---
tag: Z3DE4R4RET
kind: concept
name: Algebraic logic
msc: ["03G27", "03G05", "03G15"]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_logic
same_as:
  wikidata: Q1005222
  wikipedia: Algebraic logic
---

## Summary

Algebraic logic studies logical systems by means of algebraic structures.
Formulas are identified when they are provably equivalent, and the logical connectives become operations on the resulting classes, as in the [[Lindenbaum–Tarski algebra|QC9M3Q8B8Q]].
[[Classical propositional logic|09K9MWD2QQ]] corresponds to [[Boolean algebras|0XADJKM699]], [[intuitionistic logic|Q11N8X99FK]] to [[Heyting algebras|NG943XNQWH]], and Łukasiewicz infinite-valued logic to [[MV-algebras|8T4M7RN9AC]].
For [[predicate logic|086QKPF0A6]] there are [[cylindric algebras|1MDT9BF81H]] and polyadic algebras, and [[relation algebras|GQK4DSA11K]] treat the calculus of binary relations.

## Explanations

### L1 human

In school algebra one calculates with numbers by rules such as $a + b = b + a$.
Algebraic logic calculates with statements in the same way: "$p$ and $q$" equals "$q$ and $p$", and "not not $p$" equals "$p$" in classical logic.
The statements, up to logical equivalence, form an algebra, and questions about the logic become questions about that algebra.
For example, a formula is a theorem of classical propositional logic exactly when it takes the value $1$ in every Boolean algebra under every assignment of its variables.