Modus Ponens

In propositional logic, modus ponendo ponens (Latin for "the way that affirms by affirming"; often 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."

While it 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 . . . an inference is the dropping of a true premiss; it is the dissolution of an implication".

A justification for the "trust in inference is the belief that if the two former assertions are not in error, the final assertion 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 implies and is true, then is true. An example is:

If it's raining, I'll meet you at the movie theater.

It's raining.

Therefore, I'll meet you at the movie theater.

Modus ponens can be stated formally as:

where the rule is that whenever an instance of "" and "" appear by themselves on lines of a logical proof, "" can validly be placed on a subsequent line.

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."