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In classical deductive logic, a consistent theory is one that does not lead to a logical
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
, i.e., there exists an interpretation under which all
formulas In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a '' chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betw ...
in the theory are true. This is the sense used in traditional
Aristotelian logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...
, although in contemporary mathematical logic the term '' satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
. The completeness of the sentential calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete. A consistency proof is a
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
that a particular theory is consistent. The early development of mathematical
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, ...
was driven by the desire to provide finitary consistency proofs for all of mathematics as part of
Hilbert's program In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathe ...
. Hilbert's program was strongly impacted by the incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent). Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The
cut-elimination The cut-elimination theorem (or Gentzen's ''Hauptsatz'') is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark 1934 paper "Investigations in Logical Deduction" for ...
(or equivalently the normalization of the 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.


Consistency and completeness in arithmetic and set theory

In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, at least one of φ or ¬φ is a logical consequence of the theory.
Presburger arithmetic Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitt ...
is an axiom system for the natural numbers under addition. It is both consistent and complete. Gödel's incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and
primitive recursive arithmetic Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , reprinted in translation in as a formalization of his finitist conception of the foundations of ...
(PRA), but not to
Presburger arithmetic Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitt ...
. Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does ''not'' prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
(ZF). These set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion is interesting in set theory (and in other sufficiently expressive axiomatic systems). If ''T'' is a theory and ''A'' is an additional
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
, ''T'' + ''A'' is said to be consistent relative to ''T'' (or simply that ''A'' is consistent with ''T'') if it can be proved that if ''T'' is consistent then ''T'' + ''A'' is consistent. If both ''A'' and ¬''A'' are consistent with ''T'', then ''A'' is said to be independent of ''T''.


First-order logic


Notation

\vdash (Turnstile symbol) in the following context of mathematical logic, means "provable from". That is, a\vdash b reads: ''b'' is provable from ''a'' (in some specified formal system). See List of logic symbols. In other cases, the turnstile symbol may mean implies; permits the derivation of. See: List of mathematical symbols.


Definition

*A set of
formulas In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a '' chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betw ...
\Phi in first-order logic is consistent (written \operatorname \Phi) if there is no formula \varphi such that \Phi \vdash \varphi and \Phi \vdash \lnot\varphi. Otherwise \Phi is inconsistent (written \operatorname\Phi). *\Phi is said to be simply consistent if for no formula \varphi of \Phi, both \varphi and the negation of \varphi are theorems of \Phi. *\Phi is said to be absolutely consistent or Post consistent if at least one formula in the language of \Phi is not a theorem of \Phi. *\Phi is said to be maximally consistent if \Phi is consistent and for every formula \varphi, \operatorname (\Phi \cup \) implies \varphi \in \Phi. *\Phi is said to contain witnesses if for every formula of the form \exists x \,\varphi there exists a term t such that (\exists x \, \varphi \to \varphi ) \in \Phi, where \varphi denotes the
substitution Substitution may refer to: Arts and media *Chord substitution, in music, swapping one chord for a related one within a chord progression *Substitution (poetry), a variation in poetic scansion * "Substitution" (song), a 2009 song by Silversun Pic ...
of each x in \varphi by a t; see also
First-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
.


Basic results

# The following are equivalent: ## \operatorname\Phi ## For all \varphi,\; \Phi \vdash \varphi. # Every satisfiable set of formulas is consistent, where a set of formulas \Phi is satisfiable if and only if there exists a model \mathfrak such that \mathfrak \vDash \Phi . # For all \Phi and \varphi: ## if not \Phi \vdash \varphi, then \operatorname\left( \Phi \cup \\right); ## if \operatorname\Phi and \Phi \vdash \varphi, then \operatorname \left(\Phi \cup \\right); ## if \operatorname\Phi, then \operatorname\left( \Phi \cup \\right) or \operatorname\left( \Phi \cup \\right). # Let \Phi be a maximally consistent set of formulas and suppose it contains
witnesses In law, a witness is someone who has knowledge about a matter, whether they have sensed it or are testifying on another witnesses' behalf. In law a witness is someone who, either voluntarily or under compulsion, provides testimonial evidence, e ...
. For all \varphi and \psi : ## if \Phi \vdash \varphi, then \varphi \in \Phi, ## either \varphi \in \Phi or \lnot \varphi \in \Phi, ## (\varphi \lor \psi) \in \Phi if and only if \varphi \in \Phi or \psi \in \Phi, ## if (\varphi\to\psi) \in \Phi and \varphi \in \Phi , then \psi \in \Phi, ## \exists x \, \varphi \in \Phi if and only if there is a term t such that \varphi\in\Phi.


Henkin's theorem

Let S be a set of symbols. Let \Phi be a maximally consistent set of S-formulas containing
witnesses In law, a witness is someone who has knowledge about a matter, whether they have sensed it or are testifying on another witnesses' behalf. In law a witness is someone who, either voluntarily or under compulsion, provides testimonial evidence, e ...
. Define an equivalence relation \sim on the set of S-terms by t_0 \sim t_1 if \; t_0 \equiv t_1 \in \Phi, where \equiv denotes
equality Equality may refer to: Society * Political equality, in which all members of a society are of equal standing ** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...
. Let \overline t denote the equivalence class of terms containing t ; and let T_\Phi := \ where T^S is the set of terms based on the set of symbols S. Define the S- structure \mathfrak T_\Phi over T_\Phi , also called the term-structure corresponding to \Phi, by: # for each n-ary relation symbol R \in S, define R^ \overline \ldots \overline if \; R t_0 \ldots t_ \in \Phi; # for each n-ary function symbol f \in S, define f^ (\overline \ldots \overline ) := \overline ; # for each constant symbol c \in S, define c^:= \overline c. Define a variable assignment \beta_\Phi by \beta_\Phi (x) := \bar x for each variable x. Let \mathfrak I_\Phi := (\mathfrak T_\Phi,\beta_\Phi) be the term interpretation associated with \Phi. Then for each S-formula \varphi:


Sketch of proof

There are several things to verify. First, that \sim is in fact an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that \sim is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of t_0, \ldots ,t_ class representatives. Finally, \mathfrak I_\Phi \vDash \varphi can be verified by induction on formulas.


Model theory

In ZFC set theory with classical
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
, an inconsistent theory T is one such that there exists a closed sentence \varphi such that T contains both \varphi and its negation \varphi'. A consistent theory is one such that the following logically equivalent conditions hold #\\not\subseteq Taccording to
De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
#\varphi'\not\in T \lor \varphi\not\in T


See also

*
Cognitive dissonance In the field of psychology, cognitive dissonance is the perception of contradictory information, and the mental toll of it. Relevant items of information include a person's actions, feelings, ideas, beliefs, values, and things in the environmen ...
*
Equiconsistency In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not ...
* Hilbert's problems *
Hilbert's second problem In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems. It asks for a proof that the arithmetic is consistent – free of any internal contradictions. Hilbert stated that the axioms he consider ...
*
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. ...
* Paraconsistent logic * ω-consistency *
Gentzen's consistency proof Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a ce ...
*
Proof by contradiction In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known a ...


Footnotes


References

* * 10th impression 1991. * * * (pbk.) * * *


External links

* {{Authority control Proof theory Hilbert's problems Metalogic