Inlogic anddeductive reasoning, anargument issound if it is bothvalid in form and has no falsepremises.[1] Soundness has a related meaning inmathematical logic, wherein aformal system of logic is soundif and only if everywell-formed formula that can be proven in the system is logically valid with respect to thelogical semantics of the system.
Indeductive reasoning, a sound argument is an argument that isvalid and all of its premises are true (and as a consequence its conclusion is true as well). An argument is valid if, assuming its premises are true, the conclusionmust be true. An example of a sound argument is the following well-knownsyllogism:
Because of the logical necessity of the conclusion, this argument is valid; and because the argument is valid and its premises are true, the argument is sound.
However, an argument can be valid without being sound. For example:
This argument is valid as the conclusionmust be true assuming the premises are true. However, the first premise is false. Not all birds can fly (for example, ostriches). For an argument to be sound, the argument must be validand its premises must be true.[2]
Some authors, such asLemmon, have used the term "soundness" as synonymous with what is now meant by "validity",[3] which left them with no particular word for what is now called "soundness". But nowadays, this division of the terms is very widespread.
Inmathematical logic, alogical system has the soundness property if everyformula that can be proved in the system is logically valid with respect to thesemantics of the system.In most cases, this comes down to its rules having the property ofpreservingtruth.[4] Theconverse of soundness is known ascompleteness.
A logical system withsyntactic entailment andsemantic entailment issound if for anysequence ofsentences in its language, if, then. In other words, a system is sound when all of itstheorems arevalidities.
Soundness is among the most fundamental properties of mathematical logic. The soundness property provides the initial reason for counting a logical system as desirable. Thecompleteness 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 anaxiomatic 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). If the system allowsHilbert-style deduction, it requires only verifying the validity of the axioms and one rule of inference, namelymodus ponens (and sometimes substitution).
Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter.
Weak soundness of adeductive 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, whereS is the deductive system,L the language together with its semantic theory, andP a sentence ofL: if ⊢S P, then also ⊨L P.
Strong soundness of a deductive system is the property that any sentenceP of the language upon which the deductive system is based that is derivable from a set Γ of sentences of that language is also alogical consequence of that set, in the sense that any model that makes all members of Γ true will also makeP true. In symbols, where Γ is a set of sentences ofL: if Γ ⊢S P, then also Γ ⊨L P. Notice that in the statement of strong soundness, when Γ is empty, we have the statement of weak soundness.
IfT is a theory whose objects of discourse can be interpreted asnatural numbers, we sayT isarithmetically sound if all theorems ofT are actually true about the standard mathematical integers. For further information, seeω-consistent theory.
The converse of the soundness property is the semanticcompleteness property. A deductive system with a semantic theory is strongly complete if every sentenceP that is asemantic consequence of a set of sentences Γ can be derived in thededuction system from that set. In symbols: wheneverΓ⊨P, then alsoΓ⊢P. Completeness offirst-order logic was firstexplicitly established byGödel, though some of the main results were contained in earlier work ofSkolem.
Informally, a soundness theorem for a deductive system expresses that all provable sentences are true. Completeness states that all true sentences are provable.
Gödel'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 toisomorphism) is restricted to the intended one. The original completeness proof applies toall classical models, not some special proper subclass of intended ones.
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