Editing Soundness

{{Short description|Term in logic and deductive reasoning}}
In [[logic]] and [[deductive reasoning]], an [[argument]] is '''sound''' if it is both [[Validity (logic)|valid]] in form and has no false [[premise]]s.<ref>{{Cite web |url=http://www.logicmatters.net/resources/pdfs/ProofSystems.pdf |title=Types of proof system |last=Smith |first=Peter |date=2010 |page=5}}</ref> Soundness has a related meaning in [[mathematical logic]], wherein a [[Formal system|formal system of logic]] is sound [[if and only if]] every [[well-formed formula]] that can be proven in the system is logically valid with respect to the [[Semantics of logic|logical semantics]] of the system.

== Definition ==
In [[deductive reasoning]], a sound argument is an argument that is [[Validity (logic)|valid]] 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 conclusion ''must be'' true. An example of a sound argument is the following well-known [[syllogism]]:

: ''<small>(premises)</small>''
: All men are mortal. 
: Socrates is a man.
: ''<small>(conclusion)</small>''
: Therefore, Socrates is mortal.

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:

: All birds can fly.
: Penguins are birds.
: Therefore, penguins can fly.

This argument is valid as the conclusion ''must 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 valid ''and'' its premises must be true.<ref>{{Cite book|title=Introduction to logic|last=Gensler, Harry J., 1945-|isbn=978-1-138-91058-4|edition=Third|location=New York|oclc=957680480|date = January 6, 2017}}</ref>

Some authors, such as [[John Lemmon|Lemmon]], have used the term "soundness" as synonymous with what is now meant by "validity",<ref>{{Cite book |last=Lemmon |first=Edward John |title=Beginning logic |date=1998 |publisher=Chapman & Hall/CRC |isbn=978-0-412-38090-7 |location=Boca Raton, FL}}</ref> which left them with no particular word for what is now called "soundness". But nowadays, this division of the terms is very widespread.

== Use in mathematical logic ==

=== Logical systems ===
In [[mathematical logic]], a [[logical system]] has the soundness property if every [[formula (mathematical logic)|formula]] that can be proved in the system is logically valid with respect to the [[Formal semantics (logic)|semantics]] of the system.
In most cases, this comes down to its rules having the property of ''preserving [[truth]]''.<ref>{{Cite book|last=Mindus|first=Patricia|url=https://books.google.com/books?id=JJe_AW4jhKMC&dq=a+logical+system+has+the+soundness+property&pg=PA36|title=A Real Mind: The Life and Work of Axel Hägerström|date=2009-09-18|publisher=Springer Science & Business Media|isbn=978-90-481-2895-2|language=en}}</ref> The [[Converse (logic)#Categorical converse|converse]] of soundness is known as [[Completeness (logic)|completeness]]. 

A logical system with [[Logical consequence#Syntactic consequence|syntactic entailment]] <math>\vdash</math> and [[Logical consequence#Semantic consequence|semantic entailment]] <math>\models</math> is '''sound''' if for any [[sequence]] <math>A_1, A_2, ..., A_n</math> of [[Sentence (mathematical logic)|sentences]] in its language, if <math>A_1, A_2, ..., A_n\vdash C</math>, then <math>A_1, A_2, ..., A_n\models C</math>. In other words, a system is sound when all of its [[theorem]]s are [[Validity (logic)|validities]].

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 (logic)|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). If the system allows [[Hilbert-style deductive system|Hilbert-style deduction]], it requires only verifying the validity of the axioms and one rule of inference, namely [[modus 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====

Weak 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 ⊢<sub>''S''</sub>&nbsp;''P'', then also ⊨<sub>''L''</sub>&nbsp;''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 Γ of sentences of that language is also a [[logical consequence]] of that set, in the sense that any model that makes all members of Γ true will also make ''P'' true. In symbols, where Γ is a set of sentences of ''L'': if Γ&nbsp;⊢<sub>''S''</sub>&nbsp;''P'', then also Γ&nbsp;⊨<sub>''L''</sub>&nbsp;''P''. Notice that in the statement of strong soundness, when Γ 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 [[ω-consistent theory]].

===Relation to completeness===

The converse of the soundness property is the semantic [[Completeness (logic)|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 Γ can be derived in the [[deduction system]] from that set. In symbols: whenever {{nowrap|Γ <big>⊨</big> ''P''}}, then also {{nowrap|Γ <big>⊢</big> ''P''}}. Completeness of [[first-order logic]] was first [[Gödel's completeness theorem|explicitly established]] by [[Gödel]], 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.

[[Gödel's incompleteness theorem|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 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 ==
{{Portal|Philosophy}}

*[[Soundness (interactive proof)]]
*[[Type soundness]]

==References==
{{Reflist}}

==Bibliography==
*{{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | isbn = 1-56881-262-0}}
*{{Citation|author-link=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.

==External links==
{{Sisterprojectlinks|wikt=soundness|b=no|n=no|commons=no|q=no|s=no|v=no|d=no|species=no|voy=no}}
* [http://www.iep.utm.edu/val-snd/ Validity and Soundness] in the ''[[Internet Encyclopedia of Philosophy]].''

{{Metalogic}}
{{Mathematical logic}}

[[Category:Arguments]]
[[Category:Concepts in logic]]
[[Category:Deductive reasoning]]
[[Category:Model theory]]
[[Category:Proof theory]]