In
classical deductive logic
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
, 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 be ...
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 betwe ...
in the theory are true. This is the sense used in traditional
Aristotelian logic, 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 over ...
'' is used instead. The syntactic definition states a theory
is consistent if there is no
formula 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
for no formula
.
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 premise ...
, the logic is called
complete. The completeness of the
sentential calculus
Propositional calculus is a branch of 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 ...
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 Gove ...
in 1921, while the completeness of
predicate calculus
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
** Finitary relation, ...
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
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 proo ...
that a particular theory is consistent. 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
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, 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 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 (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 be ...
and ''A'' is an additional
axiom, ''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
* Independ ...
of ''T''.
First-order logic
Notation
(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 formal ...
, means "provable from". That is,
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
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 ...
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 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
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 ...
such that
, where
denotes the
substitution of each
in
by a
; 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:
##
## 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
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
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
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 on the set of
-terms by
if
, where
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
denote the
equivalence class of terms containing
; and let
where
is the set of terms based on the set of symbols
.
Define the
-
structure 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
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
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
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
#
[according 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 ...
#
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 p ...
*
Hilbert's problems
*
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
*
ω-consistency
*
Gentzen's consistency proof
*
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