ALGEBRAIC LOGICS
Logics individuated by the variety of algebras—the class of logical matrices—with respect to which they are complete. Where a modal logic is individuated by its accessibility relation, a substructural logic by which structural rules it drops, and a typed calculus by its judgments, an entry here is individuated by its algebra of truth values together with the designated set that fixes its consequence relation. Having an algebraic semantics is not what places a logic here—every logic in the collection has one—but having its individuation exhausted by that algebra, with no intensional operator, structural restriction, or term calculus doing the work instead.
The division is organized by the variety. Boolean entries are complete for Boolean algebras: the bivalent classical core, its fragments, its quantificational extensions, and the term traditions that codify classical validity. Heyting entries are complete for Heyting algebras: intuitionistic logic and the lattice of intermediate logics, which is the lattice of subvarieties of Heyting algebras. Orthomodular entries are complete for orthomodular lattices, where distribution fails: quantum logic. Many-Valued is the one subdivision named not for a single variety but for a family of them—MV, BL, and MTL algebras, De Morgan and Kleene lattices, and bilattices—the finitely-valued, fuzzy, partial, and paraconsistent matrices. The subdivisions are not a partition: Gödel logic is complete for the linear Heyting algebras and equally for a subvariety of BL-algebras, so entries carrying two variety individuations are cross-listed rather than duplicated.