The conception of “Anti-Diodorean” logic is based on the reversal of the definitions of Diodorus Cronus, who proposed to define modal concepts in terms of temporal ones. In case of quantum logic this is done by using the semantic interpretation of anti-Diodorean definitions, i.e. semantically defining spatio-temporal concepts by means of a quantum accessibility relation. Since Quantum Logic of Observables could be treated as the many-valued extension of the usual Quantum Logic, we can consider the Anti-Diodorean Logic of Observables in the same way.
The conception of “Anti-Diodorean” logic is based on the reversal of the definitions of Diodorus Cronus, who proposed to define modal concepts in terms of temporal ones. In case of quantum logic this is done by using the semantic interpretation of anti-Diodorean definitions, i.e. semantically defining spatio-temporal concepts by means of a quantum accessibility relation. Since Quantum Logic of Observables could be treated as the many-valued extension of the usual Quantum Logic, we can consider the Anti-Diodorean Logic of Observables in the same way.
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 <SYN,SEM>, 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.
Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.
This article employs a descriptive-analytical method with an algebraic and axiomatic approach to address quantum logic. Through several thought experiments, it is demonstrated that the distributive law for propositions does not hold in quantum mechanics. Therefore, the logic governing this domain of science is not classical logic. In fact, the algebra corresponding to the logic of quantum propositions is an orthomodular lattice, which is why the distributive law is not necessarily valid for such propositions. In quantum logic, various definitions are offered for the implication operator. Since none of these operators satisfy the deduction theorem, a suitable implication operator with this property must be defined. This objective is achieved by defining a unified quantum logic. By constructing a Lindenbaum-Tarski algebra for this unified quantum logic, it can be shown that it is indeed a quantum logic. Dishkant presented an embedding of quantum logic into the Br-modal logic system. This provides an appropriate semantics for quantum logic. A suitable model for this logic is the closed subspaces of a Hilbert space.
The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence α is identified with a quantum information quantity, represented by a density operator of a Hilbert space, whose dimension depends on the logical complexity of α. At the same time, the logical connectives of the language are interpreted as operations defined in terms of quantum logical gates. Standard quantum computational models can be described as special cases of Fock space models, where the meaning of any sentence is localized in a precise sector of a Fock space ℱ. From an intuitive point of view, the increasing number of particles described in the different sectors of ℱ can be interpreted as increasing information.
We lean upon the usual variant of the logic of quantum mechanics [1]. Here the propositions correspond to the results of the quantum experiments. A beautiful essay [2] may be connected with a conservation of semantics. But we try to broaden the semantics, admitting propositions about the possibility of results of experiments. Doing so, we fulfil the old wish of W. A. Fock, who attracts our attention to the importance of the modal categories for the interpretation of the quantum theory [3]. We begin with the formal description of the modal system Br+. The alphabet contains the signs ~, ∨ and Ↄ (negation, alternative and the sign of necessity), the set V of the prepositional variables and parentheses. The rules of formation are: if A ∈ V, then A is a proposition; if X and Y are propositions, then ~ X, X ∨ Y and Ↄ X are propositions also; there are no other propositions. ℬ will design the set of all propositions. The rules of transformation are the following: . Here X, Y ∈ ℬ. The sign ⊨ denotes the derivability in Br+ (in one step here).