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.

Symbol	Unicode	Meaning
⊥	U+22A5	orthocomplement
∧, ∨	—	meet, join
OL	—	orthologic
OML	—	orthomodular logic

Metatheory. Ortholattices. Non-distributivity. Cut-elimination. Quantum structures.

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

Limitations. Weak expressive power. No good conditional. Abstract. Limited direct application.