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Soundness

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**soundness property**if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.The converse of soundness is known as completeness.WEB,weblink Types of proof system, Smith, Peter, 2010, 5, In most cases, this comes down to its rules having the property of

*preserving truth*.{{Citation needed|date=December 2018}}

## Definition

A system with syntactic entailment vdash and semantic entailment models is**sound**if for any sequence A_1, A_2, ..., A_n of sentences in its language, if A_1, A_2, ..., A_nvdash C, then A_1, A_2, ..., A_nmodels C. In other words, a system is sound when all of its theorems are tautologies.

## Logical systems

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.{{Citation needed|date=June 2008}} 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|date=June 2008}} 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 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 âŠ¢

*S*

*P*, then also âŠ¨

*L*

*P*.

### Strong soundness

Strong soundness of a deductive system is the property that any sentence*P*of the language upon which the deductive system is based that is derivable from a set Î“ of sentences of that language is also a logical consequence of that set, in the sense that any model that makes all members of Î“ true will also make

*P*true. In symbols where Î“ is a set of sentences of

*L*: if Î“ âŠ¢

*S*

*P*, then also Î“ âŠ¨

*L*

*P*. Notice that in the statement of strong soundness, when Î“ is empty, we have the statement of weak soundness.

### Arithmetic soundness

If*T*is a theory whose objects of discourse can be interpreted as natural numbers, we say

*T*is

*arithmetically sound*if all theorems of

*T*are actually true about the standard mathematical integers. For further information, see Ï‰-consistent theory.

## Relation to completeness

The converse of the soundness property is the semantic completeness property. A deductive system with a semantic theory is strongly complete if every sentence*P*that is a semantic consequence of a set of sentences Î“ can be derived in the deduction system from that set. In symbols: whenever {{nowrap|Î“ âŠ¨

*P*}}, then also {{nowrap|Î“ âŠ¢

*P*}}. Completeness of first-order logic was first explicitly established by GÃ¶del, though some of the main results were contained in earlier work of Skolem.Informally, a soundness theorem for a deductive system expresses that all provable sentences are true. Completeness states that all true sentences are provable.GÃ¶del's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no consistent and effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language. Thus, not all sound deductive systems are complete in this special sense of completeness, in which the class of models (up to isomorphism) is restricted to the intended one. The original completeness proof applies to

*all*classical models, not some special proper subclass of intended ones.

## See also

## References

{{Reflist}}## Bibliography

- BOOK, Hinman, P., Fundamentals of Mathematical Logic, A K Peters, 2005, 1-56881-262-0,
- {{Citation|authorlink=Irving Copi|first=Irving |last=Copi |title=Symbolic Logic| edition=5th| publisher= Macmillan Publishing Co. |year=1979| isbn=0-02-324880-7}}
- Boolos, Burgess, Jeffrey.
*Computability and Logic*, 4th Ed, Cambridge, 2002.

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