Editing Consistency (section)

{{Short description|Non-contradiction of a theory}}
{{Other uses}}
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In [[deductive logic]], a '''consistent''' [[theory (mathematical logic)|theory]] is one that does not lead to a logical [[contradiction]].<ref>{{Harvnb|Tarski|1946}} states it this way: "A deductive theory is called ''consistent'' or ''non-contradictory'' if no two asserted statements of this theory contradict each other, or in other words, if of any two contradictory sentences … at least one cannot be proved," (p. 135) where Tarski defines ''contradictory'' as follows: "With the help of the word ''not'' one forms the ''negation'' of any sentence; two sentences, of which the first is a negation of the second, are called ''contradictory sentences''" (p. 20). This definition requires a notion of "proof". {{Harvnb|Gödel|1931}} defines the notion this way: "The class of ''provable formulas'' is defined to be the smallest class of formulas that contains the axioms and is closed under the relation "immediate consequence", i.e., formula ''c'' of ''a'' and ''b'' is defined as an ''immediate consequence'' in terms of ''modus ponens'' or substitution; cf {{Harvnb|Gödel|1931}}, {{Harvnb|van Heijenoort|1967|p=601}}. Tarski defines "proof" informally as "statements follow one another in a definite order according to certain principles … and accompanied by considerations intended to establish their validity [true conclusion] for all true premises – {{Harvnb|Reichenbach|1947|p=68}}]" cf {{Harvnb|Tarski|1946|p=3}}. {{Harvnb|Kleene|1952}} defines the notion with respect to either an induction or as to paraphrase) a finite sequence of formulas such that each formula in the sequence is either an axiom or an "immediate consequence" of the preceding formulas; "A ''proof is said to be a proof ''of'' its last formula, and this formula is said to be ''(formally) provable'' or be a ''(formal) theorem" cf {{harvnb|Kleene|1952|p=83}}.</ref> A theory <math>T</math> is consistent if there is no [[Formula (mathematical logic)|formula]] <math>\varphi</math> such that both <math>\varphi</math> and its negation <math>\lnot\varphi</math> are elements of the set of consequences of <math>T</math>. Let <math>A</math> be a set of [[Closed-form expression|closed sentences]] (informally "axioms") and <math>\langle A\rangle</math> the set of closed sentences provable from <math>A</math> under some (specified, possibly implicitly) formal deductive system. The set of axioms <math>A</math> is '''consistent''' when there is no formula <math>\varphi</math> such that <math>\varphi \in \langle A \rangle</math> and <math> \lnot \varphi \in \langle A \rangle</math>. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an [[principle of explosion|explosive]] [[formal system]] (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial.<ref name="Carnielli">{{cite book|last1=Carnielli|first1=Walter|last2=Coniglio|first2=Marcelo Esteban|title=Paraconsistent logic: consistency, contradiction and negation|language=en|series=Logic, Epistemology, and the Unity of Science|volume=40|publisher=Springer|location=Cham|date=2016|doi=10.1007/978-3-319-33205-5 |isbn=978-3-319-33203-1|mr=3822731|zbl=1355.03001}}</ref>{{rp|7}} Consistency of a theory is a [[syntactic]] notion, whose [[semantic]] counterpart is [[satisfiable theory|satisfiability]]. A theory is satisfiable if it has a [[Model theory#First-order logic|model]], i.e., there exists an [[interpretation (logic)|interpretation]] under which all [[axiom]]s in the theory are true.<ref>{{cite book |title=A Shorter Model Theory |first=Wilfrid |last=Hodges |page=37 |location=New York |publisher=Cambridge University Press |year=1997 |quote=Let <math>L</math> be a signature, <math>T</math> a theory in <math>L_{\infty \omega}</math> and <math>\varphi</math> a sentence in <math>L_{\infty\omega}</math>. We say that <math>\varphi</math> is a ''consequence'' of <math>T</math>, or that <math>T</math> ''entails'' <math>\varphi</math>, in symbols <math>T \vdash \varphi</math>, if every model of <math>T</math> is a model of <math>\varphi</math>. (In particular if <math>T</math> has no models then <math>T</math> entails <math>\varphi</math>.)<br>'' Warning'': we don't require that if <math>T \vdash \varphi</math> then there is a proof of <math>\varphi</math> from <math>T</math>. In any case, with infinitary languages, it's not always clear what would constitute proof. Some writers use <math>T\vdash\varphi</math> to mean that <math>\varphi</math> is deducible from <math>T</math> in some particular formal proof calculus, and they write <math>T \models \varphi</math> for our notion of entailment (a notation which clashes with our <math>A \models \varphi</math>). For first-order logic, the two kinds of entailment coincide by the completeness theorem for the proof calculus in question.<br>We say that <math>\varphi</math> is ''valid'', or is a ''logical theorem'', in symbols <math>\vdash \varphi</math>, if <math>\varphi</math> is true in every <math>L</math>-structure. We say that <math>\varphi</math> is ''consistent'' if <math>\varphi</math> is true in some <math>L</math>-structure. Likewise, we say that a theory <math>T</math> is ''consistent'' if it has a model.<br> We say that two theories S and T in L infinity omega are equivalent if they have the same models, i.e. if Mod(S) = Mod(T). }} (Please note the definition of Mod(T) on p. 30 ...)</ref> This is what ''consistent'' meant in traditional [[Term logic|Aristotelian logic]], although in contemporary mathematical logic the term ''[[satisfiable]]'' is used instead.

In a [[soundness|sound formal system]], every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive [[Mathematical logic#Formal logical systems|logic]], the logic is called '''[[Completeness (logic)#Refutation-completeness|complete]]'''.{{citation needed|date=May 2012}} The completeness of the [[propositional calculus]] was proved by [[Paul Bernays]] in 1918{{Citation needed|date=October 2009}}<ref>{{harvnb|van Heijenoort|1967|p=265}} states that Bernays determined the ''independence'' of the axioms of ''Principia Mathematica'', a result not published until 1926, but he says nothing about Bernays proving their ''consistency''.</ref> and [[Emil Post]] in 1921,<ref>Post proves both consistency and completeness of the propositional calculus of PM, cf van Heijenoort's commentary and Post's 1931 ''Introduction to a general theory of elementary propositions'' in {{harvnb|van Heijenoort|1967|pp=264ff}}. Also {{Harvnb|Tarski|1946|pp=134ff}}.</ref> while the completeness of (first order) [[predicate calculus]] was proved by [[Kurt Gödel]] in 1930,<ref>cf van Heijenoort's commentary and Gödel's 1930 ''The completeness of the axioms of the functional calculus of logic'' in {{Harvnb|van Heijenoort|1967|pp=582ff}}.</ref> and consistency proofs for arithmetics restricted with respect to the [[Mathematical induction|induction axiom schema]] were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).<ref>cf van Heijenoort's commentary and Herbrand's 1930 ''On the consistency of arithmetic'' in {{Harvnb|van Heijenoort|1967|pp=618ff}}.</ref> Stronger logics, such as [[second-order logic]], are not complete.

A '''consistency proof''' is a [[mathematical proof]] that a particular theory is consistent.<ref>A consistency proof often assumes the consistency of another theory. In most cases, this other theory is  [[Zermelo–Fraenkel set theory]] with or without the [[axiom of choice]] (this is equivalent since these two theories have been proved equiconsistent; that is, if one is consistent, the same is true for the other).</ref> The early development of mathematical [[proof theory]] was driven by the desire to provide finitary consistency proofs for all of mathematics as part of [[Hilbert's program]].  Hilbert's program was strongly impacted by the [[incompleteness theorems]], which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent).

Although consistency can be proved using model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The [[cut-elimination]] (or equivalently the [[Normalization property|normalization]] of the [[Curry-Howard|underlying calculus]] if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general.