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{{short description|A rule of logical inference}}
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In [[propositional calculus|propositional logic]], '''''modus ponens''''' ({{IPAc-en|ˈ|m|oʊ|d|ə|s|_|ˈ|p|oʊ|n|ɛ|n|z}}; '''MP'''; also '''''modus ponendo ponens''''' ([[Latin]] for "mode that by affirming affirms")<ref>{{cite book |last=Stone |first=Jon R. |year=1996 |title=Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language |location=London |publisher=Routledge |isbn=0-415-91775-1 |page=60 |url=https://books.google.com/books?id=p6_KRzXsnKIC&pg=PA60 }}</ref> or '''implication elimination''') is a [[rule of inference]].<ref>Enderton 2001:110</ref> It can be summarized as "''P [[material conditional|implies]] Q'' and ''P'' is asserted to be true, therefore ''Q'' must be true."

''Modus ponens'' is closely related to another valid form of argument, ''[[modus tollens]]''. Both have apparently similar but invalid forms such as [[affirming the consequent]],  [[denying the antecedent]], and [[evidence of absence]]. [[Constructive dilemma]] is the [[Logical disjunction|disjunctive]] version of ''modus ponens''. [[Hypothetical syllogism]] is closely related to ''modus ponens'' and sometimes thought of as "double ''modus ponens''."

The history of ''modus ponens'' goes back to [[Classical antiquity|antiquity]].<ref>[[Susanne Bobzien]] (2002). "The Development of Modus Ponens in Antiquity", ''Phronesis'' 47, No. 4, 2002.</ref> The first to explicitly describe the argument form ''modus ponens'' was [[Theophrastus]].<ref>[http://plato.stanford.edu/entries/logic-ancient/#StoSyl "Ancient Logic: Forerunners of ''Modus Ponens'' and ''Modus Tollens''"]. ''[[Stanford Encyclopedia of Philosophy]].</ref>

== Formal notation ==

The ''modus ponens'' rule may be written in [[sequent]] notation as
:<math>P \to Q,\; P\;\; \vdash\;\; Q</math>

where ''P'', ''Q'' and ''P'' → ''Q'' are statements (or propositions) in a formal language and [[⊢]] is a [[metalogic]]al symbol meaning that ''Q'' is a [[Logical consequence|syntactic consequence]] of ''P'' and ''P'' → ''Q'' in some [[formal system|logical system]].

== Explanation ==

The argument form has two premises (hypothesis). The first premise is the "if–then" or [[Material conditional|conditional]] claim, namely that ''P'' implies ''Q''. The second premise is that ''P'', the [[Antecedent (logic)|antecedent]] of the conditional claim, is true. From these two premises it can be logically concluded that ''Q'', the [[consequent]] of the conditional claim, must be true as well. In [[artificial intelligence]], ''modus ponens'' is often called [[forward chaining]].

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

:If today is Tuesday, then John will go to work.
:Today is Tuesday.
:Therefore, John will go to work.

This argument is valid, but this has no bearing on whether any of the statements in the argument are [[Truth|true]]; for ''modus ponens'' to be a sound argument, the premises must be true for any true instances of the conclusion. An [[argument]] can be valid but nonetheless [[Soundness|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 not only sound on Tuesdays (when John goes to work), but valid on every day of the week. A [[propositional calculus|propositional]] argument using ''modus ponens'' is said to be [[Deductive reasoning|deductive]].

In single-conclusion [[sequent calculus|sequent calculi]], ''modus ponens'' is the Cut rule. The [[cut-elimination theorem]] 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 [[admissible rule|admissible]].

The [[Curry–Howard correspondence]] between proofs and programs relates ''modus ponens'' to [[function application]]:  if ''f'' is a function of type ''P'' → ''Q'' and ''x'' is of type ''P'', then ''f x'' is of type ''Q''.

==Justification via truth table==
The validity of ''modus ponens'' in classical two-valued logic can be clearly demonstrated by use of a [[truth table]].
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center;" class="wikitable"
|-
! ''p''
! ''q''
! ''p'' → ''q''
|- 
| T || T || T
|- style="color:red"
| T || F || F
|- style="color:pink"
| F || T || T
|- style="color:blue"
| F || F || T
|}

In instances of ''modus ponens'' we assume as premises that ''p'' → ''q'' is true and ''p'' is true. Only one line of the truth table—the first—satisfies these two conditions (''p''  and ''p'' → ''q''). On this line, ''q'' is also true. Therefore, whenever ''p'' → ''q'' is true and ''p'' is true, ''q'' must also be true.

== Status ==

While ''modus ponens'' is one of the most commonly used [[argument form]]s 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".<ref>Alfred Tarski 1946:47. Also Enderton 2001:110ff.</ref> ''Modus ponens'' allows one to eliminate a [[material conditional|conditional statement]] from a [[formal proof|logical proof or argument]] (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'''<ref>Tarski 1946:47</ref> or the '''law of detachment'''.<ref>{{cite web|url=https://www.encyclopediaofmath.org/index.php/Modus_ponens|title=Modus ponens - Encyclopedia of Mathematics|author=|date=|website=www.encyclopediaofmath.org|accessdate=5 April 2018}}</ref> Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",<ref>Enderton 2001:111</ref> 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".<ref name="auto">Whitehead and Russell 1927:9</ref>

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".<ref name="auto"/> In other words: if one [[statement (logic)|statement]] or [[proposition]] [[material conditional|implies]] 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.<ref>{{cite book | last=Jago | first=Mark | title=Formal Logic | publisher=[http://www.humanities-ebooks.co.uk/ Humanities-Ebooks LLP] |year= 2007 |isbn=978-1-84760-041-7 }}</ref>

==Correspondence to other mathematical frameworks==
===Probability calculus===
''Modus ponens'' represents an instance of the [[Law of total probability]] which for a binary variable is expressed as:

<math>\Pr(Q)=\Pr(Q\mid P)\Pr(P)+\Pr(Q\mid \lnot P)\Pr(\lnot P)\,</math>,

where e.g. <math>\Pr(Q)</math> denotes the probability of <math>Q</math> and the [[conditional probability]] <math>\Pr(Q\mid P)</math> generalizes the logical implication <math>P \to Q</math>. Assume that <math>\Pr(Q) = 1</math> is equivalent to <math>Q</math> being TRUE, and that <math>\Pr(Q) = 0</math> is equivalent to <math>Q</math> being FALSE.  It is then easy to see that <math>\Pr(Q) = 1</math> when <math>\Pr(Q\mid P) = 1</math> and <math>\Pr(P) = 1</math>. Hence, the [[law of total probability]] represents a generalization of ''modus ponens'' <ref>Audun Jøsang 2016:2</ref>.

===Subjective logic===
''Modus ponens'' represents an instance of the binomial deduction operator in [[subjective logic]] expressed as:

<math>\omega^{A}_{Q\|P}= (\omega^{A}_{Q|P},\omega^{A}_{Q|\lnot P})\circledcirc \omega^{A}_{P}\,</math>,

where <math>\omega^{A}_{P}</math> denotes the subjective opinion about <math>P</math> as expressed by source <math>A</math>, and the conditional opinion <math>\omega^{A}_{Q|P}</math> generalizes the logical implication <math>P \to Q</math>. The deduced marginal opinion about <math>Q</math> is denoted by <math>\omega^{A}_{Q\|P}</math>. The case where <math>\omega^{A}_{P}</math> is an absolute TRUE opinion about <math>P</math> is equivalent to source <math>A</math> saying that <math>P</math> is TRUE, and the case where <math>\omega^{A}_{P}</math> is an absolute FALSE opinion about <math>P</math> is equivalent to source <math>A</math> saying that <math>P</math> is FALSE.  The deduction operator <math>\circledcirc</math> of [[subjective logic]] produces an absolute TRUE deduced opinion <math>\omega^{A}_{Q\|P}</math> when the conditional opinion <math>\omega^{A}_{Q|P}</math> is absolute TRUE and the antecedent opinion <math>\omega^{A}_{P}</math> is absolute TRUE. Hence, subjective logic deduction represents a generalization of both ''modus ponens'' and the [[Law of total probability]] <ref>Audun Jøsang 2016:92</ref>.

==Alleged cases of failure==

The philosopher and logician Vann McGee has argued that ''modus ponens'' can fail to be valid when the consequent is itself a conditional sentence.<ref>Vann McGee (1985). "A Counterexample to Modus Ponens", ''The Journal of Philosophy'' 82, 462–471.</ref> Here is an example:

:Either [[Shakespeare]] or [[Thomas Hobbes|Hobbes]] wrote ''[[Hamlet]]''.
:If either Shakespeare or Hobbes wrote ''Hamlet'', then if Shakespeare didn't do it, Hobbes did.
:Therefore, if Shakespeare didn't write ''Hamlet'', Hobbes did it.

The first premise seems reasonable enough, because Shakespeare is generally credited with writing ''Hamlet''. The second premise seems reasonable, as well, because with the set of <i>Hamlet</i>'s possible authors limited to just Shakespeare and Hobbes, eliminating one leaves only the other. But the conclusion is dubious, because if Shakespeare is ruled out as ''Hamlet'''s author, there are many more plausible alternatives than Hobbes.

The general form of McGee-type counterexamples to ''modus ponens'' is simply <math>P, P \rightarrow (Q \rightarrow R)</math>, therefore <math>Q \rightarrow R</math>; it is not essential that <math>P</math> have the form of a disjunction, as in the example given. That these kinds of cases constitute failures of ''modus ponens'' remains a minority view among logicians, but there is no consensus on how the cases should be disposed of.

==Possible fallacies==
The fallacy of [[affirming the consequent]] is a common misinterpretation of the modus ponens.{{citation needed|date=November 2016}}

==See also==
* {{annotated link|Condensed detachment}}
* {{annotated link|Latin phrases}}
* {{annotated link|Modus tollens|''Modus tollens''}}
* {{annotated link|Modus vivendi|''Modus vivendi''}}
* {{annotated link|Stoic logic}}
* "{{annotated link|What the Tortoise Said to Achilles}}"

== References ==
{{Reflist}}

== Sources ==
*Herbert B. Enderton, 2001, ''A Mathematical Introduction to Logic Second Edition'', Harcourt Academic Press, Burlington MA, {{ISBN|978-0-12-238452-3}}.
* Audun Jøsang, 2016, ''Subjective Logic; A formalism for Reasoning Under Uncertainty'' Springer, Cham, {{ISBN|978-3-319-42337-1}}
*[[Alfred North Whitehead]] and [[Bertrand Russell]] 1927 ''Principia Mathematica to *56 (Second Edition)'' paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.
*[[Alfred Tarski]] 1946 ''Introduction to Logic and to the Methodology of the Deductive Sciences'' 2nd Edition, reprinted by Dover Publications, Mineola NY. {{ISBN|0-486-28462-X}} (pbk).

== External links ==
*{{springer|title=Modus ponens|id=p/m064570}}
*{{PhilPapers|search|modus_ponens}}
* ''[http://mathworld.wolfram.com/ModusPonens.html Modus ponens]'' at Wolfram MathWorld
{{Classical logic}}
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[[Category:Rules of inference]]
[[Category:Latin logical phrases]]
[[Category:Theorems in propositional logic]]
[[Category:Classical logic]]