Quantum logic as a dynamic logic

Abstract

We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear "no". Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth's proposal of applying Tarski's semantical methods to the analysis of physical theories, with an empirical-experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operationalrealistic tradition of Jauch and Piron, i.e. as an investigation of "the logic of yes-no experiments" (or "questions"). Technically, we use the recently-developed setting of Quantum Dynamic Logic Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the "non-classicality" of quantum behavior than any perspective based on static Propositional Logic.

Johan van Benthem on Logic and Information Dynamics

“…In our own work, we gave a similar deconstruction of quantum logic [11,12,[18][19][20], showing that the so-called quantum implication (known as ''Sasaki Hook'') P ! Q is best understood semantically as a PDL-style dynamic modality ½?P Q for a ''quantum-test'' action ?P, representing a successful measurement of the quantum property P. In contrast to classical PDL tests (and classical idealized measurements), quantum tests are ''really dynamic'': they change the state of the system under observation, which explains the non-classical behavior of quantum ''implication''.…”

A branching distributed temporal logic for reasoning about entanglement-free quantum state transformations

“…Since the work of Birkhoff and von Neumann [10], various logics have been investigated as a means to formalize reasoning about propositions taking into account the principles of quantum theory, e.g., [15]. In general, it is possible to view quantum logic as a logical axiomatization of quantum theory, which provides an adequate foundation for a theory of reversible quantum processes, e.g., [29,1,2,3,4,20,21]. Research has focused also on automated reasoning (e.g., model checking for quantum systems as considered in [23]) and on formal analysis of quantum protocols (e.g., [25]).…”

“…In this way, it will be clear how we associate a mathematical denotation to each single agent. We made this strong assumption in order to provide simple and intuitive examples (see Examples 3,4,5) by means of the obvious interpretation of a propositional symbol p as the vector which encodes the mathematical state of the agent.…”

Computing the Exit Complexity of Knowledge in Distributed Quantum Computers

“…imitative computations direct to an omnipresent design of 'fault merciful' counting, of which an auxiliary observe is ascribed by Preskill [ 29]. Their claim been numerous approaches at explicating quantum schemas in provisos of formulation transformers: in [30] a synthesis of quantum assesses, based on propositional energetic deduction is constructed, while in [31], a weakest precondition semantics for quantum considerations are corrected. additionally, a propositional cogitation conceived to explicate quantum determination at an operational category is embed obtrusive in [32].…”