A Logic for Coalgebraic Simulation

AbstractBuilding on the work of L. Moss on coalgebraic logic, we study in a general setting a class of infinitary modal logics for F-coalgebras, designed to capture simulation and bisimulation. We use work by A. Thijs on coalgebraic modelling of simulation, in terms of relators Γ as extensions of functors. We prove our logics can indeed capture simulation and bisimulation, i.e. the existence of a simulation (or bisimulation) is equivalent to the preservation of (or equivalence with respect to) certain classes of sentences. Moreover, we prove that one can characterize any given coalgebra up to simulation (and, in certain conditions, up to bisimulation) by a single sentence. We show that truth for this logic can be understood as a simulation relation itself, but with respect to a richer functor F moreover, it is the the largest simulation, i.e. the similarity relation between states of the coalgebra and elements of the language. This sheds a new perspective on the classical preservation and characterizability results, and also on logic games. The two kinds of games normally used in logic (“truth games” to define the semantics dynamically, and “similarity games” between two structures) are seen to be the same kind of game at the level of coalgebras: simulation games

Logic meets Wigner's Friend (and their Friends)

We take a fresh look at Wigner's Friend thought-experiment and some of its more recent variants and extensions, such as the Frauchiger-Renner (FR) Paradox. We discuss various solutions proposed in the literature, focusing on a few questions: what is the correct epistemic interpretation of the multiplicity of state assignments in these scenarios; under which conditions can one include classical observers into the quantum state descriptions, in a way that is still compatible with traditional Quantum Mechanics?; under which conditions can one system be admitted as an additional 'observer' from the perspective of another background observer?; when can the standard axioms of multi-agent Epistemic Logic (that allow "knowledge transfer" between agents) be applied to quantum-physical observers? In the last part of the paper, we propose a new answer to these questions, sketch a particular formal implementation of this answer, and apply it to obtain a principled solution to Wigner Friend-type paradoxes.Comment: 27 page

The dynamic logic of stating and asking

Inquisitive dynamic epistemic logic (IDEL) extends standard public announcement logic incorporating ideas from inquisitive semantics. In IDEL, the standard public announcement action can be extended to a more general public utterance action, which may involve a statement or a question. While uttering a statement has the effect of a standard announcement, uttering a question typically leads to new issues being raised. In this paper, we investigate the logic of this general public utterance action. We find striking commonalities, and some differences, with standard public announcement logic. We show that dynamic modalities admit a set of reduction axioms, which allow us to turn any formula of IDEL into an equivalent formula of static inquisitive epistemic logic. This leads us to establish several complete axiomatizations of IDEL, corresponding to known axiomatizations of public announcement logic

Quantum logic as a dynamic logic

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 operational-realistic 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 (Baltag and 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

Correlated Knowledge:An Epistemic-Logic View on Quantum Entanglement

In this paper we give a logical analysis of both classical and quantum correlations We propose a new logical system to reason about the information carried by a complex system composed of several parts Our formalism is based on an extension of epistemic logic with operators for "group knowledge" (the logic GEL), further extended with atomic sentences describing the results of joint observations" (the logic LCK) As models we introduce correlation models, as a generalization of the standard representation of epistemic models as vector models We give sound and complete axiomatizations for our logics, and we use this setting to investigate the relationship between the information carried by each of the parts of a complex system and the information carried by the whole system In particular we distinguish between the "distributed information", obtainable by simply pooling together all the information that can be separately observed in any of the parts and correlated information, obtainable only by doing joint observations of the parts (and pooling together the results) Our formalism throws a new light on the difference between classical and quantum information and gives rise to an informational-logical characterization of the notion of "quantum entanglement

Algebra and Sequent Calculus for Epistemic Actions

We introduce an algebraic approach to Dynamic Epistemic Logic. This approach has the advantage that: (i) its semantics is a transparent algebraic object with a minimal set of primitives from which most ingredients of Dynamic Epistemic Logic arise, (ii) it goes with the introduction of non-determinism, (iii) it naturally extends beyond boolean sets of propositions, up to intuitionistic and non-distributive situations, hence allowing to accommodate constructive computational, information-theoretic as well as non-classical physical settings, and (iv) introduces a structure on the actions, which now constitute a quantale. We also introduce a corresponding sequent calculus (which extends Lambek calculus), in which propositions, actions as well as agents appear as resources in a resource-sensitive dynamic-epistemic logic

A Modal Logic for Supervised Learning

Formal learning theory formalizes the process of inferring a general result from examples, as in the case of inferring grammars from sentences when learning a language. In this work, we develop a general framework—the supervised learning game—to investigate the interaction between Teacher and Learner. In particular, our proposal highlights several interesting features of the agents: on the one hand, Learner may make mistakes in the learning process, and she may also ignore the potential relation between different hypotheses; on the other hand, Teacher is able to correct Learner’s mistakes, eliminate potential mistakes and point out the facts ignored by Learner. To reason about strategies in this game, we develop a modal logic of supervised learning and study its properties. Broadly, this work takes a small step towards studying the interaction between graph games, logics and formal learning theory.acceptedVersio