---
tag: C66RKJQQSC
kind: definition
name: Algebraic logic
msc: ["03G27", "03G05", "03G15"]
defines: Z3DE4R4RET
uses: [09K9MWD2QQ, 0XADJKM699, NG943XNQWH, QC9M3Q8B8Q, Q11N8X99FK]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_logic
---

## Definition

Algebraic logic is the part of mathematical logic that studies a logic through classes of algebras attached to it.
Its main objects are the following.

1. A logic, given by a formal language and a relation of consequence or provability between formulas.
2. Its algebras, often obtained as quotients of the algebra of formulas by provable equivalence, as in the [[Lindenbaum–Tarski algebra|QC9M3Q8B8Q]].
3. A translation between the two sides, under which theorems of the logic correspond to equations valid in the class of algebras.

## Remarks

Example: the completeness of [[classical propositional logic|09K9MWD2QQ]] with respect to [[Boolean algebras|0XADJKM699]] is a result of algebraic logic.
A formula is provable exactly when it is identically $1$ in every Boolean algebra, and already in the two-element Boolean algebra.

Another example: [[intuitionistic logic|Q11N8X99FK]] corresponds in the same way to the class of [[Heyting algebras|NG943XNQWH]].

Non-example: a truth-table check of a single formula is a computation in one algebra.
It is not by itself a study of the correspondence between a logic and a class of algebras.

Abstract algebraic logic studies in general which logics admit such a correspondence.