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Sep 27 Logic Colloquium: Luca Tranchini (Tuebingen)1:30pm

Logic Colloquium: Luca Tranchini (Tuebingen)

Friday, September 27th, 2019

01:30 PM - 03:00 PM

Storrs Campus
Humantities Institute, 4th Floor Homer Babbidge Library

Join us for a talk in the Logic Colloquium!

Luca Tranchini (Tuebingen)

Proof-theoretic Harmony: intensional aspects from a second-order perspective

Abstract:

The talk will focus on harmony, the widely discussed condition that two collections of introduction and of elimination rules for a certain logical constant should satisfy in order to perfectly match each other. Attention will be restricted to rules for propositional connectives in single-conclusion natural deduction format.

I will first argue that a decent criterion of harmony should be (hyper-)intensional, in the sense that taken two equivalent collections of elimination rules (i.e. such that each rule of the first collection is derivable from the rules in the second collection, and vice versa), it may well be that only one of the two collections is in harmony with a given collection of introduction rules.

To precisely formulate an intensional criterion for harmony we propose to modify the (extensional) criterion for harmony recently proposed by Schroeder-Heister (for short SH-harmony) and based on the idea that to any collection of introduction and of elimination rules one can associate a formula of intuitionistic propositional logic with propositional quantification (IPC2). Two collections of introduction and elimination rules for a certain connective are in SH-harmony if the two formulas associated to them are interderivable in IPC2.

An intensional criterion is achieved by imposing on the two formulas a condition stricter than intederivability: isomorphim. We will present the notion of isomorphism (well-known from category theory and computational study of proof systems), stress its significance for proof-theoretic semantics, and finally apply it to clarify the relationship between different ways of generating collections of elimination rules from a given collection of introduction rules known in the literature.

All welcome!

https://logic.uconn.edu/

Contact Information: https://logic.uconn.edu/about/
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Oct 4 Logic Colloquim: Ellen Lehet (Notre Dame)1:30pm

Logic Colloquim: Ellen Lehet (Notre Dame)

Friday, October 4th, 2019

01:30 PM - 03:00 PM

Storrs Campus
Humantities Institute, 4th Floor Homer Babbidge Library

Join us for a talk by Ellen Lehet on category theory.

Ellen Lehet (University of Notre Dame)

"The Explanatory Value of Category Theory"

Abstract

Category theory has proven to be applicable across all of mathematics. In some sense this is not surprising because category theory was created for the purpose of application (specifically, application to algebraic topology). But I will argue that the significance of category theory extends past its applicability — in particular, there is a significant explanatory benefit. The question of what constitutes a mathematical explanation is of perennial interest to philosophers. Reflection on category theory’s unique role in mathematics unearths some features of mathematical explanation that are not often made explicit and that philosophers have tended not to notice.

There are many ways that category theoretic methods provide explanations. For instance, important results in different areas of mathematics are unified by the fact that they are all corollaries of the same category theoretic theorem, such as the theorem that right adjoints preserve limits. Or consider the ways of defining structures in category theory with universal properties — the whole perspective sheds light on how constructions from different domains are related to one another. The categorical product for instance, unites many seemingly unrelated mathematical constructions such as the Cartesian product, union, and conjunction. Such examples introduce both generalization and unification within mathematics. Moreover, this unification allows for meaningful and surprising mathematical analogies to arise. These gen- eralizations and analogies are explanatory and result from the structural features of category theory.

In order to highlight the explanatory value of category theory, I will first provide a characterization of the structure unique to category theory. It is this structure that makes category theory apt for producing explanations. With a clear picture of category theoretic structure, I will present a few examples that illustrate how category theory proves to be explanatory — in particular, how the structural features of category theory are explanatory.

All welcome!

https://logic.uconn.edu/calendar/

Contact Information: https://logic.uconn.edu/about/
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Oct 11 Logic Colloquium: Ethan Brauer (OSU)1:30pm

Logic Colloquium: Ethan Brauer (OSU)

Friday, October 11th, 2019

01:30 PM - 03:00 PM

Storrs Campus
Humantities Institute, 4th Floor Homer Babbidge Library

Join us for a talk in the Logic Colloquium!

Ethan Brauer (OSU):

A Modal Theory of Free Choice Sequences

Free choice sequences (also called 'infinitely proceeding sequences') are a concept from intuitionistic mathematics that are central to the development of the intuitionistic theory of the continuum. Free choice sequences have, however, been largely rejected or ignored both by classical and other constructive mathematicians. In this talk I argue that the objections to free choice sequences can be overcome by grounding the concept in our temporal intuition and formalizing the theory in modal logic. I will present a theory of free choice sequences as a modal extension of classical second-order arithmetic. The resulting theory is able to prove modal versions of the intuitionist's axioms for so-called lawless sequences, suffices for the development of a theory of real number generators, and captures many results distinctive of intuitionistic analysis including the non-existence of functions on real numbers with definable discontinuities.

All welcome.

https://logic.uconn.edu/

Contact Information: https://logic.uconn.edu/about/
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