The validity of a logical argument refers to whether or not the conclusion follows logically from the premises, i.e., whether it is possible to deduce the conclusion from the premises and the allowable syllogisms of the logical system being used. If it is possible to do so, the argument is said to be valid; otherwise it is invalid. A classical example of a valid argument is the following:

All men are mortal.

Socrates is a man.

Therefore Socrates is mortal.

Truth and validity are different notions. An argument may be valid and yet the conclusion may be false if one or more of the premises is false, as the following example shows:

All men are registered voters.

Moby Dick is a man.

Therefore Moby Dick is a registered voter.

On the other hand, an argument may be invalid and yet the conclusion may be true, as the following example shows:

All men are mortal.

Oxygen is a chemical element.

Therefore, some men can run a mile in four minutes.

Mathematical proofs are also said to be valid or invalid. A mathematical proof is valid if the conclusion follows from the assumptions by applying legal mathematical operations to arrive at the conclusion.