Modus ponens

Transformation rules
Propositional calculus
Rules of inference
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / hypothetical syllogism Constructive / destructive dilemma Absorption / modus tollens / modus ponendo tollens
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology Negation introduction
Predicate logic
Universal generalization / instantiation Existential generalization / instantiation

In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP; also modus ponendo ponens (Latin for "mode that affirms by affirming") or implication elimination) is a rule of inference. It can be summarized as "P 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 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 antiquity. The first to explicitly describe the argument form modus ponens was Theophrastus.

Formal notation

The modus ponens rule may be written in sequent notation as

$P\to Q,\;P\;\;\vdash \;\;Q$

where P, Q and P → Q are statements (or propositions) in a formal language and ⊢ is a metalogical symbol meaning that Q is a syntactic consequence of P and P → Q in some logical system.

Explanation

The argument form has two premises (hypothesis). The first premise is the "if–then" or conditional claim, namely that P implies Q. The second premise is that P, the 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 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 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 argument using modus ponens is said to be deductive.

In single-conclusion 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.

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.

p	q	p → q
T	T	T
T	F	F
F	T	T
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". Modus ponens allows one to eliminate a conditional statement from a 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 or the law of detachment. Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones", 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 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". In other words: if one statement or proposition 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.

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:

$\Pr(Q)=\Pr(Q\mid P)\Pr(P)+\Pr(Q\mid \lnot P)\Pr(\lnot P)\,$,

where e.g. $\Pr(Q)$ denotes the probability of $Q$ and the conditional probability $\Pr(Q\mid P)$ generalizes the logical implication $P\to Q$. Assume that $\Pr(Q)=1$ is equivalent to $Q$ being TRUE, and that $\Pr(Q)=0$ is equivalent to $Q$ being FALSE. It is then easy to see that $\Pr(Q)=1$ when $\Pr(Q\mid P)=1$ and $\Pr(P)=1$. Hence, the law of total probability represents a generalization of modus ponens .

Subjective logic

Modus ponens represents an instance of the binomial deduction operator in subjective logic expressed as:

$\omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\circledcirc \omega _{P}^{A}\,$,

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

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. Here is an example:

Either Shakespeare or 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 range of Hamlet '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 $P,P\rightarrow (Q\rightarrow R)$, therefore $Q\rightarrow R$; it is not essential that $P$ 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.

See also

• Condensed detachment
• Latin phrases
• Modus tollens
• Modus vivendi
• Stoic logic
• "What the Tortoise Said to Achilles"

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

• Hazewinkel, Michiel, ed. (2001) [1994], "Modus ponens", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Modus ponens at PhilPapers
• Modus ponens at Wolfram MathWorld