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In Savage's classic decision-theoretic framework, actions are formally defined as functions from states to outcomes. But where do the state space and outcome space come from? Expanding on recent work by Blume, Easley, and Halpern (BEH), we consider a language-based framework in which actions are identified with (conditional) descriptions in a simple underlying language, while states and outcomes (along with probabilities and utilities) are constructed as part of a representation theorem. Our work expands the role of language from that of BEH by using it not only for the conditions that determine which actions are taken, but also the effects. More precisely, we take the set of actions to be built from those of the form "do(phi)", for formulas phi in the underlying language. This presents a problem: how do we interpret the result of do(phi) when phi is underspecified (i.e., compatible with multiple states)? We answer this using tools familiar from the semantics of counterfactuals: roughly speaking, do(phi) maps each state to the "closest" phi-state. This notion of "closest" is also something we construct as part of the representation theorem; in effect, then, we prove that (under appropriate assumptions) the agent is acting as if each underspecified action is first made definite and then evaluated (i.e., by maximizing expected utility). Of course, actions in the real world are often not presented in a fully precise manner, yet agents reason about and form preferences among them all the same. Our work brings the abstract tools of decision theory into closer contact with such real-world scenarios., In Proceedings TARK 2021, arXiv:2106.10886

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, and using frameworks developed in previous work by Bjorndahl and Baltag et al., 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 (roughly speaking) 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., Comment: 26 pages

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, and using frameworks developed by Bjorndahl and Baltag et al., 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 (roughly speaking) 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., Comment: In Proceedings TARK 2017, arXiv:1707.08250. The full version of this paper, including the longer proofs, is at arXiv:1612.02055

We reformulate a key definition given by Wang and Agotnes [18] to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of \(\varphi \) to be the “local truth” of \(\varphi \), semantically rendered via an interior operator. This is closely related to the notion of \(\varphi \) 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.

Halpern and Pass [8] introduce a logic of justified belief and go on to prove that strong rationalizability is characterized in this logic in terms of common justified belief of rationality (CJBR). Their paper provides semantics for this logic but no axiomatization. We correct this deficiency by reformulating the definition of justified belief and providing a complete axiomatization of this new system. We then prove a result analogous to the characterization of strong rationalizability in terms of CJBR, and analyze the additional assumptions needed to do so.

A single counterfactual conditional can have a multitude of interpretations that differ, intuitively, in the connection between antecedent and consequent. Using structural equation models (SEMs) to represent event dependencies, we illustrate various types of explanation compatible with a given counterfactual. We then formalize in the SEM framework the notion of an acceptable explanation, identifying the class of event dependencies compatible with a given counterfactual. Finally, by incorporating SEMs into possible worlds, we provide an update semantics with the enriched structure necessary for the evaluation of counterfactual conditionals.