First-order frames for orthomodular quantum logic

摘要

One of the main problems of the orthoframe approach to quantum logic was that orthomodularity could not be captured by any first-order condition. This paper studies an elementary and natural class of orthomodular frames that can work around this limitation. Set-theoretically, the frames (X,≤, ⊥,M) we propose form a natural subclass of the orthoframes (X,⊥), where ⊥ is an irreflexive and symmetric relation on X. More specifically, they are partially-ordered orthoframes with a designated subset M is contained in X. Our frame class contains the canonical orthomodular frame of the logic and therefore characterises orthomodular quantum logic (OQL). To prove soundness, a further restriction is placed on admissible valuations which requires that they be point-generated in addition to the requirement that they return stable sets.