View source for Modus ponens

{{short description|Rule of logical inference}}
{{Italic title}}
{{Infobox mathematical statement
| name = <em>Modus ponens</em>
| type = {{Plainlist|
* [[Deductive reasoning|Deductive]] argument form
* [[Rule of inference]]
}}
| field = {{Plainlist|
* [[Philosophy:Classical logic|Classical logic]]
* [[Propositional calculus]]
}}
| statement = <math>P</math> implies <math>Q</math>. <math>P</math> is true. Therefore, <math>Q</math> must also be true.
| symbolic statement = <math>P \to Q,\; P\; \vdash\ Q</math>
}}
{{Transformation rules}}

In [[Propositional calculus|propositional logic]], '''{{lang|la|modus ponens}}''' ({{IPAc-en|ˈ|m|oʊ|d|ə|s|_|ˈ|p|oʊ|n|ɛ|n|z}}; '''MP'''), also known as '''{{lang|la|modus ponendo ponens}}''' ({{ety|la||method of putting by placing}}),<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=[https://archive.org/details/latinforillitera0000ston/page/60 60] |url=https://archive.org/details/latinforillitera0000ston |url-access=registration }}</ref> '''implication elimination''', or '''affirming the antecedent''',<ref>[https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095354544 "Oxford reference: affirming the antecedent"]. ''Oxford Reference''.</ref> is a [[Deductive reasoning|deductive]] argument form and [[Rule of inference|rule of inference]].<ref>Enderton 2001:110</ref> It can be summarized as "''P'' [[Material conditional|implies]] ''Q.'' ''P'' is true. Therefore, ''Q'' must also be true."

''Modus ponens'' is a mixed [[Hypothetical syllogism|hypothetical syllogism]] and is closely related to another [[Philosophy:Validity (logic)|valid]] form of argument, ''[[Modus tollens|modus tollens]]''. Both have apparently similar but invalid forms: [[Philosophy:Affirming the consequent|affirming the consequent]] and [[Denying the antecedent|denying the antecedent]]. [[Philosophy:Constructive dilemma|Constructive dilemma]] is the [[Logical disjunction|disjunctive]] version of ''modus ponens''.

The history of ''modus ponens'' goes back to [[History:Classical antiquity|antiquity]].<ref>[[Biography:Susanne Bobzien|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 [[Biography:Theophrastus|Theophrastus]].<ref>[http://plato.stanford.edu/entries/logic-ancient/#StoSyl "Ancient Logic: Forerunners of ''Modus Ponens'' and ''Modus Tollens''"]. ''[[Stanford Encyclopedia of Philosophy]]''.</ref> It, along with ''[[Modus tollens|modus tollens]]'', is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal.

== Explanation ==

The form of a ''modus ponens'' argument is a mixed [[Hypothetical syllogism|hypothetical syllogism]], with two premises and a conclusion:

# If ''P'', then ''Q''.
# ''P''.
# Therefore, ''Q''.

The first premise is a [[Material conditional|conditional]] ("if–then") claim, namely that ''P'' implies ''Q''. The second premise is an assertion that ''P'', the [[Philosophy:Antecedent (logic)|antecedent]] of the conditional claim, is the case. From these two premises it can be logically concluded that ''Q'', the [[Consequent|consequent]] of the conditional claim, must be the case as well.

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 [[Philosophy:Validity (logic)|valid]], but this has no bearing on whether any of the statements in the argument are actually [[Truth|true]]; for ''modus ponens'' to be a [[Soundness|sound]] argument, the premises must be true for any true instances of the conclusion. An [[Philosophy:Argument|argument]] 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 [[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|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|function application]]:  if ''f'' is a function of type ''P'' → ''Q'' and ''x'' is of type ''P'', then ''f x'' is of type ''Q''.

In [[Artificial intelligence|artificial intelligence]], ''modus ponens'' is often called [[Forward chaining|forward chaining]].

== Formal notation ==

The ''modus ponens'' rule may be written in [[Sequent|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|metalogic]]al symbol meaning that ''Q'' is a [[Logical consequence|syntactic consequence]] of ''P'' and ''P'' → ''Q'' in some [[Formal system|logical system]].

==Justification via truth table==
The validity of ''modus ponens'' in classical two-valued logic can be clearly demonstrated by use of a [[Truth table|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 forms 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|website=encyclopediaofmath.org|access-date=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 [[Philosophy:Statement (logic)|statement]] or [[Proposition|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==

===Algebraic semantics===

In mathematical logic, [[Algebraic semantics (mathematical logic)|algebraic semantics]] treats every sentence as a name for an element in an ordered set. Typically, the set can be visualized as a [[Lattice (order)|lattice]]-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>\neg{(P \wedge Q)}</math> and <math>\neg{P} \vee \neg{Q}</math>, for instance, are equivalent (as is standard), then <math>\neg{(P \wedge Q)} = \neg{P} \vee \neg{Q}</math>. Logical implication becomes a matter of relative position: <math>P</math> logically implies <math>Q</math> just in case <math>P \leq Q</math>, i.e., when either <math>P = Q</math> or else <math>P</math> lies below <math>Q</math> and is connected to it by an upward path.

In this context, to say that <math display="inline">P</math> and <math>P \rightarrow  Q</math> together imply <math>Q</math>—that is, to affirm ''modus ponens'' as valid—is to say that the highest point which lies below both <math>P</math> and <math>P \rightarrow  Q</math> lies below <math>P</math>, i.e., that <math>P \wedge (P \rightarrow  Q) \leq Q</math>.{{efn|The highest point that lies below both <math>X</math> and <math>Y</math> is the "[[Join and meet|meet]]" of <math>X</math> and <math>Y</math>, denoted by <math>X \wedge Y</math>.}} In the semantics for basic propositional logic, the algebra is [[Boolean algebra (structure)|Boolean]], with <math>\rightarrow</math> construed as the [[Material conditional|material conditional]]: <math>P \rightarrow  Q = \neg{P} \vee Q</math>. Confirming that <math>P \wedge (P \rightarrow  Q) \leq Q</math> is then straightforward, because <math>P \wedge (P \rightarrow  Q) = P \wedge Q</math> and <math>P \wedge Q \leq Q</math>. With other treatments of <math>\rightarrow</math>, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted.

===Probability calculus===
If <math>\Pr(P \rightarrow Q) = x</math> and  <math>\Pr(P) = y</math>, then <math>\Pr(Q)</math> must lie in the interval <math>[x + y - 1, x]</math>.{{efn|Since <math>\neg P</math> implies  <math>P \rightarrow Q</math>, <math>x</math> must always be greater than or equal to <math>1 - y</math>, and therefore <math>x+y-1</math> will be greater than or equal to <math>0</math>. And since <math>y</math> must always be less than or equal to <math>1</math>, <math>x+y-1</math> must always be less than or equal to <math>x</math>.}}<ref name="Hailperin, T. 1996">{{cite book |last1=Hailperin |first1=Theodore |title=Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications |year=1996 |page=203 |publisher=Associated University Presses|location=London|isbn=0934223459}}</ref> For the special case <math>x = y = 1</math>, <math>\Pr(Q)</math> must equal <math>1</math>.

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

<math display="block">\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|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==

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.<ref>Vann McGee (1985). "A Counterexample to Modus Ponens", ''The Journal of Philosophy'' 82, 462–471.</ref> The following is an example:

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

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 [[Paradoxes of material implication|paradoxes of material implication]]).

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> 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.<ref>Sinnott-Armstrong, Moor, and Fogelin (1986). "A Defense of Modus Ponens", ''The Journal of Philosophy'' 83, 296–300.</ref><ref>D. E. Over (1987). "Assumption and the Supposed Counterexamples to Modus Ponens", ''Analysis'' 47, 142–146.</ref><ref>Bledin (2015). "Modus Ponens Defended", ''The Journal of Philosophy'' 112, 462–471.</ref>

In [[Philosophy:Deontic logic|deontic logic]], 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."<ref name="SEP_Deontic_Logic">{{cite web |url=https://plato.stanford.edu/entries/logic-deontic/#4.5 |title=Deontic Logic |access-date=January 30, 2020 |date=April 21, 2010}} ''[[Stanford Encyclopedia of Philosophy]]''.</ref> It would appear to follow that if Doe is in fact gently murdering his mother, then by ''modus ponens'' he is doing exactly what he should, unconditionally, be doing. Here again, ''modus ponens'' failure is not a popular diagnosis but is sometimes argued for.<ref>E.g., by Kolodny and MacFarlane (2010). "Ifs and Oughts", ''The Journal of Philosophy'' 107, 115–143.</ref>

==Possible fallacies==
The fallacy of [[Philosophy:Affirming the consequent|affirming the consequent]] is a common misinterpretation of the ''modus ponens''.<ref>{{Cite web|url=https://www.iep.utm.edu/fallacy/|title=Fallacies {{!}} Internet Encyclopedia of Philosophy|website=iep.utm.edu|access-date=2020-03-06}}</ref>

==See also==
* {{annotated link|Philosophy:Condensed detachment}}
* {{annotated link|Philosophy:Import-Export (logic)}}
* {{annotated link|Modus tollens|''Modus tollens''}}
* {{annotated link|Stoic logic}}
* {{annotated link|What the Tortoise Said to Achilles}}

== Notes ==
{{Notelist}}

== 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}}
*[[Biography:Alfred North Whitehead|Alfred North Whitehead]] and [[Biography:Bertrand Russell|Bertrand Russell]] 1927 ''Principia Mathematica to *56 (Second Edition)'' paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.
*[[Biography:Alfred Tarski|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}}

{{DEFAULTSORT:Modus Ponens}}
[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]
[[Category:Classical logic]]

{{Sourceattribution|Modus ponens}}