Modus ponens

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In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP):

If P, then Q.

P.

Therefore, Q.

or in logical operator notation:

P → Q

P

⊢ Q

where ⊢ represents the logical assertion.

or may also be written:

P     P → Q

       Q

In Metalogics the modus ponens is the cut-rule. The cut-elimination theorem says that the cut is valid (admissible rule) in some logical calculus (sequent calculus).

The argument form has two premises. 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.

Here is an example of an argument that fits the form modus ponens:

If democracy is the best system of government, then everyone should vote.

Democracy is the best system of government.

Therefore, everyone should vote.

The fact that the argument is valid cannot assure us that any of the statements in the argument are true; the validity of modus ponens tells us that the conclusion must be true if all the premises are true. It is wise to recall that a valid argument within which one or more of the premises are not true is called an unsound argument, whereas if all the premises are true, then the argument is sound. In most logical systems, Modus ponens is considered to be valid. However, the instances of its use may be either sound or unsound:

If the argument is modus ponens and its premises are true, then it is sound.

The premises are true.

Therefore, it is a sound argument.

A propositional argument using modus ponens is said to be deductive.

Modus ponens can also be referred to as affirming the antecedent or "Law of Detachment".

For an amusing dialog that problematizes modus ponens, see Lewis Carroll's "What the Tortoise Said to Achilles."

See also

• Hypothetical syllogism
• Modus tollens
• Affirming the consequent
• Denying the antecedent
• Inference ruleca:Modus ponens

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