Epistemic logics for sceptical agents

Abstract

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.

From positive PDL to its non-classical extensions

“…The canonical PDL ∼ -structure is defined just as the canonical PDL +structure where ∼ C is defined as in the proof of Theorem 5.1, with the exception that t ∈ P are non-trivial prime theories. 5 Claim 5.4. If t is a non-trivial prime theory, then so is t ∼ C = {X ; ∼X ∈ t}.…”

“…According to the informational interpretation of Lambek models, states are seen as bodies of information (or information states) and R represents merging of information states; we may read Ruvw as "merging u with v might result in information state w". A similar interpretation of relational Lambek models is the basis of several applications of substructural logics in epistemic logic; see [5,36,39] where versions of DF N L e are used. ( [39] argues for the need to use non-associative commutative structures, but in that paper an operational version-where R is a binary operation-is used; [5,36] use relational models.)…”

“…A similar interpretation of relational Lambek models is the basis of several applications of substructural logics in epistemic logic; see [5,36,39] where versions of DF N L e are used. ( [39] argues for the need to use non-associative commutative structures, but in that paper an operational version-where R is a binary operation-is used; [5,36] use relational models.) P DL \ and its extensions can be seen as formalisms for reasoning about structured modifications of information states, with actions representing types of such modifications.…”

Substructural Inquisitive Logics

“…This logic is equivalent to Došen's system E + from (Došen, 1988) enriched with the axiom A2 for ⊥ and the rules R7, R8, R9 for t and ¬. The system can be viewed also as a nondistributive and noncommutative version of the Hilbert system for Full Lambek logic with a paraconsistent negation used in (Bílkova, Majer, & Peliš, 2016) and (Sedlár, 2015). The original Lambek logic was introduced in (Lambek, 1958) as a logic of syntactic types.…”

First Degree Entailment with Group Attitudes and Information Updates

“…An FDE-based group epistemic logic with universal and common knowledge is a fragment of paraconistent Propositional Dynamic Logic studied in [31,32]. Bílková et al [10] outline an extension of their substructural epistemic framework with common knowledge, but completeness is left for future research. The relation between substructural logic and classical information dynamics is studied in [5,7] and [1], for example; [15,27] discuss an informationdynamic interpretation of the Routley-Meyer semantics for some substructural logics.…”