What is "modus ponens"

n. (context philosophy logic English) A valid form of argument in which the antecedent of a conditional proposition is affirmed, thereby entailing the affirmation of the consequent. Modus ponens has the form:

In propositional logic, modus ponendo ponens ( Latin for "the way that affirms by affirming"; generally abbreviated to MP or modus ponens) or implication elimination is a valid, simple argument form and rule of inference. It can be summarized as "P implies Q; P is asserted to be true, so therefore Q must be true." The history of modus ponens goes back to antiquity.

While modus ponens is one of the most commonly used concepts 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. 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. An example is:

If it is raining, I will meet you at the theater. It is raining. Therefore, I will meet you at the theater.

Modus ponens can be stated formally as:

$$\frac{P \to Q,\; P}{\therefore Q}$$

where the rule is that whenever an instance of "P → Q" and "P" appear by themselves on lines of a logical proof, Q can validly be placed on a subsequent line; furthermore, the premise P and the implication "dissolves", their only trace being the symbol Q that is retained for use later e.g. in a more complex deduction.

It 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."