🤖 AI Summary
This study investigates the structural and proof-theoretic issues arising from forcing conditions in relational quantum modal logic, with a focus on the coherence of modal transitions across compatible worlds. By introducing a notion of compatibility component structures, the work establishes, for the first time, a connection between such structures and central propositional approximations, characterizing upper and lower approximations of stable propositions within superselection models. Leveraging relational semantics, Boolean algebras, and local validity semantics—combined with connectedness constructions and maximal consistent pair techniques—the authors build a canonical model and formulate a unified logical system. The paper proves that the multi-conclusion sequent calculi extended with axioms T, 4, and B are complete with respect to component frames, equivalence frames, and connected compatibility frames, thereby establishing a comprehensive proof theory for component modal logics.
📝 Abstract
This paper determines the structural and proof-theoretic consequences of the forcing condition in relational quantum modal logic, under which every modal transition available at a world is also available at every world compatible with it. We prove that modal successor sets are constant on compatibility components, so boxed truth sets belong to the Boolean algebra of unions of these components. The relation holding exactly between worlds in the same component assigns to each stable proposition its greatest lower and least upper approximations by unions of components, and in hard superselection models these are exactly the approximations by central propositions. We adopt local validity for sequents with multiple conclusions to give a semantics for modal excluded middle on frames with several components. A connectedization obtained by adding one point then shows that component frames, equivalence frames satisfying the forcing condition, and connected compatibility frames with universal modal accessibility have the same logic for sequents with one conclusion. Finally, maximal consistent pairs yield a canonical model, and the calculus obtained by adding T, 4, and B is proved sound and complete for the three frame classes. These results characterize the logical scope of the forcing condition and establish a complete proof theory for component modalities.