In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if and only if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.

If these semantic and syntactic definitions are equivalent for a particular logic, the logic is complete. The completeness of sentential calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of 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 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. Hilbert's program was strongly impacted by incompleteness theorems, 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 (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.

Read more about Consistency:  Consistency and Completeness in Arithmetic and Set Theory, First-order Logic

Other articles related to "consistency":

Concrete Slump Test
... More specifically, it measures the consistency of the concrete in that specific batch ... This test is performed to check the consistency of freshly made concrete ... Consistency is a term very closely related to workability ...
Covariation Model - Consistency
... Consistency is the covariation of behavior across time ... If Jane is generous all the time, she shows high consistency ... or is generous only at specific times, perhaps around the holidays, she shows low consistency ...
Convention Of Consistency
... The convention of consistency means that same accounting principles should be used for preparing financial statements for different periods ... The concept of consistency does not mean that no change should be made in accounting procedures ... Whenever, consistency is not followed this fact may be fully disclosed ...
Prescott Lecky
... Lecky's self-consistency theory, that self-consistency is a primary motivating force in human behavior ... self and the self's overall need for a "master" motive that serves to maintain for the self a consistency in ideas ... Self-consistency theory remains relevant to contemporary personality and clinical psychologists ...

Famous quotes containing the word consistency:

    All religions have honored the beggar. For he proves that in a matter at the same time as prosaic and holy, banal and regenerative as the giving of alms, intellect and morality, consistency and principles are miserably inadequate.
    Walter Benjamin (1892–1940)

    The lawyer’s truth is not Truth, but consistency or a consistent expediency.
    Henry David Thoreau (1817–1862)

    People who love only once in their lives are ... shallow people. What they call their loyalty, and their fidelity, I call either the lethargy of custom or their lack of imagination. Faithfulness is to the emotional life what consistency is to the life of the intellect—simply a confession of failures.
    Oscar Wilde (1854–1900)