Orthologic

Origin. Goldblatt, "Semantic analysis of orthologic" (1974). Ortholattices — not orthomodular lattices; adding the orthomodular law gives the stronger system OML. Quantum-inspired. Weaker than Boolean. Minimal quantum logic.

Models. Ortholattices. Orthocomplementation. Weaker distributivity. Quantum event structures.

Formalism.

Ortholattice: ⟨L, ∧, ∨, ⊥, 0, 1⟩ where: ∧, ∨: meet, join. ⊥: orthocomplementation. a⊥⊥ = a (involution); a ≤ b implies b⊥ ≤ a⊥ (antitone). a ∧ a⊥ = 0, a ∨ a⊥ = 1.

Orthomodular law — NOT assumed by orthologic: a ≤ b implies b = a ∨ (b ∧ a⊥). Weaker than distributivity, stronger than mere orthocomplementation. Adding it to orthologic gives orthomodular logic OML; the closed subspaces of a Hilbert space satisfy it, so orthologic is sound but not complete for that model.

Orthologic (minimal): Propositional logic of ortholattices. Drops distributivity. Keeps orthocomplementation. Weaker than classical.

Sequent system: Goldblatt's calculus relates single formulas: sequents are α ⊢ β, with no side context. Contraction and weakening are not what is restricted — the absence of a context is. Distribution fails because no context can be carried across ∨-elimination. Cut-elimination holds.

Invalidity: A ∧ (B ∨ C) ⊬ (A ∧ B) ∨ (A ∧ C). Distribution fails. Non-Boolean.

Semantic interpretation: Complete for orthoframes ⟨X, ⊥⟩ with ⊥ irreflexive and symmetric, propositions being the ⊥-closed sets (Goldblatt 1974). The concrete quantum model — closed subspaces of Hilbert space — validates more than orthologic proves, since it is orthomodular. Meet = intersection. Join = closed span. Orthocomplement = orthogonal subspace.

Relation to quantum logic: Orthologic ⊂ orthomodular logic ⊂ Boolean. Minimal quantum. Varying strengths.

Symbols.

Metatheory. Orthologic is sound and complete for the variety of ortholattices, and, by Goldblatt (1974), for orthoframes ⟨X, ⊥⟩ with ⊥ irreflexive and symmetric, propositions being the ⊥-closed subsets. Since the complement of such a ⊥ is reflexive and symmetric, an orthoframe is exactly a frame for the modal system KTB, and Goldblatt's translation embeds orthologic faithfully into B, with quantum propositions as the □◇-stable sets. Goldblatt's calculus relates single formulas rather than sequents with side contexts; it admits cut elimination and has the finite model property, so orthologic is decidable. Ortholattices form a variety, being equationally axiomatized, and an ortholattice is Boolean iff it is distributive, so adding distribution to orthologic yields classical logic outright. An ortholattice is orthomodular iff it contains no copy of the benzene ring O6 as a subalgebra, which is why the orthomodular law is independent of orthologic. The closed subspaces of a Hilbert space form an orthomodular lattice, so orthologic is sound but not complete for the quantum model.

Applies to. Quantum mechanics. Non-Boolean structures. Substructural logic. Lattice theory.

Limitations. Orthologic has no implication connective and no deduction theorem: modus ponens for the Sasaki hook, a ∧ (a⊥ ∨ (a ∧ b)) ≤ b, is equivalent to the orthomodular law and so fails in ortholattices generally, while modus ponens for the material candidate a⊥ ∨ b is equivalent to distributivity and fails in every non-Boolean ortholattice. Goldblatt's sequents carry no side context, so multi-premise reasoning cannot be expressed internally and the calculus proves only entailments between single formulas. Distribution fails, so proof by cases and reasoning under a disjunction are unavailable. The logic is incomplete for its intended model: the closed subspaces of a Hilbert space validate the orthomodular law, which orthologic does not derive, so orthologic underdetermines the quantum lattice. Ortholattices also lack a general tensor product, so composite systems have no orthological construction.

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