Announcement as effort on topological spaces

Abstract

We propose a multi-agent logic of knowledge, public announcements and arbitrary announcements, interpreted on topological spaces in the style of subset space semantics. The arbitrary announcement modality functions similarly to the effort modality in subset space logics, however, it comes with intuitive and semantic differences. We provide axiomatizations for three logics based on this setting, with S5 knowledge modality, and demonstrate their completeness. We moreover consider the weaker axiomatizations of three logics with S4 type of knowledge and prove soundness and completeness results for these systems.

A dynamic logic for learning theory

“…More recent papers, closely related to our logic, are Bjorndahl [8], van Ditmarsch et alia [20,21], and Baltag et alia [7]. Bjorndahl [8] introduces dynamic modalities [ϕ] for arbitrary formulas (rather than restricting to observational variables [o], as we do), though with a different semantics (according to which [ϕ] restricts the space to the interior of ϕ, in contrast to our simpler semantics, that follows the standard definition of update or "public announcement').…”

“…His syntax does not contain the effort modality, or any other form of quantifying over observations. The work of van Ditmarsch et alia [20,21] uses Bjorndahl-style dynamic modalities in combination with a topological version of the so-called "arbitrary public announcement" operator, which is a more syntacticdriven relative of the effort modality. This syntactic nature comes with a price: the logic of arbitrary public announcements is much less well-behaved than SSL (or our logic), in particular it has non-compositional features (-the meaning of a formula may depend on the meaning of all atomic variables, including the ones that do not occur in that formula!).…”

“…Of course, we are not interested in cases of convergence to the truth 'by accident', but in determining whether or not a given learner is guaranteed to eventually track the truth; hence, the focus on 'The Logic of Reliable Inquiry'. 5 We propose a formalism that combines ideas from: Subset Space Logics, as introduced by Moss and Parikh [17], investigated further by Dabrowski et al [9] and already merged with the DEL tradition in prior work [25,3,20,22,8,7]; the topological approach to FLT in [16,5]; and the general agenda of bridging DEL and FLT in [13]. Semantically, we take intersection spaces (a type of subset spaces that are closed under finite non-empty intersections), with points interpreted as possible worlds and neighbourhoods interpreted as observations (or information states) (see, e.g., [18] for a survey on subset space logics).…”

Uncertainty About Evidence

“…Consider a public announcement style update mechanic in which knowability plays the role of the precondition of the corresponding announcement, as in [11]. The effect of an announcement is then manifested as a transition from the initial evidence state to a more informative one, without requiring global changes in the given model, as in logics of information dynamics interpreted on subset space models [28,12,16,6,3]. The enriched structure owing to the evidence states and their variable interpretations raises the question of whether such a dynamic logic can be reduced to a weaker, static logic, as is often the case in similar settings.…”

Logic and Topology for Knowledge, Knowability, and Belief

“…More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an "evidence-inhand" conception of knowledge and a weaker "evidence-out-there" notion of what could come to be known. As such, we find it quite natural to study Stalnaker's principles in the richer semantic setting developed in [11], which is based on topological subset spaces, a class of epistemic models of growing interest in recent years [27,15,11,34,35]. These models support a careful reworking of Stalnaker's system in a manner that respects the distinction described above, yielding a trimodal logic of knowledge, knowability, and belief that is our main object of study.…”