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In classical deductive logic, a consistent
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either
semantic Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
or
syntactic In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency) ...
terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an
interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...
under which all formulas 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 In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable ove ...
'' is used instead. The syntactic definition states 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 \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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, 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 Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of ...
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 Gove ...
in 1921, while the completeness of predicate calculus was proved by
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
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 In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
, 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 pr ...
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 part ...
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 mathema ...
. Hilbert's program was strongly impacted by the
incompleteness theorems 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 ...
, 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 In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...
, 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, omit ...
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 phil ...
show that any sufficiently strong
recursively enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...
theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...
(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, omit ...
. 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 ...
(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 rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
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 Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
of ''T''.


First-order logic


Notation

\vdash (Turnstile symbol) in the following context of
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, 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 logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subs ...
. In other cases, the turnstile symbol may mean implies; permits the derivation of. See:
List of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
.


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 complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
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 Term may refer to: * Terminology, or term, a noun or compound word used in a specific context, in particular: **Technical term, part of the specialized vocabulary of a particular field, specifically: ***Scientific terminology, terms used by scient ...
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 quantifie ...
.


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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
\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 elit ...
. Let \overline t denote the
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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 a ...
\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 Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...
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 quantifie ...
, 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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
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 math ...
#\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 problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...
*
Hilbert's second problem In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his Hilbert's problems, 23 problems. It asks for a proof that the arithmetic is consistency proof, consistent – free of any internal contradictions. Hilber ...
*
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 A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syst ...
* 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 ...


Footnotes


References

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


External links

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