An argument is

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

So suppose we have a sound argument:

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

*and*, second, it so happens that the premises

*are*all true. It follows that the conclusion must be true. That is the nice thing about soundness: if you

*know*an argument is sound, then you

*know*that its conclusion is true. By definition, all sound arguments have true conclusions. So soundness is a very good quality for an argument to have.

In mathematical logic, a formal deduction calculus is said to be

**sound**with respect to a given logic (i.e. wrt 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.