Three Fixed-Dimension Satisfiability Semantics for Quantum Logic: Implications and an Explicit Separator

Abstract

We compare three satisfiability notions for propositional formulas in the language {not, and, or} over a fixed finite-dimensional Hilbert space H=F^d with F in {R, C}. The first is the standard Hilbert-lattice semantics on the subspace lattice L(H), where meet and join are total operations. The second is a global commuting-projector semantics, where all atoms occurring in the formula are interpreted by a single pairwise-commuting projector family. The third is a local partial-Boolean semantics, where binary connectives are defined only on commeasurable pairs and definedness is checked nodewise along the parse tree. We prove, for every fixed d >= 1, Sat_COM^d(phi) implies Sat_PBA^d(phi) implies Sat_STD^d(phi) for every formula phi. We then exhibit the explicit formula SEP-1 := (p and (q or r)) and not((p and q) or (p and r)) which is satisfiable in the standard semantics for every d >= 2, but unsatisfiable under both the global commuting and the partial-Boolean semantics. Consequently, for every d >= 2, the satisfiability classes satisfy SAT_COM^d subseteq SAT_PBA^d subset SAT_STD^d and SAT_COM^d subset SAT_STD^d, while the exact relation between SAT_COM^d and SAT_PBA^d remains open. The point of the paper is semantic comparison, not a new feasibility reduction or a generic translation theorem.