Concept

Consistency

Overview

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory

T

{\displaystyle T}

is consistent if there is no formula

φ

{\displaystyle \varphi }

such that both

φ

{\displaystyle \varphi }

and its negation

¬

φ

{\displaystyle \lnot \varphi }

are elements of the set of consequences of

T

{\displaystyle T}

. Let

A

{\displaystyle A}

be a set of closed sentences (informally "axioms") and

⟨

A

⟩

{\displaystyle \langle A\rangle }

the set of closed sentences provable from

A

{\displaystyle A}

under some (specified, possibly implicitly) formal deductive system. The set of axioms

A

{\displaystyle A}

is consistent when there is no formula

φ

{\displaystyle \varphi }

such that

φ

∈

⟨

A

⟩

{\displaystyle \varphi \in \langle A\rangle }

and

¬

φ

∈

⟨

A

⟩

{\displaystyle \lnot \varphi \in \langle A\rangle }

. A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive 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 counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant in traditional Aristotelian logic, 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, the logic is called complete. The completeness of the propositional calculus was proved by Paul Bernays in 1918 and Emil Post 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 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 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.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

Found by search and not yet checked by a person; they may only mention the idea in passing.

1. what are the specific conditions which it has to fulfil? what is its use; its relevancy; its purport in present concrete experiences? The more it is insisted that the theoretical standard--consistency--is final within theory, the more germane and the more urgent is the question: What then in the concrete is theory? and of what nature _is_ the material consistency which is the test of its formal consistency?[23]

   John Dewey, The Influence of Darwin on Philosophy, and other essays in contemporary thought

   Names Consistency several times; opens on Consistency.

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2. The Logic of Consistency is a necessary auxiliary to the logic of truth, not only because what is inconsistent with itself or with other truths cannot be true, but also because truth can only be successfully pursued by drawing inferences from experience, which, if warrantable at all, admit of being generalized, and, to test their warrantableness, require to be exhibited in a generalized form; after which the…

   John Stuart Mill, A System of Logic

   States what Consistency is; names Consistency twice.

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3. But, however that may be, I should not regard Hegel's reasoning, even if it were valid, as properly belonging to logic: it would rather be an application of logic to the actual world. Logic itself would be concerned rather with such questions as what self-consistency is, which Hegel, so far as I know, does not discuss. And though he criticises the traditional logic, and professes to replace it by an improved logic…

   Bertrand Russell, Our Knowledge of the External World as a Field for Scientific Method in Philosophy

   States what Consistency is; names Consistency.

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Philosophers who discussed Consistency

• Per Martin-Löf
• Crispin Wright
• William Lycan
• James Feibleman

Related ideas

• First-order logic
• Independence (mathematical logic)
• Interpretation (logic)
• Zermelo–Fraenkel set theory
• Negation
• Kurt Gödel
• Logical equivalence
• Propositional logic
• Proof theory
• Mathematical induction
• Mathematical logic
• Model theory
• Peano axioms
• Soundness
• Syntax
• Semantics
• Completeness (logic)
• Term logic
• Computably enumerable set
• Second-order logic
• Emil Leon Post
• Formal system
• Well-formed formula
• Structure (mathematical logic)

Opposing perspectives

• Theory

Place in the tradition

Part of First-order logic.

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