Some Computationalconstraints In Epistemic Logic

Advances in Modal Logic, Vol. 8, 2010

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is the Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas.

2008

The interpreted system model offers a computationally grounded model, in terms of the states of computer processes, to S5 epistemic logics. This paper extends the interpreted system model, and provides a computationally grounded one, called the interpreted perception system model, to those epistemic logics other than S5. It is usually assumed, in the interpreted system model, that those parts of the environment that are visible to an agent are correctly perceived by the agent as a whole. The essential idea of the interpreted perception system model is that an agent may have incorrect perception or observations to the visible parts of the environment and the agent may not be aware of this. The notion of knowledge can be defined so that an agent knows a statement iff the statement holds in those states that the agent can not distinguish (from the current state) by using only her correct observations. We establish a logic of knowledge and certainty, called KC logic, with a sound and complete proof system. The knowledge modality in this logic is S4 valid. It becomes S5 if we assume an agent always has correct observations; and more interestingly, it can be S4.2 or S4.3 under other natural constraints on agents and their sensors to the environment.

Syntactic logics do not suffer from the problems of logical omniscience but are often thought to lack interesting properties relating to epistemic notions. By focusing on the case of rule based agents, I develop a framework for modelling resource-bounded agents and show that the resulting models have a number of interesting properties.

This paper argues that propositional modal logics based on Kripke-structures cannot be accepted by epistemologists as a minimal framework to describe propositional knowledge. In fact, many authors have raised doubts over the validity of the so-called principle of epistemic closure, which is always valid in normal modal logics. This paper examines how this principle might be criticized and discusses one possible way to obtain a modal logic where it does not hold, namely through the introduction of impossible worlds.

2015

The aim of epistemic logics is to formalize epistemic states and actions of (possibly human) rational agents. A traditional means for representing these states and actions employs the framework of modal logics, where knowledge corresponds to some necessity operator. Modal axioms (K, T, 4, 5,...) then

Lecture Notes in Computer Science, 1998

We give a general approach to characterizing minimal information in a modal context. Our modal treatment can be used for many applications, but is especially relevant under epistemic interpretations of the operator . Relative to an arbitrary modal system, we give three characterizations of minimal information and provide conditions under which these characterizations are equivalent. We then study information orders based on bisimulations and Ehrenfeucht-Fraïssé games. Moving to the area of epistemic logics, we show that for one of these orders almost all systems trivialize the notion of minimal information. Another order which we present is much more promising as it permits to minimize with respect to positive knowledge. In S5, the resulting notion of minimal knowledge coincides with well-established proposals. For S4 we compare the two orders.

Mind, 2007

Epistemic modal operators give rise to something very like, but also very unlike, Moore's paradox. I set out the puzzling phenomena, explain why a standard relational semantics for these operators cannot handle them, and recommend an alternative semantics. A pragmatics appropriate to the semantics is developed and interactions between the semantics, the pragmatics, and the definition of consequence are investigated. The semantics is then extended to probability operators. Some problems and prospects for probabilistic representations of content and context are explored.

Reasoning with Relevant Epistemic Logics, PhD Dissertation, Scuola Normale Superiore, 2025

The present dissertation investigates the relationships between classical and nonclassical logics within a common logical platform, in which it is possible to address philosophical questions. More specifically, the focus of the dissertation lies in applications of relevant logics to problems in formal epistemology, such as the problem of logical omnisicence, the distinction between explicit and implicit beliefs, and the modeling of belief revision. As the main contribution of the present work, I devise so-called contextual modal logics, i.e. modal logics where modal operators individuate the range of application of a given logic, to be chosen from the class of extensions of weak relevant logics. In this way, I argue that it is possible to adequately formalise the context-sensitivity of reasoning, a feature which arises when we confront reasoning about the factual and the epistemic domain. Contextual modal logics are presented both semantically and proof-theoretically, by means of Hilbert-style axiom systems. The language of the logics is modularly extended, from the basic relevant language, so as to include topic sensitive operators, implicit belief operators and dynamic operators. Characterisation results are obtained for all extensions of the logics in the contextual modal family.

Acts of Knowledge: History, Philosophy and Logic, 2009

Journal of Logic and Computation, 2015

In this article, we introduce an epistemic modal operator modelling knowledge over distributive non-associative full Lambek calculus with a negation. Our approach is based on the relational semantics for substructural logics: we interpret the elements of a relational frame as information states consisting of collections of data. The principal epistemic relation between the states is the one of being a reliable source of information, on the basis of which we explicate the notion of knowledge as information confirmed by a reliable source. From this point of view it is natural to define the epistemic operator formally as the backward-looking diamond modality. The framework is a generalization and extension of the system of relevant epistemic logic proposed by Majer and Peliš (2009, college Publications, 123-135) and developed by Bílková et al. (2010, college Publications, 22-38). The system is modular in the sense that the axiomatization of the epistemic operator is sound and complete with respect to a wide class of background logics, which makes the system potentially applicable to a wide class of epistemic contexts. Our system admits a weak form of logical omniscience (the monotonicity rule), but avoids stronger ones (a necessitation rule and a K-axiom) as well as some closure properties discussed in normal epistemic logics (like positive and negative introspection). For these properties we provide characteristic frame conditions, so that they can be present in the system if they are considered to be appropriate for some specific epistemic context. We also prove decidability of the weakest epistemic logic we consider, using a filtration method. Finally, we outline further extensions of our framework to a multiagent system.