)">entails,<br>proves</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Double_turnstile" title="Double turnstile">⊨</a> </div> <a href="Double_turnstile" title="Double turnstile">entails,<br>therefore</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Therefore_sign" title="Therefore sign">∴</a> </div> <a href="Logical_consequence" title="Logical consequence">therefore</a>

</div></td><td class="navbox-list" style="border-left:2px solid #fdfdfd;padding:0px;padding-top:0.85em;text-align:center;white-space:nowrap;padding-bottom:0.85em;width:10em;"><div>

<div style="font-size:150%;margin-bottom:0.55em;"> <a href="Therefore_sign#Similar_signs" title="Therefore sign">∵</a> </div> <a href="Therefore_sign#Similar_signs" title="Therefore sign">because</a>

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<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>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<li>Set operations:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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