Thomas F. Icard

A Topological Study of the Closed Fragment of GLP

In this article, we study the canonical model for the closed fragment of GLP and establish its precise relationship with a universal model constructed by Ignatiev. In particular, we effectively characterize the canonical model in terms of a coordinate system based on sequences of ordinals up to e0.We then define a simple topological model of this logic by defining a natural polytopology on the ordinal e0 itself.

Inclusion and Exclusion in Natural Language

We present a formal system for reasoning about inclusion and exclusion in natural language, following work by MacCartney and Manning. In particular, we show that an extension of the Monotonicity Calculus, augmented by six new type markings, is sufficient to derive novel inferences beyond monotonicity reasoning, and moreover gives rise to an interesting logic of its own. We prove soundness of the resulting calculus and discuss further logical and linguistic issues, including a new connection to the classes of weak, strong, and superstrong negative polarity items.

Schematic validity in dynamic epistemic logic: decidability

Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. The classic example is Public Announcement Logic (PAL), an extension of epistemic logic based on the idea of information acquisition as elimination of possibilities. In this paper, we address the open question of whether the set of schematic validities of PAL, the set of formulas all of whose substitution instances are valid, is decidable. We obtain positive answers for multi-agent PAL, as well as its extension with relativized common knowledge, PAL-RC. The conceptual significance of substitution failure is also discussed.

Information dynamics and uniform substitution

The picture of information acquisition as the elimination of possibilities has proven fruitful in many domains, serving as a foundation for formal models in philosophy, linguistics, computer science, and economics. While the picture appears simple, its formalization in dynamic epistemic logic reveals subtleties: given a valid principle of information dynamics in the language of dynamic epistemic logic, substituting complex epistemic sentences for its atomic sentences may result in an invalid principle. In this article, we explore such failures of uniform substitution. First, we give epistemic examples inspired by Moore, Fitch, and Williamson. Second, we answer affirmatively a question posed by van Benthem: can we effectively decide when every substitution instance of a given dynamic epistemic principle is valid? In technical terms, we prove the decidability of this schematic validity problem for public announcement logic (PAL and PAL-RC) over models for finitely many fully introspective agents, as well as models for infinitely many arbitrary agents. The proof of this result illuminates the reasons for the failure of uniform substitution.

Pragmatic Considerations on Comparative Probability

While pragmatic arguments for numerical probability axioms have received much attention, justifications for axioms of qualitative probability have been less discussed. We offer an argument for the requirement that an agent’s qualitative (comparative) judgments be probabilistically representable, inspired by, but importantly different from, the Money Pump argument for transitivity of preference and Dutch book arguments for quantitative coherence. The argument is supported by a theorem, to the effect that a subject is systematically susceptible to dominance given her preferred acts, if and only if the subject’s comparative judgments preclude representation by a standard probability measure (or set of such measures).

Provability and Interpretability Logics with Restricted Realizations

The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set

Inferring Probability Comparisons

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On the Conditional Logic of Simulation Models

. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set

Bayes, Bounds, and Rational Analysis

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A Monotonicity Calculus and Its Completeness.

, where each sentence in