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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] , 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.

** Of arguments **

An [argument] is ** sound ** if and only if

** 1 ** The argument is [valid] .

** 2 ** All of its premises are [true] .

For instance,

:All men are mortal.

:Socrates is a man.

:Therefore, Socrates is mortal.

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:

:All organisms with wings can fly.

:Penguins have wings.

:Therefore, penguins can fly.

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. 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.

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