This paper uses the notion of Galois-connection to examine the relation between valuation-spaces and logics. Every valuation-space gives rise to a logic, and every logic gives rise to a valuation space, where the resulting pair of functions form a Galois-connection, and the composite functions are closure-operators. A valuation-space (resp., logic) is said to be complete precisely if it is Galois-closed. Two theorems are proven. A logic is complete if and only if it is reflexive and transitive. A valuation-space is complete (...) if and only if it is closed under formation of super-valuations. (shrink)

Logical matrices for orthomodular logic are introduced. The underlying algebraic structures are orthomodular lattices, where the conditional connective is the Sasaki arrow. An axiomatic calculusOMC is proposed for the orthomodular-valid formulas.OMC is based on two primitive connectives — the conditional, and the falsity constant. Of the five axiom schemata and two rules, only one pertains to the falsity constant. Soundness is routine. Completeness is demonstrated using standard algebraic techniques. The Lindenbaum-Tarski algebra ofOMC is constructed, and it is shown to be (...) an orthomodular lattice whose unit element is the equivalence class of theses ofOMC. (shrink)

This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

The notion of relative compability is introduced, according to which compatibility is construed as relative to individual quantum states. The compatibility domain of two observablesA, B is defined to be the set com(A, B) of states relative to whichA andB are compatible. Three basic categories of relative compatibility are then defined according to the character of com(A, B): absolute compatibility (ordinary compatibility), absolute incompatibility, and partial compatibility. Then com(A, B) is seen to be a subspace of Hilbert space invariant underA (...) andB; being a subspace, it corresponds to a quantum binary observable (projection), denotedc(A, B). IfA andB are themselves binary observables, thenc(A, B) is expressible as a quantum logical compound ofA andB using the lattice operations meet, join, and orthocomplement. This suggests extending the notion of relative compatibility to the theory of orthomodular lattices, as well as to more general lattice-theoretic formulations of quantum mechanics. (shrink)

Part 1 – Formal Development of Theory .................................................................................................4..

The interpretation of quantum mechanics is discussed from the viewpoint of quantum logic (QL). QL is understood to concern the possible properties that can be ascribed to a physical system SYS. The micro-state of SYS at any given moment t is identified with the set of all properties actualized by SYS at time t. Minimal adequacy requirements are proposed for all interpretations of micro-states. A strict interpretation is defined to be one according to which the properties ascribable to SYS are (...) individuated by the projection operators on the associated Hilbert space. Two strict interpretations are examined. Kochen's interpretation is also discussed, and it is argued that it is not a strict interpretation. (shrink)

In this paper, I present a modal system called ∏ (Pi), characterizing it both axiomatically and algebraically, the latter being in terms of structures called π-algebras (pi-algebras). Pi-algebras are a natural generalization of Boolean algebras with operators – a generalization in which equality is replaced by congruence in the characterizing conditions. The resulting system of modal logic is "sub- Lewis", in the sense that it is properly contained in the weakest Lewis system, S1.

In the present paper I describe a general formal semantic scheme for the interpretation of quantum mechanics (QM), and on the basis of this scheme I examine the modal interpretation of QM — both the Copenhagen and the anti-Copenhagen variants — proposed by van Fraassen [19, 20, 21], This is intended to be a fragment of a larger work [12] which additionally investigates a number of closely related interpretations, including ones proposed by Bub and Demopoulos [1, 2, 3, 4], Fine (...) [5, 6], and Krips [16, 17].Formal semantically speaking (see, e.g., Thomason [18]), a logic L may be characterized as an ordered pair, where SYN is the underlying syntax (language) of L, and SEM is the semantics of L, which consists of a class of semantic assignments on SYN. (shrink)

This paper presents a general formal semantic scheme for the interpretation of quantum mechanics, in terms of which van Fraassen's Copenhagen and anti-Copenhagen variants of the modal interpretation are examined. The general character of the modal interpretation is motivated in a discussion of classical statistical mechanics, the distinction being made between statistical states and micro-states. The notion of a quasi-classical (micro) state is introduced in a discussion of the theorem of Gleason and Kochen and Specker. It is shown that, according (...) to the anti-Copenhagen variant, the class of micro-states coincides with a special class of quasi-classical states. The paper concludes with two general criticisms of the anti-Copenhagen variant. (shrink)