Soundness

A logical argument is sound if and only if


 * 1) the argument is valid
 * 2) all of its premises are true.

A proof procedure (e.g. natural deduction) of a logical system is sound if it proves only valid formulas (also tautologies). In notation, a logical system is sound if $$X_1,\ldots,X_n \vdash Y$$ implies $$X_1,\ldots,X_n \models Y$$.

Sound arguments
Suppose we have a sound argument (in this case a syllogism):


 * All men are mortal.
 * Socrates is a man.
 * Therefore, Socrates is mortal.

The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.

The following argument is valid but not sound:


 * All animals can fly.
 * Pigs are animals.
 * Therefore, pigs can fly.

Since the first premise is actually false, the argument, though valid, is not sound.