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, but this is not the case in general. The word derives from the Germanic 'Sund' as in Gesundheit, meaning health. Thus to say that an argument is sound means, following the etymology, to say that the argument is healthy.
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Of arguments
An argument is sound if and only if
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.
Of logical systems
Soundness is among the most fundamental properties in mathematical logic. A 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.^{[citation needed]} 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),^{[citation needed]} 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
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 models 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. In other words, a system is sound if each of its theorems (i.e. formulas provable from the empty set) is valid in every structure of the language.
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