Algebras As Models of Logics
Algebraic logic treats algebraic structures, often bounded lattices, as models (interpretations) of certain logics, making logic a branch of the order theory.
In algebraic logic:
- Variables are tacitly universally quantified over some universe of discourse. There are no existentially quantified variables or open formulas;
- Terms are built up from variables using primitive and defined operations. There are no connectives;
- Formulas, built from terms in the usual way, can be equated if they are logically equivalent. To express a tautology, equate a formula with a truth value;
- The rules of proof are the substitution of equals for equals, and uniform replacement. Modus ponens remains valid, but is seldom employed.
In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators."
Algebraic formalisms going beyond first-order logic in at least some respects include:
- Combinatory logic, having the expressive power of set theory;
- Relation algebra, arguably the paradigmatic algebraic logic, can express Peano arithmetic and most axiomatic set theories, including the canonical ZFC.
| Logical system | Its models |
| Classical sentential logic | Lindenbaum-Tarski algebra
Two-element Boolean algebra |
| Intuitionistic propositional logic | Heyting algebra |
| Łukasiewicz logic | MV-algebra |
| Modal logic K | Modal algebra |
| Lewis's S4 | Interior algebra |
| Lewis's S5; Monadic predicate logic | Monadic Boolean algebra |
| First-order logic | complete Boolean algebra
Cylindric algebra Predicate functor logic |
| Set theory | Combinatory logic
Relation algebra |
Read more about this topic: Algebraic Logic
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