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.

Of arguments

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.

sounds and unsounds

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.

Soundness (interactive proof)

Soundness is a property of interactive proof systems that requires that no prover can make the verifier accept for a wrong statement except with some small probability. The upper bound of this probability is referred to as the soundness error of a proof system.

More formally, for every prover , and every :

The above definition uses the somewhat arbitrary soundness error 2⁻⁸⁰. As long as the soundness error is bounded by a polynomial fraction of the potential running time of the verifier (i.e. ), it is always possible to amplify soundness until the soundness error becomes negligible relative to the running time of the verifier. This is achieved by repeating the proof and accepting only if all proofs verify. After repetitions, a soundness error will be reduced to .

See also

References