Soundness

A logical argument is sound if and only if, (1) the argument is valid and (2) all of its premises are true.

Suppose we have a sound argument:

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

In this case we have an argument where, first, if the premises are all true, then the conclusion must be true (i.e., the argument is valid); and, second, it so happens that the premises are all true. It follows that the conclusion must be true. If you know an argument is sound, then you know that its conclusion is true. By definition, all sound arguments have true conclusions.

---

In mathematical logic, a formal deduction calculus is said to be sound with respect to a given logic (i.e. with respect to its semantics) if every statement that can be derived within this calculus is a tautology of the logic. Stated differently, this says that everything that can be formally (syntactically) calculated is semantically true. The reverse condition is called completeness.