Suppose we have a sound argument (in this case a syllogism):
- 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.
The following argument is valid but not sound:
- All animals can fly.
- A pig is an animal.
- Therefore, a pig can fly.
This is a logical argument, but its first premise is not true.
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) demonstrated is semantically valid. The reverse condition is called completeness.