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Modus ponens

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</style><div role="note" class="hatnote navigation-not-searchable">"Forward reasoning" redirects here. For other uses, see <a href="Forward_chaining" title="Forward chaining">Forward chaining</a>.</div>

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</style><table class="infobox vcard"><caption class="infobox-title fn" style="padding-bottom:0.2em;">Modus ponens</caption><tbody><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">

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<ul><li><a href="Deductive_reasoning" title="Deductive reasoning">Deductive</a> <a href="Argument_form" class="mw-redirect" title="Argument form">argument form</a></li>

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

</div></td></tr><tr><th scope="row" class="infobox-label">Field</th><td class="infobox-data"><div class="plainlist">

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

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

</div></td></tr><tr><th scope="row" class="infobox-label">Statement</th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><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"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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

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</math><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">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><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"> is true. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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

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</math><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"> must also be true.</td></tr><tr><th scope="row" class="infobox-label">Symbolic statement</th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to Q,\;P\;\vdash \ Q}">

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

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</math><img src="./eab15e54594e644f0ead345aa509ddd4d234d837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.397ex; height:2.509ex;" alt="{\displaystyle P\to Q,\;P\;\vdash \ Q}" loading="lazy"></td></tr></tbody></table>

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</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="Rule_of_inference" title="Rule of inference">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="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;">

<a href="Rule_of_inference" title="Rule of inference">Rules of inference</a> (<a href="List_of_rules_of_inference" title="List of rules of inference">List</a>)</th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;">

<ul><li><a href="Conditional_proof" title="Conditional proof">Implication introduction</a> / </li>

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

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

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

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

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

<li><a href="Absorption_(logic)" title="Absorption (logic)">Absorption</a> / <a href="Modus_tollens" title="Modus tollens">modus tollens</a> / <a href="Modus_ponendo_tollens" title="Modus ponendo tollens">modus ponendo tollens</a></li>

<li><a href="Modus_non_excipiens" title="Modus non excipiens">Modus non excipiens</a></li>

<li><a href="Negation_introduction" title="Negation introduction">Negation introduction</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="background:#eaeaff;">

<a href="Rule_of_replacement" title="Rule of replacement">Rules of replacement</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;">

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

<li><a href="Commutative_property#Propositional_logic" title="Commutative property">Commutativity</a></li>

<li><a href="Distributive_property#Propositional_logic" title="Distributive property">Distributivity</a></li>

<li><a href="Double_negation" title="Double negation">Double negation</a></li>

<li><a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a></li>

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

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

<li><a href="Exportation_(logic)" title="Exportation (logic)">Exportation</a></li>

<li><a href="Tautology_(rule_of_inference)" title="Tautology (rule of inference)">Tautology</a></li></ul>

</div></td>

</tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%;">

<a href="First-order_logic" title="First-order logic">Predicate logic</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;">

<a href="Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;">

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

<li><a href="Existential_generalization" title="Existential generalization">Existential generalization</a> / <a href="Existential_instantiation" title="Existential instantiation">instantiation</a></li></ul></td>

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In <a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">propositional logic</a>, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP), also known as modus ponendo ponens (from <a href="Latin_language" class="mw-redirect" title="Latin language">Latin</a> 'mode that by affirming affirms'),<a href="#cite_note-1">[1]</a> implication elimination, or affirming the antecedent,<a href="#cite_note-2">[2]</a> is a <a href="Deductive_reasoning" title="Deductive reasoning">deductive</a> <a href="Argument_form" class="mw-redirect" title="Argument form">argument form</a> and <a href="Rule_of_inference" title="Rule of inference">rule of inference</a>.<a href="#cite_note-3">[3]</a> It can be summarized as "P <a href="Material_conditional" title="Material conditional">implies</a> Q. P is true. Therefore, Q must also be true."

Modus ponens is a mixed <a href="Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a> and is closely related to another <a href="Validity_(logic)" title="Validity (logic)">valid</a> form of argument, <a href="Modus_tollens" title="Modus tollens">modus tollens</a>. Both have apparently similar but invalid forms: <a href="Affirming_the_consequent" title="Affirming the consequent">affirming the consequent</a> and <a href="Denying_the_antecedent" title="Denying the antecedent">denying the antecedent</a>. <a href="Constructive_dilemma" title="Constructive dilemma">Constructive dilemma</a> is the <a href="Logical_disjunction" title="Logical disjunction">disjunctive</a> version of modus ponens.

The history of modus ponens goes back to <a href="Classical_antiquity" title="Classical antiquity">antiquity</a>.<a href="#cite_note-4">[4]</a> The first to explicitly describe the argument form modus ponens was <a href="Theophrastus" title="Theophrastus">Theophrastus</a>.<a href="#cite_note-5">[5]</a> It, along with <a href="Modus_tollens" title="Modus tollens">modus tollens</a>, is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal.

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

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The form of a modus ponens argument is a mixed <a href="Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a>, with two premises and a conclusion:

<li>Therefore, Q.</li></ol>

The first premise is a <a href="Material_conditional" title="Material conditional">conditional</a> ("if–then") claim, namely that P implies Q. The second premise is an assertion that P, the <a href="Antecedent_(logic)" title="Antecedent (logic)">antecedent</a> of the conditional claim, is the case. From these two premises it can be logically concluded that Q, the <a href="Consequent" title="Consequent">consequent</a> of the conditional claim, must be the case as well.

An example of an argument that fits the form modus ponens:

<ol><li>If today is Tuesday, then John will go to work.</li>

<li>Today is Tuesday.</li>

<li>Therefore, John will go to work.</li></ol>

This argument is <a href="Validity_(logic)" title="Validity (logic)">valid</a>, but this has no bearing on whether any of the statements in the argument are actually <a href="Truth" title="Truth">true</a>; for modus ponens to be a <a href="Soundness" title="Soundness">sound</a> argument, the premises must be true for any true instances of the conclusion. An <a href="Argument" title="Argument">argument</a> can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound. For example, John might be going to work on Wednesday. In this case, the reasoning for John's going to work (because it is Wednesday) is unsound. The argument is only sound on Tuesdays (when John goes to work), but valid on every day of the week. A <a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">propositional</a> argument using modus ponens is said to be <a href="Deductive_reasoning" title="Deductive reasoning">deductive</a>.

In single-conclusion <a href="Sequent_calculus" title="Sequent calculus">sequent calculi</a>, modus ponens is the Cut rule. The <a href="Cut-elimination_theorem" title="Cut-elimination theorem">cut-elimination theorem</a> for a calculus says that every proof involving Cut can be transformed (generally, by a constructive method) into a proof without Cut, and hence that Cut is <a href="Admissible_rule" title="Admissible rule">admissible</a>.

The <a href="Curry%E2%80%93Howard_correspondence" title="Curry–Howard correspondence">Curry–Howard correspondence</a> between proofs and programs relates modus ponens to <a href="Function_application" title="Function application">function application</a>: if f is a function of type P → Q and x is of type P, then f x is of type Q.

In <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, modus ponens is often called <a href="Forward_chaining" title="Forward chaining">forward chaining</a>.

<div class="mw-heading mw-heading2"><h2 id="Formal_notation">Formal notation</h2></div>

<th>p

</th>

<th>q

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<th>p → q

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<td>T

</td></tr>

<tr>

<td>F

</td></tr>

<tr>

<td>T

</td></tr>

<tr>

<td>T

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

The modus ponens rule may be written in <a href="Sequent" title="Sequent">sequent</a> notation as

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

</semantics>

</math><img src="./9cfe9d41c26581c2a6fb6d4dd49275f53d5df619.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.752ex; height:2.509ex;" alt="{\displaystyle P\to Q,\;P\;\;\vdash \;\;Q}" loading="lazy"></dd></dl>

where P, Q and P → Q are statements (or propositions) in a formal language and <a href="%E2%8A%A2" class="mw-redirect" title="⊢">⊢</a> is a <a href="Metalogic" title="Metalogic">metalogical</a> symbol meaning that Q is a <a href="Logical_consequence" title="Logical consequence">syntactic consequence</a> of P and P → Q in some <a href="Formal_system" title="Formal system">logical system</a>.

In classical two-valued logic, modus ponens can be interpreted from the <a href="Material_conditional" title="Material conditional">material conditional</a> (implication) operator. The notion of implication as a rigorous mathematical construct can be traced back to around the turn of the 19th century with early works of mathematical logic, such as <a href="Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a> and <a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a>. Given two variables p and q that can either be true or false, implication (p → q) is defined as all cases being true except where p is true and q is false.<a href="#cite_note-6">[6]</a> By the assumptions of modus ponens, p → q and p are true, and by the definition of implication, it can be deduced that q must also be true.

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

While modus ponens is one of the most commonly used <a href="Argument_form" class="mw-redirect" title="Argument form">argument forms</a> in logic, it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution".<a href="#cite_note-7">[7]</a> Modus ponens allows one to eliminate a <a href="Material_conditional" title="Material conditional">conditional statement</a> from a <a href="Formal_proof" title="Formal proof">logical proof or argument</a> (the antecedents) and thereby not carry these antecedents forward in an ever-lengthening string of symbols; for this reason modus ponens is sometimes called the rule of detachment<a href="#cite_note-8">[8]</a> or the law of detachment.<a href="#cite_note-9">[9]</a> Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",<a href="#cite_note-10">[10]</a> and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q [the consequent] ... an inference is the dropping of a true premise; it is the dissolution of an implication".<a href="#cite_note-auto-11">[11]</a>

A justification for the "trust in inference is the belief that if the two former assertions [the antecedents] are not in error, the final assertion [the consequent] is not in error".<a href="#cite_note-auto-11">[11]</a> In other words: if one <a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">statement</a> or <a href="Proposition" title="Proposition">proposition</a> <a href="Material_conditional" title="Material conditional">implies</a> a second one, and the first statement or proposition is true, then the second one is also true. If P implies Q and P is true, then Q is true.<a href="#cite_note-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Correspondence_to_other_mathematical_frameworks">Correspondence to other mathematical frameworks</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Algebraic_semantics">Algebraic semantics</h3></div>

In mathematical logic, <a href="Algebraic_semantics_(mathematical_logic)" title="Algebraic semantics (mathematical logic)"> algebraic semantics</a> treats every sentence as a name for an element in an ordered set. Typically, the set can be visualized as a <a href="Lattice_(order)" title="Lattice (order)"> lattice</a>-like structure with a single element (the "always-true") at the top and another single element (the "always-false") at the bottom. Logical equivalence becomes identity, so that when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {(P\wedge Q)}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4be161515235d3370fbe696fd3139d4e796b1e2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.526ex; height:2.843ex;" alt="{\displaystyle \neg {(P\wedge Q)}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {P}\vee \neg {Q}}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b2742b9536b178ed58f26cf33e741c6e370992a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.267ex; height:2.509ex;" alt="{\displaystyle \neg {P}\vee \neg {Q}}" loading="lazy">, for instance, are equivalent (as is standard), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c68e6647f92a05127918c8dda35f56184c5b1e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.891ex; height:2.843ex;" alt="{\displaystyle \neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}}" loading="lazy">. Logical implication becomes a matter of relative position: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> logically implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> just in case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\leq Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./988a9d9c9076581ea7a51e0a16cdd755d5bf3bb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P\leq Q}" loading="lazy">, i.e., when either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2abc7e2c5a78e9e6cb7a2a907279953f9b4a3f52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P=Q}" loading="lazy"> or else <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> lies below <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> and is connected to it by an upward path.

In this context, to say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./038590207af1024a629c1a08c855e9ac46bf5610.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="{\textstyle P}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86439ea857adc8eaec93c4d14270b8ba6bd2a6a9.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\rightarrow Q}" loading="lazy"> together imply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

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

</semantics>

</math><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">—that is, to affirm modus ponens as valid—is to say that the highest point which lies below both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86439ea857adc8eaec93c4d14270b8ba6bd2a6a9.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\rightarrow Q}" loading="lazy"> lies below <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, i.e., that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e39e185d682e36e2200579ab31b26b303a9b70ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.272ex; height:2.843ex;" alt="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}" loading="lazy">.<a href="#cite_note-13">[a]</a> In the semantics for basic propositional logic, the algebra is <a href="Boolean_algebra_(structure)" title="Boolean algebra (structure)"> Boolean</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./53e574cc3aa5b4bf5f3f5906caf121a378eef08b.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"> construed as the <a href="Material_conditional" title="Material conditional">material conditional</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q=\neg {P}\vee Q}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2d18fcbec1d1f3b4670d59b70f19089639e3a915.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\rightarrow Q=\neg {P}\vee Q}" loading="lazy">. Confirming that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e39e185d682e36e2200579ab31b26b303a9b70ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.272ex; height:2.843ex;" alt="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}" loading="lazy"> is then straightforward, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge (P\rightarrow Q)=P\wedge Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./50828a331ead18c5b93547b2ef79722650bce3c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.6ex; height:2.843ex;" alt="{\displaystyle P\wedge (P\rightarrow Q)=P\wedge Q}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge Q\leq Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c680779a5b9efbdd7043b58c1aa393ce05f1eed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.103ex; height:2.509ex;" alt="{\displaystyle P\wedge Q\leq Q}" loading="lazy">. With other treatments of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./53e574cc3aa5b4bf5f3f5906caf121a378eef08b.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">, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted.

<div class="mw-heading mw-heading3"><h3 id="Probability_calculus">Probability calculus</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(P\rightarrow Q)=x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ab2f7dc3d5d4fa14d812eae3cbd60b0afc89abec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.93ex; height:2.843ex;" alt="{\displaystyle \Pr(P\rightarrow Q)=x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(P)=y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c7962bbc6ffff74f479141205a52a92472bcced8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.303ex; height:2.843ex;" alt="{\displaystyle \Pr(P)=y}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Q)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9c629eb750321a80c64ad3529aded1a33e35f0bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.142ex; height:2.843ex;" alt="{\displaystyle \Pr(Q)}" loading="lazy"> must lie in the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x+y-1,x]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x+y-1,x]}</annotation>

</semantics>

</math><img src="./451d2bceec60acd6fc3f90b5f28bf3a51cfb0b47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.986ex; height:2.843ex;" alt="{\displaystyle [x+y-1,x]}" loading="lazy">.<a href="#cite_note-14">[b]</a><a href="#cite_note-Hailperin,_T._1996-15">[13]</a> For the special case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y=1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./52620e42f1d8b888bc7664321a6e8bf6318d6df8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.845ex; height:2.509ex;" alt="{\displaystyle x=y=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Q)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9c629eb750321a80c64ad3529aded1a33e35f0bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.142ex; height:2.843ex;" alt="{\displaystyle \Pr(Q)}" loading="lazy"> must equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Subjective_logic">Subjective logic</h3></div>

Modus ponens represents an instance of the binomial deduction operator in <a href="Subjective_logic" title="Subjective logic">subjective logic</a> expressed as:

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\circledcirc \omega _{P}^{A}\,,}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}" loading="lazy"> denotes the subjective opinion about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> as expressed by source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, and the conditional opinion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q|P}^{A}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{Q|P}^{A}}</annotation>

</semantics>

</math><img src="./1fbcadc50e7dbdbb35fea79b4b71a2c00305c529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.67ex; height:3.676ex;" alt="{\displaystyle \omega _{Q|P}^{A}}" loading="lazy"> generalizes the logical implication <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><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">. The deduced marginal opinion about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q\|P}^{A}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{Q\|P}^{A}}</annotation>

</semantics>

</math><img src="./20dad6bcf5e3ab6771b6c23ac043f94082495db3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.034ex; height:3.676ex;" alt="{\displaystyle \omega _{Q\|P}^{A}}" loading="lazy">. The case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}" loading="lazy"> is an absolute TRUE opinion about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is equivalent to source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> saying that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is TRUE, and the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}" loading="lazy"> is an absolute FALSE opinion about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is equivalent to source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> saying that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> is FALSE. The deduction operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circledcirc }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cf70a53592b87a725eabcbb2dffc880e9aa9b66c.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 \circledcirc }" loading="lazy"> of <a href="Subjective_logic" title="Subjective logic">subjective logic</a> produces an absolute TRUE deduced opinion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q\|P}^{A}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{Q\|P}^{A}}</annotation>

</semantics>

</math><img src="./20dad6bcf5e3ab6771b6c23ac043f94082495db3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.034ex; height:3.676ex;" alt="{\displaystyle \omega _{Q\|P}^{A}}" loading="lazy"> when the conditional opinion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q|P}^{A}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{Q|P}^{A}}</annotation>

</semantics>

</math><img src="./1fbcadc50e7dbdbb35fea79b4b71a2c00305c529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.67ex; height:3.676ex;" alt="{\displaystyle \omega _{Q|P}^{A}}" loading="lazy"> is absolute TRUE and the antecedent opinion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}" loading="lazy"> is absolute TRUE. Hence, subjective logic deduction represents a generalization of both modus ponens and the <a href="Law_of_total_probability" title="Law of total probability">Law of total probability</a>.<a href="#cite_note-16">[14]</a>

<div class="mw-heading mw-heading2"><h2 id="Alleged_cases_of_failure">Alleged cases of failure</h2></div>

Philosophers and linguists have identified a variety of cases where modus ponens appears to fail. Vann McGee, for instance, argued that modus ponens can fail for conditionals whose consequents are themselves conditionals.<a href="#cite_note-17">[15]</a> The following is an example:

<ol><li>Either <a href="Shakespeare" class="mw-redirect" title="Shakespeare">Shakespeare</a> or <a href="Thomas_Hobbes" title="Thomas Hobbes">Hobbes</a> wrote <a href="Hamlet" title="Hamlet">Hamlet</a>.</li>

<li>If either Shakespeare or Hobbes wrote Hamlet, then if Shakespeare did not do it, Hobbes did.</li>

<li>Therefore, if Shakespeare did not write Hamlet, Hobbes did it.</li></ol>

Since Shakespeare did write Hamlet, the first premise is true. The second premise is also true, since starting with a set of possible authors limited to just Shakespeare and Hobbes and eliminating one of them leaves only the other. However, the conclusion is doubtful, since ruling out Shakespeare as the author of Hamlet would leave numerous possible candidates, many of them more plausible alternatives than Hobbes (if the if-thens in the inference are read as material conditionals, the conclusion comes out true simply by virtue of the false antecedent. This is one of the <a href="Paradoxes_of_material_implication" title="Paradoxes of material implication">paradoxes of material implication</a>).

The general form of McGee-type counterexamples to modus ponens is simply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P,P\rightarrow (Q\rightarrow R)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./efa231e26c737e13fdd3488af510cbd9a5afcdbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.165ex; height:2.843ex;" alt="{\displaystyle P,P\rightarrow (Q\rightarrow R)}" loading="lazy">, therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\rightarrow R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b76e444a574635586201a7ce3424d8777e3d6e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.216ex; height:2.509ex;" alt="{\displaystyle Q\rightarrow R}" loading="lazy">; it is not essential that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> be a disjunction, as in the example given. That these kinds of cases constitute failures of modus ponens remains a controversial view among logicians, but opinions vary on how the cases should be disposed of, and some authors have rejected McGee's argument.<a href="#cite_note-18">[16]</a><a href="#cite_note-19">[17]</a><a href="#cite_note-20">[18]</a><a href="#cite_note-21">[19]</a><a href="#cite_note-22">[20]</a>

In <a href="Deontic_logic" title="Deontic logic">deontic logic</a>, some examples of conditional obligation also raise the possibility of modus ponens failure. These are cases where the conditional premise describes an obligation predicated on an immoral or imprudent action, e.g., "If Doe murders his mother, he ought to do so gently," for which the dubious unconditional conclusion would be "Doe ought to gently murder his mother."<a href="#cite_note-SEP_Deontic_Logic-23">[21]</a> It would appear to follow that if Doe is in fact gently murdering his mother, then by modus ponens it appears that he is doing exactly what he should, unconditionally, be doing. Here again, modus ponens' failure is not a popular diagnosis and is criticized by some authors, but is sometimes argued for.<a href="#cite_note-24">[22]</a>

<div class="mw-heading mw-heading2"><h2 id="Possible_fallacies">Possible fallacies</h2></div>

The fallacy of <a href="Affirming_the_consequent" title="Affirming the consequent">affirming the consequent</a> is a common misinterpretation of the modus ponens.<a href="#cite_note-25">[23]</a>

<ul><li><a href="Condensed_detachment" title="Condensed detachment">Condensed detachment</a></li>

<li><a href="Import-export_(logic)" class="mw-redirect" title="Import-export (logic)">Import-export (logic)</a> – Principle of classical logicPages displaying short descriptions of redirect targets</li>

<li><a href="Latin_phrases" class="mw-redirect" title="Latin phrases">Latin phrases</a></li>

<li><a href="Modus_tollens" title="Modus tollens">Modus tollens</a> – Rule of logical inference</li>

<li><a href="Modus_vivendi" title="Modus vivendi">Modus vivendi</a> – Arrangement that allows conflicting parties to coexist in peace</li>

<li><a href="Stoic_logic" class="mw-redirect" title="Stoic logic">Stoic logic</a> – Virtue-focused philosophical systemPages displaying short descriptions of redirect targets</li>

<li><a href="What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a> – 1895 allegorical dialogue by Lewis Carroll</li></ul>

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

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</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-13"><a href="#cite_ref-13">^</a> The highest point that lies below both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> is the "<a href="Join_and_meet" title="Join and meet">meet</a>" of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6edc6915b42026ef5d46c585f7e44955f2d15ecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\wedge Y}" loading="lazy">.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}" loading="lazy"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86439ea857adc8eaec93c4d14270b8ba6bd2a6a9.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\rightarrow Q}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> must always be greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f2914ce99dee5c9e92ccedf35d1f455d319ddf36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.158ex; height:2.509ex;" alt="{\displaystyle 1-y}" loading="lazy">, and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y-1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9a43835d4c846620b3e473907aaa125cccd5dffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.328ex; height:2.509ex;" alt="{\displaystyle x+y-1}" loading="lazy"> will be greater than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">. And since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> must always be less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y-1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9a43835d4c846620b3e473907aaa125cccd5dffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.328ex; height:2.509ex;" alt="{\displaystyle x+y-1}" loading="lazy"> must always be less than or equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">.

</li>

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

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

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</style><cite id="CITEREFStone1996" class="citation book cs1">Stone, Jon R. (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/latinforillitera0000ston">Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language</a>. London: Routledge. p. <a rel="nofollow" class="external text" href="https://archive.org/details/latinforillitera0000ston/page/60">60</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-415-91775-1</bdi>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a rel="nofollow" class="external text" href="https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095354544">"Oxford reference: affirming the antecedent"</a>. <a href="Oxford_Reference" title="Oxford Reference">Oxford Reference</a>.

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> Enderton 2001:110

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="Susanne_Bobzien" title="Susanne Bobzien">Susanne Bobzien</a> (2002). "The Development of Modus Ponens in Antiquity", Phronesis 47, No. 4, 2002.

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBobzien2020" class="citation encyclopaedia cs1">Bobzien, Susanne (15 April 2020). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-ancient/">"Ancient Logic"</a>. In <a href="Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFArlo-Costa2021" class="citation encyclopaedia cs1">Arlo-Costa, Horacio (3 July 2021). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-conditionals/">"The Logic of Conditionals"</a>. In <a href="Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> Alfred Tarski 1946:47. Also Enderton 2001:110ff.

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> Tarski 1946:47

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Modus_ponens">"Modus ponens - Encyclopedia of Mathematics"</a>. encyclopediaofmath.org. Retrieved 5 April 2018.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> Enderton 2001:111

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<li id="cite_note-auto-11">^ <a href="#cite_ref-auto_11-0">a</a> <a href="#cite_ref-auto_11-1">b</a> Whitehead and Russell 1927:9

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFJago2007" class="citation book cs1">Jago, Mark (2007). Formal Logic. Humanities-Ebooks LLP. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84760-041-7</bdi>.</cite>

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<li id="cite_note-Hailperin,_T._1996-15"><a href="#cite_ref-Hailperin,_T._1996_15-0">^</a> <cite id="CITEREFHailperin1996" class="citation book cs1">Hailperin, Theodore (1996). Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications. London: Associated University Presses. p. 203. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0934223459</bdi>.</cite>

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<li id="cite_note-16"><a href="#cite_ref-16">^</a> Audun Jøsang 2016:92

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> Vann McGee (1985). "A Counterexample to Modus Ponens", The Journal of Philosophy 82, 462–471.

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<li id="cite_note-18"><a href="#cite_ref-18">^</a> Sinnott-Armstrong, Moor, and Fogelin (1986). "A Defense of Modus Ponens", The Journal of Philosophy 83, 296–300.

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<li id="cite_note-19"><a href="#cite_ref-19">^</a> D. E. Over (1987). "Assumption and the Supposed Counterexamples to Modus Ponens", Analysis 47, 142–146.

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<li id="cite_note-20"><a href="#cite_ref-20">^</a> Bledin (2015). "Modus Ponens Defended", The Journal of Philosophy 112, 462–471.

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<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFPiller1996" class="citation journal cs1">Piller, Christian (1 January 1996). <a rel="nofollow" class="external text" href="https://www.academia.edu/23974622/Vann_McGees_counterexample_to_modus_ponens">"Vann McGee's counterexample to modus ponens"</a>. Philosophical Studies. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00355293">10.1007/BF00355293</a>.</cite>

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<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFPriest1992" class="citation journal cs1">Priest, Graham (1992). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2253915">"Review of Truth, Vagueness and Paradox: An Essay on the Logic of Truth"</a>. Mind. 101 (403): 586–590. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-4423">0026-4423</a>.</cite>

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<li id="cite_note-SEP_Deontic_Logic-23"><a href="#cite_ref-SEP_Deontic_Logic_23-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-deontic/#4.5">"Deontic Logic"</a>. 21 April 2010. Retrieved 30 January 2020.</cite> <a href="Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.

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<li id="cite_note-24"><a href="#cite_ref-24">^</a> E.g., by Kolodny and MacFarlane (2010). "Ifs and Oughts", The Journal of Philosophy 107, 115–143.

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.iep.utm.edu/fallacy/">"Fallacies | Internet Encyclopedia of Philosophy"</a>. iep.utm.edu. Retrieved 6 March 2020.</cite>

</li>

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

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

<ul><li>Herbert B. Enderton, 2001, A Mathematical Introduction to Logic Second Edition, Harcourt Academic Press, Burlington MA, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-238452-3</bdi>.</li>

<li>Audun Jøsang, 2016, Subjective Logic; A formalism for Reasoning Under Uncertainty Springer, Cham, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-42337-1</bdi></li>

<li><a href="Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> and <a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> 1927 Principia Mathematica to *56 (Second Edition) paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.</li>

<li><a href="Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> 1946 Introduction to Logic and to the Methodology of the Deductive Sciences 2nd Edition, reprinted by Dover Publications, Mineola NY. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-28462-X</bdi> (pbk).</li></ul>

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

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

<li><a rel="nofollow" class="external text" href="https://philpapers.org/s/modus_ponens">Modus ponens</a> at <a href="PhilPapers" title="PhilPapers">PhilPapers</a></li>

<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ModusPonens.html">Modus ponens</a> at Wolfram MathWorld</li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Classical_logic61" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Classical_logic61" style="font-size:114%;margin:0 4em"><a href="Classical_logic" title="Classical logic">Classical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classical logics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

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

<ul><li><a href="Commutativity_of_conjunction" title="Commutativity of conjunction">Commutativity of conjunction</a></li>

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

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

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

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

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

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

<ul><li><a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a></li>

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

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

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

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

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

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

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

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

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

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

<ul><li><a href="Negation_introduction" title="Negation introduction">Negation</a></li>

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

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

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

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

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

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

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

<ul><li><a href="Double_negation_elimination" class="mw-redirect" title="Double negation elimination">Double negation</a></li>

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

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

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

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

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

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

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

<ul><li><a href="Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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