# Soundness

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 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.
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.