# Dictionary Definition

inconsistency

### Noun

1 the relation between propositions that cannot
both be true at the same time [syn: incompatibility,
mutual
exclusiveness, repugnance]

2 the quality of being inconsistent and lacking a
harmonious uniformity among things or parts [ant: consistency]

# User Contributed Dictionary

## English

### Noun

- The state of being inconsistent
- an incompatibility between two propositions that cannot both be true

# Extensive Definition

In traditional Aristotelian
logic, consistency is a semantic
concept meaning that two
or more propositions
are simultaneously true under some interpretation.

In modern logic there is a syntactic
definition that also fits the complex mathematical
theories developed since Frege's
Begriffsschrift
(1879): a set of statements
are called consistent with respect to a certain logical
calculus (also called a logical system or a formal system), if
no
formula of the form 'P and not-P' is derivable
from those statements by the rules of
the calculus. That is to say that the
theory is free from contradictions and that P and not-P are not
both theorems of that
system.

If these two definitions are equivalent
for a particular logical calculus, then the system is said to have
a complete set of
rules. The crucial step in the proofs
of completeness of the sentential
calculus by Paul Bernays
in 1918 and Emil Post in
1921, and the proof of the completeness of predicate
calculus by Kurt Godel in
1930 is to show that the system's syntactic consistency implies its
semantic consistency.

A consistency proof is a mathematical
proof that a logical system 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. Hilbert's program fell to Gödel's insight, as
expressed in his two
incompleteness theorems, that sufficiently strong proof
theories cannot prove their own consistency.

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
(or equivalently the normalization
of the underlying
calculus if there is one) implies the consistency of the
calculus: since there is obviously no cut-free proof of falsity,
there is no contradiction in general.

## Consistency and completeness

The fundamental results relating consistency and
completeness were
proven by Kurt
Gödel:

- Gödel's completeness theorem shows that any consistent first-order theory is complete with respect to a maximal consistent set of formulae which are generated by means of a proof search algorithm.
- Gödel's incompleteness theorems show that theories capable of expressing their own provability relation and of carrying out a diagonal argument are capable of proving their own consistency only if they are inconsistent. Such theories, if consistent, are known as essentially incomplete theories.

By applying these ideas, we see that we can find
first-order theories of the following four kinds:

- Inconsistent theories, which have no models;
- Theories which cannot talk about their own provability relation, such as Tarski's axiomatisation of point and line geometry, and Presburger arithmetic. Since these theories are satisfactorily described by the model we obtain from the completeness theorem, such systems are complete;
- Theories which can talk about their own consistency, and which include the negation of the sentence asserting their own consistency. Such theories are complete with respect to the model one obtains from the completeness theorem, but contain as a theorem the derivability of a contradiction, in contradiction to the fact that they are consistent;
- Essentially incomplete theories.

In addition, it has recently been discovered that
there is a fifth class of theory, the self-verifying
theories, which are strong enough to talk about their own
provability relation, but are too weak to carry out Gödelian
diagonalisation, and so which can consistently prove their own
consistency. However as with any theory, a theory proving its own
consistency provides us with no interesting information, since
inconsistent theories also prove their own consistency.

## Formulas

A set of formulas \Phi in first-order logic is consistent (written Con\Phi) if and only if there is no formula \phi such that \Phi \vdash \phi and \Phi \vdash \lnot\phi. Otherwise \Phi is inconsistent and is written Inc\Phi.\Phi is said to be simply consistent iff for no formula \phi of \Phi are
both \phi and the negation of \phi theorems of
\Phi.

\Phi is said to be absolutely consistent or Post
consistent iff at least one formula of \Phi is not a theorem of
\Phi.

\Phi is said to be maximally consistent if and
only if for every formula \phi, if Con \Phi \cup \phi then \phi \in
\Phi.

\Phi is said to contain witnesses if and only if
for every formula of the form \exists x \phi there exists a term t
such that (\exists x \phi \to \phi ) \in \Phi. See First-order
logic.

### Basic results

1. The following are equivalent:(a) Inc\Phi

(b) For all \phi,\; \Phi \vdash \phi.

2. 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
.

3. For all \Phi and \phi:

(a) if not \Phi \vdash \phi, then Con \Phi \cup
\;

(b) if Con \Phi and \Phi \vdash \phi, then Con
\Phi \cup \;

(c) if Con \Phi, then Con \Phi \cup \ or Con \Phi
\cup \.

4. Let \Phi be a maximally consistent set of
formulas and contain witnesses. For all \phi and \psi :

(a) if \Phi \vdash \phi, then \phi \in
\Phi,

(b) either \phi \in \Phi or \lnot \phi \in
\Phi,

(c) (\phi \or \psi) \in \Phi if and only if \phi
\in \Phi or \psi \in \Phi,

(d) if (\phi\to\psi) \in \Phi and \phi \in \Phi ,
then \psi \in \Phi,

(e) \exists x \phi \in \Phi if and only if there
is a term t such that \phi\in\Phi.

### Henkin's theorem

Let \Phi be a maximally consistent set of
formulas containing witnesses.

Define a binary relation on the set of S-terms
t_0 \sim t_1 \! if and only if \; t_0 = t_1 \in \Phi; and let
\overline t \! denote the equivalence class of terms containing t
\!; and let T_ := \ where T^S \! is the set of terms based on the
symbol set S \!.

Define the S-structure \mathfrak T_ over T_ \!
the term-structure corresponding to \Phi by:

(1) For n-ary R \in S, R^ \overline \ldots
\overline if and only if \; R t_0 \ldots t_ \in \Phi,

(2) For n-ary f \in S, f^ (\overline \ldots
\overline ) := \overline ,

(3) For c \in S, c^:= \overline c.

Let \mathfrak I_ := (\mathfrak T_,\beta_) be the
term interpretation associated with \Phi, where \beta _ (x) := \bar
x.

(*) \; For all \phi,\; \mathfrak I_ \vDash \phi
if and only if \; \phi \in \Phi.

### Sketch of proof

There are several things to verify. First, that \sim is 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_ \vDash \Phi can be verified by induction on formulas.## See also

## References

- The Cambridge Dictionary of Philosophy, consistency
- H.D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic
- Jevons, W.S., Elementary Lessons in Logic, 1870

portalpar Logic

inconsistency in German:
Widerspruchsfreiheit

inconsistency in Hebrew: עקביות (לוגיקה)

inconsistency in Russian:
Непротиворечивость

inconsistency in Chinese: 形式系統相容性

# Synonyms, Antonyms and Related Words

aloofness, ambiguity, ambivalence, anarchy, antagonism, antilogy, antinomy, antipathy, antithesis, asymmetry, capriciousness, changeability, changeableness, chaos, choppiness, clashing, collision, conflict, confrontation, confusion, contradiction, contradistinction,
contraindication,
contraposition,
contrariety,
contrast, counterposition,
cross-purposes, dappleness, departure, desultoriness, deviability, deviation, difference, differentiation,
diffusion, disaccord, disaccordance, disagreement, disconformity, discongruity, discontinuity, discordance, discrepancy, discreteness, disjunction, dislocation, disorder, disparity, dispersal, dispersion, disproportion, disproportionateness,
dissent, dissimilarity, dissolution, dissonance, distinction, distinctness, divarication, divergence, divergency, diversification,
diversity, eccentricity, entropy, equivocality, erraticism, far cry, fickleness, flightiness, freakishness, heresy, heterodoxy, heterogeneity, hostility, illogic, illogicality, illogicalness, impulsiveness, inaccordance, incoherence, incommensurability,
incompatibility,
inconclusiveness,
incongruity,
inconsonance,
inconstancy,
inequality, inharmoniousness,
inharmony, inimicalness, instability, invalidity, irrationality, irreconcilability,
irregularity,
jerkiness, mercuriality, mixture, moodiness, motleyness, mutability, nonadhesion, noncohesion, noncompliance, nonconcurrence, nonconformability,
nonconformance,
nonconformism,
nonconformity,
nonobservance,
nonstandardization,
nonuniformity,
odds, oppositeness, opposition, opposure, oppugnance, oppugnancy, originality, otherness, oxymoron, paradox, perversity, pluralism, polarity, protest, raggedness, reasonlessness, recalcitrance, recusance, recusancy, refractoriness, repugnance, restlessness, scattering,
self-contradiction, senselessness, separateness, shiftiness, showdown, unadherence, unadhesiveness, uncertainty, unconformability,
unconformism,
unconformity,
undependability,
unevenness, unfixedness, unlikeness, unorthodoxy, unpredictability,
unreasonableness,
unreliability,
unscientificness,
unsettledness,
unsoundness,
unstableness,
unsteadfastness,
unsteadiness,
untenableness,
untenacity, variability, variance, variation, variegation, variety, variousness, versatility, wantonness, wavering, waywardness, whimsicality