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.
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.