Quantum Team Logic and Bell’s Inequalities

Abstract

A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [7]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this end we generalise the team semantics of dependence logic [11] first to probabilistic team semantics, and then to what we call quantum team semantics.

Propositional team logics

“…In team semantics the truth of a propositional formula is evaluated in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility of considering probabilities of formulas, as in [19], and the meaning of concepts such as dependence, independence and inclusion, as in [34]. It is the latter possibility that is our focus in this paper.…”

A logic for arguing about probabilities in measure teams

“…Inequalities of this form (7) are of great importance in foundations of quantum mechanics, see [1] and [7]. Because of the completeness result presented in the next section, we will see that this inequalities are provable in our logic.…”

Logics for First-Order Team Properties

“…Finding an effective axiomatization in the more general case of Ind enriched with the logical constants in FOT is left as future work. Apart from theoretical significance, our results also provide new logical tools for the applications of team-based logics in other related areas; such applications have been studied in recent years, e.g., in database theory [19], formal semantics of natural language [4,5], Bayesian statistics [18,7], social choice theory [30], and quantum information theory [23]. In particular, inquisitive logic [6] adopts, independently, also the team semantics to provide formal semantics of questions in natural language, and the first-order version of inquisitive logic can be viewed as a team-based logic (in a slightly different setting) with the weak disjunction \\/ and the weak quantifiers ∀ 1 , ∃ 1 .…”