---
tag: VM8TQ70CVJ
kind: concept
name: Consistency
aliases: [absolute-consistency, consistancy, consistencies, consistency-mathematical-logic, consistency-proof, consistent, consistent-set, consistent-theory, henkins-theorem, inconsistancy, inconsistency, inconsistency-principle, inconsistent, inconsistent-theory, incosistent, internal-logic, logical-consistency, logically-consistent, relative-consistency, self-consistent]
sources:
  - wikipedia: en:75802@r1369696261
    locator: Consistency
    note: adopted
    mention: wikipedia://en/75802
    sections: [Summary]
same_as:
  wikidata: Q1319773
  wikipedia: Consistency
inventory: [SYA32FTV69]
---

## Summary

In deductive logic, a **consistent** theory is one that does not lead to a logical contradiction. A theory $T$ is consistent if there is no formula $\varphi$ such that both $\varphi$ and its negation $\lnot\varphi$ are elements of the set of consequences of $T$. Let $A$ be a set of closed sentences (informally "axioms") and $\langle A\rangle$ the set of closed sentences provable from $A$ under some (specified, possibly implicitly) formal deductive system. The set of axioms $A$ is **consistent** when there is no formula $\varphi$ such that $\varphi \in \langle A \rangle$ and $\lnot \varphi \in \langle A \rangle$. 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.