In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker’s core insights, we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and what is knowable; we argue that the foundational axioms of Stalnaker’s system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the (...) principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an “evidence-in-hand” conception of knowledge and a weaker “evidence-out-there” notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker’s postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics. (shrink)

We reformulate a key definition given by Wáng and Ågotnes to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ???? to be the “local truth” of ????, semantically rendered via an interior operator. This is closely related to the notion of ???? being “knowable”. We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial (...) role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness. (shrink)

This paper proposes new semantics for propositional dynamic logic (PDL), replacing the standard relational semantics. Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) (Kremer and Mints in Ann Pure Appl Logic (...) 131:133–158, 2005). We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic (Dabrowski et al. in Ann Pure Appl Logic 78:73–110, 1996), and show that the topological interpretation of public announcements as given in Bjorndahl (in: van Ditmarsch and Sandu (eds) Jaakko Hintikka on knowledge and game theoretical semantics, outstanding contributions to Logic, vol 12, Springer, 2018) coincides exactly with a natural interpretation of test programs. (shrink)

This lively and accessible textbook provides a comprehensive and unified introduction to classical and modal logics, treating them with the same level of rigour and detail and showing how they fit together. A fully self-contained learning resource, it will be ideal for upper-level university courses.

Through a series of examples, we illustrate some important drawbacks that the action model logic framework suffers from in its ability to represent the dynamics of information updates. We argue that these problems stem from the fact that the action model, a central construct designed to encode agents’ uncertainty about actions, is itself effectively common knowledge amongst the agents. In response to these difficulties, we motivate and propose an alternative semantics that avoids them by endogenizing the action model. We discuss (...) the relationship between this new framework and action model logic, and provide a sound and complete axiomatization of several new logics that naturally arise. (shrink)

Classical philosophical analyses seek to explain knowledge as deriving from more basic notions. The influential “knowledge first” program in epistemology reverses this tradition, taking knowledge as its starting point. From the perspective of epistemic logic, however, this is not so much a reversal as it is the default—the field arguably begins with the specialization of “necessity” to “epistemic necessity”—that is, it begins with knowledge. In this context, putting knowledge second would be the reversal. This article motivates, develops, and explores such (...) a “knowledge second” approach in epistemic logic, founded on distinguishing what a body of evidence actually entails from what it is (merely) believed to entail. We import a logical framework that captures exactly this distinction, use it to define formal notions of (internal and external) justification and knowledge, and investigate applications to the KK principle, the “strong belief” postulate, and the regress problem. (shrink)