Revisiting Epistemic Logic with Names

Abstract

This paper revisits the multi-agent epistemic logic presented in [10], where agents and sets of agents are replaced by abstract, intensional "names". We make three contributions. First, we study its model theory, providing adequate notions of bisimulation and frame morphisms, and use them to study the logic's expressive power and definability. Second, we show that the logic has a natural neighborhood semantics, which in turn allows to show that the axiomatization in [10] does not rely on possibly controversial introspective properties of knowledge. Finally, we extend the logic with common and distributed knowledge operators, and provide a sound and complete axiomatization for each of these extensions. These results together put the original epistemic logic with names in a more modern context and opens the door for a logical analysis of epistemic phenomena where group membership is uncertain or variable.In [10], Grove and Halpern studied a generalized version of multi-agents epistemic logic where the usual labels for agents and sets of agents are replaced by abstract names whose extension might vary from state to state. 1 Despite being interpreted in standard multi-agents epistemic models, the resulting language does away with the familiar K i modalities, and instead contains two families of epistemic operators: S n , standing for "someone with name n knows", and E n , standing for "everyone with name n knows". This generalization is conceptually important. The "names" that index the S n and the E n modalities can refer intensionally to both individuals and groups. Since these extensions are not fixed in a given model, the logic allows to study social-epistemic phenomena that involve uncertainty or variability in the agents' identities or group membership. These phenomena are pervasive. [19,10] already provide convincing examples for distributed systems. Massive coordinated actions or social movements, especially online, also provide contemporary cases [1,4], where for instance we refer to group labels like "Trump supporters" or "trolls" without knowing exactly who the members of these groups are or even failing to know whether we, ourselves, are members of those groups. 2 The study in [10], however, focuses on the two modalities mentioned above, and in particular leaves aside notions of common and distributed knowledge. These notions are, however, central to theories of social conventions [16,2] and collective action [25].[19] study a closely related notion of common knowledge, to which we come back briefly in Section 4, but do not provide an axiomatization. Distributed knowledge for intensional or indexical group names has been studied in [20], but in a more expressive language with explicit quantification.

Epistemic Logics of Structured Intensional Groups

“…The modalities 〈 ] and [ ] are monotone modalities of the ∃∀ and ∀∀ type and can therefore be studied in terms of monotone neighborhood semantics, if we understand sets {r(w) | r ∈ a(w)} as so called core neighborhood sets [7,14]. Relational Σ-frames generalize relational frames for epistemic logic with names [2,6] (Example 1), which are categorialy equivalent to monotone neighborhood frames with neighborhood sets indexed by the set of names. Not surprisingly, a closely related connection arises between relational Σ-frames of this paper and monotone neighborhood frames where neighborhoods are indexed with algebraic terms.…”

“…Given the completeness of the basic Σ-logic with respect to relational Σ-frames (Theorem 1), completeness with respect to all neighborhood Σ-frames follows 6 With the complex algebra/ultrafilter frame construction at hand, we can describe the algebraic duality, and obtain a definability theorem characterizing modally definable classes of neighborhood Σ-frames (cf. Theorem 2 of [2]).…”

“…Grove and Halpern's work is enjoying a recent resurgence of interest in the epistemic logic community. [2] identifies a monotone neighborhood-style semantics for Grove and Halpern's language and, building directly on the S n and E n modalities, considers expansions with non-rigid versions of common and distributed knowledge. Distributed or common knowledge for intensional group names has also been studied by [12,13].…”

“…where (the "interpretation function") is a homomorphism from T m Σ ∪ Fm Σ to F +2 , that is, Example 1. Consider a relational frame for epistemic logic with names [2]. Let N ("names"), A ("agents") and W ("worlds") be three non-empty sets.…”

Propositional Dynamic Logic with Quantification over Regular Computation Sequences

“…It is interesting to note that there is a relation between the framework presented here and epistemic logic with names [2,5], containing epistemic operators E n and S n , where n is a "name", possibly referring to a number of agents, reading "all agents named n know that..." and "some agent named by n knows that...". Our framework generalizes this construction to the setting where "names" are regular expressions (i.e.…”