Math 2.0

Mathematical logic and foundations Definition

Algebraic logic

Builds on Propositional logicBoolean algebraHeyting algebraLindenbaum–Tarski algebraIntuitionistic 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.
  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 with respect to Boolean algebras is a result of algebraic logic. A formula is provable exactly when it is identically 11 in every Boolean algebra, and already in the two-element Boolean algebra.

Another example: intuitionistic logic corresponds in the same way to the class of Heyting algebras.

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.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Boolean algebra, concept, no evidence levelBoolean algebraconceptHeyting algebra, concept, no evidence levelHeyting algebraconceptIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptLindenbaum–Tarski algebra, concept, no evidence levelLindenbaum–Tarski alg…conceptProposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptAlgebraic logic, definition, no evidence levelAlgebraic logicdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boolean algebra concept
  2. Heyting algebra concept
  3. Intuitionistic logic concept
  4. Lindenbaum–Tarski algebra concept
  5. Proposition concept
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Negation definition
  10. Biconditional definition
  11. Propositional logic concept
  12. Algebraic logic definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_logic
About this page
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URIm2://C66RKJQQSC
Kinddefinition
WritingW0 stub
MSC03G27, 03G05, 03G15
DefinesAlgebraic logic
UsesPropositional logic, Boolean algebra, Heyting algebra, Lindenbaum–Tarski algebra, Intuitionistic logic
Changed2 November 2026

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