In propositional logic, modus ponens (; MP), also known as modus ponendo ponens (from Latin 'mode that by affirming affirms'), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q. P is true. Therefore, Q must also be true." Modus ponens is a mixed hypothetical syllogism and is closely related to another valid form of argument, modus tollens. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of modus ponens. The history of modus ponens goes back to antiquity. The first to explicitly describe the argument form modus ponens was Theophrastus. It, along with 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, with two premises and a conclusion:
If P, then Q. P. Therefore, Q. The first premise is a conditional ("if–then") claim, namely that P implies Q. The second premise is an assertion that P, the antecedent of the conditional claim, is the case. From these two premises it can be logically concluded that Q, the 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 valid, but this has no bearing on whether any of the statements in the argument are actually 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 Tuesday) is unsound. The argument is 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. Also, for example, for understanding:
( x ) ( P x ⊃ Q x ) {\displaystyle (x)(Px\supset Qx)}
P a {\displaystyle Pa}
Q a {\displaystyle Qa}
The first statement represents a general rule that holds for every object x. The second specifies that a particular object a possesses property P. From these two premises, we can logically infer the third statement, namely that object a also possesses property Q. 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. In artificial intelligence, modus ponens is often called forward chaining.
Formal notation
The modus ponens rule may be written in sequent notation as
P → Q , P ⊢ Q {\displaystyle 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. In classical two-valued logic, modus ponens can be interpreted from the material conditional (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 Begriffsschrift and Principia Mathematica. 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. 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.
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
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