Consistency
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In deductive logic, a consistent
theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
is one that does not lead to a logical contradiction. A theory T is consistent if there is no
formula 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 betwe ...
\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 there is no formula \varphi such that \varphi \in \langle A \rangle and \lnot \varphi \in \langle A \rangle. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an
explosive An explosive (or explosive material) is a reactive substance that contains a great amount of potential energy that can produce an explosion if released suddenly, usually accompanied by the production of light, heat, sound, and pressure. An ex ...
formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose
semantic Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
counterpart is satisfiability. A theory is satisfiable 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 , . Models can be divided in ...
, i.e., there exists an interpretation under which all
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 ...
s in the theory are true. This is what ''consistent'' meant in traditional
Aristotelian logic In logic and formal semantics, 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 b ...
, although in contemporary mathematical logic the term '' satisfiable'' is used instead. In a 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
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, the logic is called complete. The completeness of the
propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
was proved by Paul Bernays in 1918 and
Emil Post Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Govern ...
in 1921, while the completeness of (first order) 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 a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...
that a particular theory is consistent. The early development of mathematical
proof theory Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
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 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 is an axiom system for the natural numbers under addition. It is both consistent and complete.
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
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 (PRA), but not to Presburger arithmetic. 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 suc ...
(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 A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
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 ...
, ''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

In the following context of
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, the turnstile symbol \vdash means "provable from". That is, a\vdash b reads: ''b'' is provable from ''a'' (in some specified formal system).


Definition

*A set of formulas \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 In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
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 of each x in \varphi by a t; see also First-order logic.


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. 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. 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. 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 A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
\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, 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 #\varphi'\not\in T \lor \varphi\not\in T


See also

*
Cognitive dissonance In the field of psychology, cognitive dissonance is described as a mental phenomenon in which people unknowingly hold fundamentally conflicting cognitions. Being confronted by situations that challenge this dissonance may ultimately result in some ...
*
Equiconsistency In mathematical logic, two theory (mathematical logic), theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and Vice-versa, vice versa. In this case, they are, roughly speaking, "as consistent ...
* Hilbert's problems * Hilbert's second problem *
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 logi ...
* Paraconsistent logic * ω-consistency * Gentzen's consistency proof *
Proof by contradiction In logic, 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. Although it is quite freely used in mathematical pr ...


Notes


References

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


External links

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