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
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 ...
such that both
and its negation
are elements of the set of consequences of
. Let
be a set of
closed sentences (informally "axioms") and
the set of closed sentences provable from
under some (specified, possibly implicitly) formal deductive system. The set of axioms
is consistent when there is no formula
such that
and
. 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 means "provable from". That is,
reads: ''b'' is provable from ''a'' (in some specified formal system).
Definition
*A set of
formulas in first-order logic is consistent (written
) if there is no formula
such that
and
. Otherwise
is inconsistent (written
).
*
is said to be simply consistent if for no formula
of
, both
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
are theorems of
.
*
is said to be absolutely consistent or Post consistent if at least one formula in the language of
is not a theorem of
.
*
is said to be maximally consistent if
is consistent and for every formula
,
implies
.
*
is said to contain witnesses if for every formula of the form
there exists a
term such that
, where
denotes the
substitution of each
in
by a
; see also
First-order logic.
Basic results
# The following are equivalent:
##
## For all
# Every satisfiable set of formulas is consistent, where a set of formulas
is satisfiable if and only if there exists a model
such that
.
# For all
and
:
## if not
, then
;
## if
and
, then
;
## if
, then
or
.
# Let
be a maximally consistent set of formulas and suppose it contains
witnesses. For all
and
:
## if
, then
,
## either
or
,
##
if and only if
or
,
## if
and
, then
,
##
if and only if there is a term
such that
.
Henkin's theorem
Let
be a
set of symbols. Let
be a maximally consistent set of
-formulas containing
witnesses.
Define an
equivalence relation on the set of
-terms by
if
, where
denotes
equality. Let
denote the
equivalence class of terms containing
; and let
where
is the set of terms based on the set of symbols
.
Define the
-
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 ...
over
, also called the term-structure corresponding to
, by:
# for each
-ary relation symbol
, define
if
# for each
-ary function symbol
, define
# for each constant symbol
, define
Define a variable assignment
by
for each variable
. Let
be the term
interpretation associated with
.
Then for each
-formula
:
Sketch of proof
There are several things to verify. First, that
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
is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of
class representatives. Finally,
can be verified by induction on formulas.
Model theory
In
ZFC set theory with classical
first-order logic, an inconsistent theory
is one such that there exists a closed sentence
such that
contains both
and its negation
. A consistent theory is one such that the following
logically equivalent conditions hold
#
[according to De Morgan's laws]
#
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