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interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Regla_d%27infer%C3%A8ncia" title="Regla d'inferència – Catalan" lang="ca" hreflang="ca" data-title="Regla d'inferència" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Schlussregel" title="Schlussregel – German" lang="de" hreflang="de" data-title="Schlussregel" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tuletusreegel" title="Tuletusreegel – Estonian" lang="et" hreflang="et" data-title="Tuletusreegel" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BC%CF%80%CE%B5%CF%81%CE%B1%CF%83%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%BA%CE%B1%CE%BD%CF%8C%CE%BD%CE%B1%CF%82" title="Συμπερασματικός κανόνας – Greek" lang="el" hreflang="el" data-title="Συμπερασματικός κανόνας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Regla_de_inferencia" title="Regla de inferencia – Spanish" lang="es" hreflang="es" data-title="Regla de inferencia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D8%B9%D8%AF%D9%87_%D8%A7%D8%B3%D8%AA%D9%86%D8%AA%D8%A7%D8%AC" title="قاعده استنتاج – Persian" lang="fa" hreflang="fa" data-title="قاعده استنتاج" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A8gle_d%27inf%C3%A9rence" title="Règle d'inférence – French" lang="fr" hreflang="fr" data-title="Règle d'inférence" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B6%94%EB%A1%A0_%EA%B7%9C%EC%B9%99" title="추론 규칙 – Korean" lang="ko" hreflang="ko" data-title="추론 규칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aturan_penarikan_kesimpulan" title="Aturan penarikan kesimpulan – Indonesian" lang="id" hreflang="id" data-title="Aturan penarikan kesimpulan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Regola_di_inferenza" title="Regola di inferenza – Italian" lang="it" hreflang="it" data-title="Regola di inferenza" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%99%D7%A1%D7%A7" title="כלל היסק – Hebrew" lang="he" hreflang="he" data-title="כלל היסק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Afleidingsregel" title="Afleidingsregel – Dutch" lang="nl" hreflang="nl" data-title="Afleidingsregel" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8E%A8%E8%AB%96%E8%A6%8F%E5%89%87" title="推論規則 – Japanese" lang="ja" hreflang="ja" data-title="推論規則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Regu%C5%82a_dedukcyjna" title="Reguła dedukcyjna – Polish" lang="pl" hreflang="pl" data-title="Reguła dedukcyjna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Regra_de_infer%C3%AAncia" title="Regra de inferência – Portuguese" lang="pt" hreflang="pt" data-title="Regra de inferência" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%B8%D0%BB%D0%BE_%D0%B2%D1%8B%D0%B2%D0%BE%D0%B4%D0%B0" title="Правило вывода – Russian" lang="ru" hreflang="ru" data-title="Правило вывода" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Rule_of_inference" title="Rule of inference – Simple English" lang="en-simple" hreflang="en-simple" data-title="Rule of inference" data-language-autonym="Simple English" data-language-local-name="Simple 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li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Transformation rules</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%; border-bottom:1px #fefefe solid;"> <a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a class="mw-selflink selflink">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Conditional_proof" title="Conditional proof"><span>Implication introduction</span></a> / <a href="/wiki/Modus_ponens" title="Modus ponens"><span title="A→B,   A   ⊢   B">elimination (<i>modus ponens</i>)</span></a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction"><span title="A→B,   B→A   ⊢   A↔B">Biconditional introduction</span></a> / <a href="/wiki/Biconditional_elimination" title="Biconditional elimination"><span title="A↔B   ⊢   A→B">elimination</span></a></li> <li><a href="/wiki/Conjunction_introduction" title="Conjunction introduction"><span title="A,   B   ⊢   A∧B">Conjunction introduction</span></a> / <a href="/wiki/Conjunction_elimination" title="Conjunction elimination"><span title="A∧B   ⊢   A">elimination</span></a></li> <li><a href="/wiki/Disjunction_introduction" title="Disjunction introduction"><span title="A   ⊢   A∨B">Disjunction introduction</span></a> / <a href="/wiki/Disjunction_elimination" title="Disjunction elimination"><span title="A∨B,   A→C,   B→C   ⊢   C">elimination</span></a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism"><span title="A∨B,   ¬A   ⊢   B">Disjunctive</span></a> / <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism"><span title="A→B,   B→C   ⊢   A→C">hypothetical syllogism</span></a></li> <li><a href="/wiki/Constructive_dilemma" title="Constructive dilemma"><span title="A→P,   B→Q,   A∨B   ⊢   P∨Q">Constructive</span></a> / <a href="/wiki/Destructive_dilemma" title="Destructive dilemma"><span title="A→P,   B→Q,   ¬P∨¬Q   ⊢   ¬A∨¬B">destructive dilemma</span></a></li> <li><a href="/wiki/Absorption_(logic)" title="Absorption (logic)"><span title="A→B   ⊢   A→A∧B">Absorption</span></a> / <a href="/wiki/Modus_tollens" title="Modus tollens"><span title="A→B,   ¬B   ⊢   ¬A"><i>modus tollens</i></span></a> / <a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens"><span title="¬(A∧B),   A   ⊢   ¬B"><i>modus ponendo tollens</i></span></a></li> <li><a href="/wiki/Negation_introduction" title="Negation introduction">Negation introduction</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_replacement" title="Rule of replacement">Rules of replacement</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <div class="hlist"> <ul><li><a href="/wiki/Associative_property#Propositional_logic" title="Associative property"><span title="A∨(B∨C)   =   (A∨B)∨C">Associativity</span></a></li> <li><a href="/wiki/Commutative_property#Propositional_logic" title="Commutative property"><span title="A∨B   =   B∨A">Commutativity</span></a></li> <li><a href="/wiki/Distributive_property#Propositional_logic" title="Distributive property"><span title="A∧(B∨C)   =   (A∧B)∨(A∧C)">Distributivity</span></a></li> <li><a href="/wiki/Double_negation" title="Double negation"><span title="¬¬A   =   A">Double negation</span></a></li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li> <li><a href="/wiki/Material_implication_(rule_of_inference)" title="Material implication (rule of inference)"><span title="A→B   ⊢   ¬A∨B">Material implication</span></a></li> <li><a href="/wiki/Exportation_(logic)" title="Exportation (logic)"><span title="(A∧B)→C   ⊢   A→(B→C)">Exportation</span></a></li> <li><a href="/wiki/Tautology_(rule_of_inference)" title="Tautology (rule of inference)"><span title="A∨A   =   A">Tautology</span></a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%;"> <a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a class="mw-selflink selflink">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Universal_generalization" title="Universal generalization">Universal generalization</a> / <a href="/wiki/Universal_instantiation" title="Universal instantiation">instantiation</a></li> <li><a href="/wiki/Existential_generalization" title="Existential generalization">Existential generalization</a> / <a href="/wiki/Existential_instantiation" title="Existential instantiation">instantiation</a></li></ul></td> </tr></tbody></table> <p>In <a href="/wiki/Logic" title="Logic">logic</a> and the <a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">philosophy of logic</a>, specifically in <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive reasoning</a>, a <b>rule of inference</b>, <b>inference rule</b> or <b>transformation rule</b> is a <a href="/wiki/Logical_form" title="Logical form">logical form</a> consisting of a function which takes premises, analyzes their <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a>, and returns a conclusion (or <a href="/wiki/Multiple-conclusion_logic" title="Multiple-conclusion logic">conclusions</a>). </p><p>For example, the rule of inference called <i><a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a></i> takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> with respect to the semantics of <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a> (as well as the semantics of many other <a href="/wiki/Non-classical_logic" title="Non-classical logic">non-classical logics</a>), in the sense that if the premises are true (under an interpretation), then so is the conclusion. </p><p>Typically, a rule of inference preserves truth, a semantic property. In <a href="/wiki/Many-valued_logic" title="Many-valued logic">many-valued logic</a>, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are <a href="/wiki/Recursion" title="Recursion">recursive</a> are important; i.e. rules such that there is an <a href="/wiki/Effective_procedure" class="mw-redirect" title="Effective procedure">effective procedure</a> for determining whether any given formula is the conclusion of a given set of formulae according to the rule. An example of a rule that is not effective in this sense is the infinitary <a href="/wiki/%CE%A9-consistent_theory" title="Ω-consistent theory">ω-rule</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Popular rules of inference in <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> include <i><a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a></i>, <i><a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a></i>, and <a href="/wiki/Contraposition" title="Contraposition">contraposition</a>. First-order <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a> uses rules of inference to deal with <a href="/wiki/Logical_quantifier" class="mw-redirect" title="Logical quantifier">logical quantifiers</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Standard_form">Standard form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rule_of_inference&action=edit&section=1" title="Edit section: Standard form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Formal_logic" class="mw-redirect" title="Formal logic">formal logic</a> (and many related areas), rules of inference are usually given in the following standard form: </p><p>  Premise#1 <br />  Premise#2 <br />        <b>...</b> <br /><u>  Premise#n   </u> <br />  Conclusion </p><p>This expression states that whenever in the course of some logical derivation the given premises have been obtained, the specified conclusion can be taken for granted as well. The exact formal language that is used to describe both premises and conclusions depends on the actual context of the derivations. In a simple case, one may use logical formulae, such as in: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\to B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8dd84619daff17b52a08b77d15db2b9ad6c2a" 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}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {A\quad \quad \quad }}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>A</mi> <mspace width="1em" /> <mspace width="1em" /> <mspace width="1em" /> </mrow> <mo>_</mo> </munder> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {A\quad \quad \quad }}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae14f3e7411747757d5ca86f6816926fd30500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-right: -0.387ex; margin-bottom: -0.827ex; width:9.1ex; height:3.176ex;" alt="{\displaystyle {\underline {A\quad \quad \quad }}\,\!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0862a1de92638c6dbf56966deeb873becc27ec3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.38ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle B\!}"></span></dd></dl> <p>This is the <i><a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a></i> rule of <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>. Rules of inference are often formulated as <a href="/wiki/Schema_(logic)" class="mw-redirect" title="Schema (logic)">schemata</a> employing <a href="/wiki/Metavariable" title="Metavariable">metavariables</a>.<sup id="cite_ref-Reynolds2009_2-0" class="reference"><a href="#cite_note-Reynolds2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the rule (schema) above, the metavariables A and B can be instantiated to any element of the universe (or sometimes, by convention, a restricted subset such as <a href="/wiki/Proposition" title="Proposition">propositions</a>) to form an <a href="/wiki/Infinite_set" title="Infinite set">infinite set</a> of inference rules. </p><p>A proof system is formed from a set of rules chained together to form proofs, also called <i>derivations</i>. Any derivation has only one final conclusion, which is the statement proved or derived. If premises are left unsatisfied in the derivation, then the derivation is a proof of a <i>hypothetical</i> statement: "<i>if</i> the premises hold, <i>then</i> the conclusion holds." </p> <div class="mw-heading mw-heading2"><h2 id="Example:_Hilbert_systems_for_two_propositional_logics">Example: Hilbert systems for two propositional logics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rule_of_inference&action=edit&section=2" title="Edit section: Example: Hilbert systems for two propositional logics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert system</a>, the premises and conclusion of the inference rules are simply formulae of some language, usually employing metavariables. For graphical compactness of the presentation and to emphasize the distinction between axioms and rules of inference, this section uses the <a href="/wiki/Sequent" title="Sequent">sequent</a> notation (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c0d30cf8cb7dba179e317fcde9583d842e80f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }"></span>) instead of a vertical presentation of rules. In this notation, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}{\text{Premise }}1\\{\text{Premise }}2\\\hline {\text{Conclusion}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="none solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Premise </mtext> </mrow> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Premise </mtext> </mrow> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Conclusion</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{\text{Premise }}1\\{\text{Premise }}2\\\hline {\text{Conclusion}}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2ca0736b56770850a9c2149ac9bfb27d339d15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:13.732ex; height:10.176ex;" alt="{\displaystyle {\begin{array}{c}{\text{Premise }}1\\{\text{Premise }}2\\\hline {\text{Conclusion}}\end{array}}}"></span> </p><p>is written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\text{Premise }}1),({\text{Premise }}2)\vdash ({\text{Conclusion}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Premise </mtext> </mrow> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Premise </mtext> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⊢</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Conclusion</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\text{Premise }}1),({\text{Premise }}2)\vdash ({\text{Conclusion}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edc721df013604df1859e8767f065450d844214c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.898ex; height:2.843ex;" alt="{\displaystyle ({\text{Premise }}1),({\text{Premise }}2)\vdash ({\text{Conclusion}})}"></span>. </p><p>The formal language for classical <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> can be expressed using just negation (¬), implication (→) and propositional symbols. A well-known axiomatization, comprising three axiom schemata and one inference rule (<i>modus ponens</i>), is: </p> <pre>(CA1) ⊢ <i>A</i> → (<i>B</i> → <i>A</i>)<br /> (CA2) ⊢ (<i>A</i> → (<i>B</i> → <i>C</i>)) → ((<i>A</i> → <i>B</i>) → (<i>A</i> → <i>C</i>))<br /> (CA3) ⊢ (¬<i>A</i> → ¬<i>B</i>) → (<i>B</i> → <i>A</i>)<br /> (MP) <i>A</i>, <i>A</i> → <i>B</i> ⊢ <i>B</i> </pre> <p>It may seem redundant to have two notions of inference in this case, ⊢ and →. In classical propositional logic, they indeed coincide; the <a href="/wiki/Deduction_theorem" title="Deduction theorem">deduction theorem</a> states that <i>A</i> ⊢ <i>B</i> if and only if ⊢ <i>A</i> → <i>B</i>. There is however a distinction worth emphasizing even in this case: the first notation describes a <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deduction</a>, that is an activity of passing from sentences to sentences, whereas <i>A</i> → <i>B</i> is simply a formula made with a <a href="/wiki/Logical_connective" title="Logical connective">logical connective</a>, implication in this case. Without an inference rule (like <i>modus ponens</i> in this case), there is no deduction or inference. This point is illustrated in <a href="/wiki/Lewis_Carroll" title="Lewis Carroll">Lewis Carroll</a>'s dialogue called "<a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a>",<sup id="cite_ref-ChiaraDoets1996_3-0" class="reference"><a href="#cite_note-ChiaraDoets1996-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> as well as later attempts by <a href="/wiki/What_the_Tortoise_Said_to_Achilles#Discussion" title="What the Tortoise Said to Achilles">Bertrand Russell and Peter Winch</a> to resolve the paradox introduced in the dialogue. </p><p>For some non-classical logics, the deduction theorem does not hold. For example, the <a href="/wiki/Three-valued_logic" title="Three-valued logic">three-valued logic</a> of <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Łukasiewicz</a> can be axiomatized as:<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> </p> <pre>(CA1) ⊢ <i>A</i> → (<i>B</i> → <i>A</i>)<br /> (LA2) ⊢ (<i>A</i> → <i>B</i>) → ((<i>B</i> → <i>C</i>) → (<i>A</i> → <i>C</i>))<br /> (CA3) ⊢ (¬<i>A</i> → ¬<i>B</i>) → (<i>B</i> → <i>A</i>)<br /> (LA4) ⊢ ((<i>A</i> → ¬<i>A</i>) → <i>A</i>) → <i>A</i><br /> (MP) <i>A</i>, <i>A</i> → <i>B</i> ⊢ <i>B</i> </pre> <p>This sequence differs from classical logic by the change in axiom 2 and the addition of axiom 4. The classical deduction theorem does not hold for this logic, however a modified form does hold, namely <i>A</i> ⊢ <i>B</i> if and only if ⊢ <i>A</i> → (<i>A</i> → <i>B</i>).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Admissibility_and_derivability">Admissibility and derivability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rule_of_inference&action=edit&section=3" title="Edit section: Admissibility and derivability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Admissible_rule" title="Admissible rule">Admissible rule</a></div> <p>In a set of rules, an inference rule could be redundant in the sense that it is <i>admissible</i> or <i>derivable</i>. A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is one whose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the difference, consider the following set of rules for defining the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> (the <a href="/wiki/Natural_deduction" title="Natural deduction">judgment</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,\,{\mathsf {nat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\,\,{\mathsf {nat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56836978b9bf5e4a172bb62d6a808b7bdb9056de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.329ex; height:1.843ex;" alt="{\displaystyle n\,\,{\mathsf {nat}}}"></span> asserts the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is a natural number): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}{\begin{array}{c}\\\hline {\mathbf {0} \,\,{\mathsf {nat}}}\end{array}}&{\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\end{array}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\begin{array}{c}\\\hline {\mathbf {0} \,\,{\mathsf {nat}}}\end{array}}&{\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\end{array}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb37bbe426f552b53a29aff1992099c599c0d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.027ex; height:7.176ex;" alt="{\displaystyle {\begin{matrix}{\begin{array}{c}\\\hline {\mathbf {0} \,\,{\mathsf {nat}}}\end{array}}&{\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\end{array}}\end{matrix}}}"></span></dd></dl> <p>The first rule states that <b>0</b> is a natural number, and the second states that <b>s(</b><i>n</i><b>)</b> is a natural number if <i>n</i> is. In this proof system, the following rule, demonstrating that the second successor of a natural number is also a natural number, is derivable: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(s(} n\mathbf {))} \,\,{\mathsf {nat}}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">s</mi> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">)</mo> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(s(} n\mathbf {))} \,\,{\mathsf {nat}}}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b14176857073903c03c73cb41b8f87f10c775f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.207ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}{n\,\,{\mathsf {nat}}}\\\hline {\mathbf {s(s(} n\mathbf {))} \,\,{\mathsf {nat}}}\end{array}}}"></span></dd></dl> <p>Its derivation is the composition of two uses of the successor rule above. The following rule for asserting the existence of a predecessor for any nonzero number is merely admissible: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}{\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\\\hline {n\,\,{\mathsf {nat}}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}{\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\\\hline {n\,\,{\mathsf {nat}}}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e03dacbbf021cffc457fc83e164942f9143fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.073ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}{\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}\\\hline {n\,\,{\mathsf {nat}}}\end{array}}}"></span></dd></dl> <p>This is a true fact of natural numbers, as can be proven by <a href="/wiki/Mathematical_induction" title="Mathematical induction">induction</a>. (To prove that this rule is admissible, assume a derivation of the premise and induct on it to produce a derivation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\,\,{\mathsf {nat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\,\,{\mathsf {nat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56836978b9bf5e4a172bb62d6a808b7bdb9056de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.329ex; height:1.843ex;" alt="{\displaystyle n\,\,{\mathsf {nat}}}"></span>.) However, it is not derivable, because it depends on the structure of the derivation of the premise. Because of this, derivability is stable under additions to the proof system, whereas admissibility is not. To see the difference, suppose the following nonsense rule were added to the proof system: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{c}\\\hline {\mathbf {s(-3)} \,\,{\mathsf {nat}}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em" rowlines="solid"> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mo mathvariant="bold">−</mo> <mn mathvariant="bold">3</mn> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c}\\\hline {\mathbf {s(-3)} \,\,{\mathsf {nat}}}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d283949be3ec29e8e975ad039ab73599261d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.092ex; height:7.176ex;" alt="{\displaystyle {\begin{array}{c}\\\hline {\mathbf {s(-3)} \,\,{\mathsf {nat}}}\end{array}}}"></span></dd></dl> <p>In this new system, the double-successor rule is still derivable. However, the rule for finding the predecessor is no longer admissible, because there is no way to derive <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {-3} \,\,{\mathsf {nat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">−</mo> <mn mathvariant="bold">3</mn> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">a</mi> <mi mathvariant="sans-serif">t</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {-3} \,\,{\mathsf {nat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88aceb4e1d50e19e019223220992aa6ad01c8d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.348ex; height:2.176ex;" alt="{\displaystyle \mathbf {-3} \,\,{\mathsf {nat}}}"></span>. The brittleness of admissibility comes from the way it is proved: since the proof can induct on the structure of the derivations of the premises, extensions to the system add new cases to this proof, which may no longer hold. </p><p>Admissible rules can be thought of as <a href="/wiki/Theorem" title="Theorem">theorems</a> of a proof system. For instance, in a <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculus</a> where <a href="/wiki/Cut_elimination" class="mw-redirect" title="Cut elimination">cut elimination</a> holds, the <i>cut</i> rule is admissible. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rule_of_inference&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Argumentation_scheme" title="Argumentation scheme">Argumentation scheme</a></li> <li><a href="/wiki/Immediate_inference" title="Immediate inference">Immediate inference</a></li> <li><a href="/wiki/Inference_objection" class="mw-redirect" title="Inference objection">Inference objection</a></li> <li><a href="/wiki/Law_of_thought" title="Law of thought">Law of thought</a></li> <li><a href="/wiki/List_of_rules_of_inference" title="List of rules of inference">List of rules of inference</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Structural_rule" title="Structural rule">Structural rule</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rule_of_inference&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBoolosBurgessJeffrey2007" class="citation book cs1">Boolos, George; Burgess, John; Jeffrey, Richard C. (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/computabilitylog0000bool/page/364"><i>Computability and logic</i></a>. Cambridge: Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/computabilitylog0000bool/page/364">364</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-87752-7" title="Special:BookSources/978-0-521-87752-7"><bdi>978-0-521-87752-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computability+and+logic&rft.place=Cambridge&rft.pages=364&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-87752-7&rft.aulast=Boolos&rft.aufirst=George&rft.au=Burgess%2C+John&rft.au=Jeffrey%2C+Richard+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcomputabilitylog0000bool%2Fpage%2F364&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARule+of+inference" class="Z3988"></span></span> </li> <li id="cite_note-Reynolds2009-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Reynolds2009_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_C._Reynolds2009" class="citation book cs1">John C. Reynolds (2009) [1998]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2OwlTC4SOccC&pg=PA12"><i>Theories of Programming Languages</i></a>. Cambridge University Press. p. 12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-10697-9" title="Special:BookSources/978-0-521-10697-9"><bdi>978-0-521-10697-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theories+of+Programming+Languages&rft.pages=12&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-10697-9&rft.au=John+C.+Reynolds&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2OwlTC4SOccC%26pg%3DPA12&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARule+of+inference" class="Z3988"></span></span> </li> <li id="cite_note-ChiaraDoets1996-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-ChiaraDoets1996_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKosta_Dosen1996" class="citation book cs1">Kosta Dosen (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TCthvF8xLIAC&pg=PA290">"Logical consequence: a turn in style"</a>. In <a href="/wiki/Maria_Luisa_Dalla_Chiara" title="Maria Luisa Dalla Chiara">Maria Luisa Dalla Chiara</a>; Kees Doets; Daniele Mundici; Johan van Benthem (eds.). <i>Logic and Scientific Methods: Volume One of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995</i>. Springer. p. 290. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-4383-7" title="Special:BookSources/978-0-7923-4383-7"><bdi>978-0-7923-4383-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Logical+consequence%3A+a+turn+in+style&rft.btitle=Logic+and+Scientific+Methods%3A+Volume+One+of+the+Tenth+International+Congress+of+Logic%2C+Methodology+and+Philosophy+of+Science%2C+Florence%2C+August+1995&rft.pages=290&rft.pub=Springer&rft.date=1996&rft.isbn=978-0-7923-4383-7&rft.au=Kosta+Dosen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTCthvF8xLIAC%26pg%3DPA290&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARule+of+inference" class="Z3988"></span> <a rel="nofollow" class="external text" href="http://www.mi.sanu.ac.rs/~kosta/LOGCONS.pdf">preprint (with different pagination)</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBergmann2008" class="citation book cs1">Bergmann, Merrie (2008). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoma00mber"><i>An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems</i></a></span>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoma00mber/page/n113">100</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88128-9" title="Special:BookSources/978-0-521-88128-9"><bdi>978-0-521-88128-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+many-valued+and+fuzzy+logic%3A+semantics%2C+algebras%2C+and+derivation+systems&rft.pages=100&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-88128-9&rft.aulast=Bergmann&rft.aufirst=Merrie&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoma00mber&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARule+of+inference" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBergmann2008" class="citation book cs1">Bergmann, Merrie (2008). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoma00mber"><i>An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems</i></a></span>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoma00mber/page/n127">114</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88128-9" title="Special:BookSources/978-0-521-88128-9"><bdi>978-0-521-88128-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+many-valued+and+fuzzy+logic%3A+semantics%2C+algebras%2C+and+derivation+systems&rft.pages=114&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-88128-9&rft.aulast=Bergmann&rft.aufirst=Merrie&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoma00mber&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARule+of+inference" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/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="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_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="/wiki/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="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/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="/wiki/Propositional_calculus" 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="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/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="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/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="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/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="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/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="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/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="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/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="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/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="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/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="/wiki/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="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a class="mw-selflink selflink">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/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"> 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