# Online Encyclopedia

# 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 (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.