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.

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