Modeling correlated information change: from conditional beliefs to quantum conditionals

Abstract

In this paper, we propose a unified logical framework for representing and analyzing various forms of correlated information change. Our main thesis is that "logical dynamics," in the sense of van Benthem (Exploring logical dynamics. CSLI Publications, Stanford, 1996; Logical dynamics of information and interaction. Cambridge University Press, Cambridge, 2011), and in particular dynamic epistemic notions of conditional, as developed in Baltag and Smets (Electron Notes Theor Comput Sci 165:5-21, 2006a; Stud Log 89:185-209, 2008a; Texts in logic and games. Amsterdam University Press, Amsterdam, pp 9-58, 2008b), play a central role in understanding and modeling a wide range of apparently very different information-gathering phenomena which do have one specific feature in common, namely the very act of learning new information may directly change the reality that is being learned. On the one hand, we focus on the way in which an introspective agent changes her beliefs when learning new higher-order information, i.e., information that may refer to her own beliefs. On the other hand, we analyze situations in which an observer learns about a phenomenon by performing observations that may perturb the very phenomenon under study, as in the case of quantum measurements, or observations in social sciences, psychology and medicine. Our formal techniques are based on ideas Communicated by M.L. Dalla Chiara, R. Giuntini, E. Negri. from dynamic logic and on the modeling of "dynamic conditionals." We offer a semantics based on "test frames," i.e., Kripke frames labeled by propositional formulae which yields a unified setting for the two types of correlated information change under study. We show how this framework can be used to analyze the ontic and epistemic-informational aspects of quantum measurements and to compare them with other types of observation, testing, belief revision, counterfactual conditionals, etc.

Dynamic Logic of Quantum Field Theory

“…That is, by means of the quantum test [A? q ] (see [3] for precise definition of [A? q ]), the truth condition of quantum formulas ∼A, A ∧ B, A B and A B are defined as follows [2,3]:…”

Implicit and Explicit Stances in Logic

“…However, in what follows, we stick with concrete examples, always keeping an open eye for intensional differences between even tightly mutually translated systems. 7 From Translating to Merging Finally, moving away from an emphasis on reduction, there is a weaker but still significant contact between explicit and implicit logics, namely that of compatibility. Can such systems be merged in meaningful ways?…”

“…But in this version, pluralism is much less striking 18. An additional virtue is the analogy with our earlier dynamic-epistemic logics of knowledge change, making it possible to add epistemic considerations of agency to quantum information theory,[7].…”

“…Theories that are equivalent under translation, perhaps an inefficient one, may have different computational properties. We will not pursue this angle here, but complexity is a natural way of driving a finer wedge 7. Also, translations only work with the right deductive power on both sides.…”

Automated Proof Search System for Logic of Correlated Knowledge

“…One of the latest development in this way is quantum modal logics. The research have been done by A. Baltag and S. Smets [4,5], F. Boge [8], V. Vilasini [22] and N. Nurgalieva [17]. Our study covers field of logic of correlated knowledge which is part of quantum modal logics.…”