This one-day workshop will feature three graduate student speakers:

 

Modeling Chisholm's Logic of Obligation, Requirement, and Defeat

With Marian J.R. Gilton (Graduate Student, LPS) 

This talk investigates the foundations of Chisholm's deontic logic as an alternative to standard deontic logic. Over the course of his career, Chisholm articulated a set of axioms for a deontic logic, and he gestured toward its connection with his work with Sosa on a logic of preference. Although Chisholm's work discussed the motivations and applications of his axioms, he never provided a model for them. This talk’s chief aim is to construct a model for Chisholm's deontic logic defined in terms of the Chisholm-Sosa preference logic. This is done by combining techniques from modal logic and social choice theory. Since Chisholm's system is less familiar today than standard deontic logic, the paper begins with a thorough reconstruction of his axiomatization. Finally, having shown that Chisholm's logic is a viable alternative to standard deontic logic, this talk shows that his system evades Chisholm's paradox in a novel way.

 

A Model Theoretic Discussion of Statistical Learning

With Kino Zhao (Graduate Student, LPS)

One of the primary theoretical tools in machine learning is Vapnik-Chervonenkis dimension (VC dimension), which measures the maximum number of distinct data points a hypothesis set can distinguish. An important result of Laskowski showed that finite VC dimension corresponds to a logical notion independently developed by Shelah, known as the non-independence property. In subsequent decades much work has been done on finite VC dimension within model theory under the aegis of so-called NIP theories. In this talk, it’s proven the following: against the standard model of arithmetic with the ring structure, the problem of determining whether a formula has finite VC dimension is at least as hard as computing the second Turing jump, and thus is in an obvious sense computationally intractable. Zhao will also discuss what is known about the feasibility of calculating the VC dimension in cases, like the real numbers with the field signature, where it is known that this quantity is in-principle computable.

 

Isaacson's Thesis and Wilkie's Theorem

With Stella Moon (Masters Student, University of Amsterdam)

This talk explores Isaacson's thesis and Wilkie's theorem, providing philosophical and formal results on how they relate to each other. At a first approximation, Isaacson's thesis claims that Peano arithmetic is sound and complete with respect to genuinely arithmetical statements. Moon will give a survey of Isaacson's philosophical views leading up to his thesis and, using internalist notions familiar from recent work on internal categoricity theorems, will provide a formal definition of genuinely arithmetical. As for Wilkie’s theorem, it roughly says that, from an external perspective, Peano arithmetic is minimal, in that it is entailed by all categorical axiomatisations of the natural numbers satisfying a certain syntactic restriction. After expositing Wilkie's theorem and the relation of its proof to other known techniques, Moon will discuss its relation to Isaacson's thesis, in particular by reference to maximal and orthogonal such axiomatisations.

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