Foundations of Mathematics

Axiomatic System

What is modus ponens?

The Latin term modus ponens means “mode that affirms,” or in the case of logic, stands for the rule of detachment. This rule (also known as a rule of inference) pertains to the “if…then” statement and forms the basis of most proofs: “If p then q,” or if p is true, then the conclusion q is true. Or simply, it is often seen as the following:

If p, then q.

p. Therefore, q.

To see this another way:

p ⇒ q: ‘If it is raining, then there are clouds in the sky.”

p: “It is raining.”

q: “There are clouds in the sky.”

There are several ways to break down the modus ponens. The argument form has two premises: The “if-then” (or conditional claim), or namely that pimplies q; and that p (called the antecedent of the conditional claim) is true. From these two premises it can be logically concluded that q (called the consequent of the conditional claim) must be true as well; in other words, if the antecedent of a conditional is true, then the consequent must be true.


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