Orthologic

Origin. Goldblatt (1974), various. Orthomodular lattices. 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⊥⊥ = a. a ∧ a⊥ = 0, a ∨ a⊥ = 1.

Orthomodular law: a ≤ b implies b = a ∨ (b ∧ a⊥). Weaker than distributivity. Quantum compatible.

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

Sequent system: Modified structural rules. Contraction restricted. Cut-elimination holds.

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

Semantic interpretation: Closed subspaces of Hilbert space. Meet = intersection. Join = closed span. Orthocomplement = orthogonal subspace.

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

Symbols.

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.