Soundness

In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth.

An argument is sound if and only if

1. The argument is valid, and 2. All of its premises are true.

For instance,

The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.

The following argument is valid but not sound:

Since the first premise is actually false, the argument, though valid, is not sound.

Soundness is among the most fundamental properties of mathematical logic. The soundness property provides the initial reason for counting a logical system as desirable. The completeness property means that every validity (truth) is provable. Together they imply that all and only validities are provable.

Most proofs of soundness are trivial. For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). Most axiomatic systems have only the rule of modus ponens (and sometimes substitution), so it requires only verifying the validity of the axioms and one rule of inference.

Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter.

Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based. In symbols, where S is the deductive system, L the language together with its semantic theory, and P a sentence of L: if ⊢_S P, then also ⊨_L P.

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