Steve Awodey

Homotopy theoretic models of identity types

This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.

Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics so as to shed new light on the relevant strengths and limits of higher-order logic.

Carnap, completeness, and categoricity:The gabelbarkeitssatz OF 1928

In 1929 Carnap gave a paper in Prague on “Investigations in General Axiomatics”; a briefsummary was published soon after. Its subject lookssomething like early model theory, and the mainresult, called the Gabelbarkeitssatz, appears toclaim that a consistent set of axioms is complete justif it is categorical. This of course casts doubt onthe entire project. Though there is no furthermention of this theorem in Carnaps publishedwritings, his Nachlass includes a largetypescript on the subject, Investigations inGeneral Axiomatics. We examine this work here,showing that it provides important insights intoCarnaps development during this critical period, thetransition from Aufbau to Syntax,especially regarding the nature and motivation ofCarnaps logicism. Moreover, we show how theAxiomatics influenced Carnaps student Gödel inreaching the fundamental logical results that soonafterwards undermined Carnaps project.

TOPOLOGY AND MODALITY: THE TOPOLOGICAL INTERPRETATION OF FIRST-ORDER MODAL LOGIC

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this paper the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 firstorder modal logic is complete with respect to such topological semantics. It has been known since the work of McKinsey and Tarski [?] that, by extending the Stone representation theorem for Boolean algebras, topological spaces provide semantics to propositional modal logic. Specifically, a necessity operator obeying the rules of the system S4 can be interpreted by the interior operation in a topological space. This result, however, is limited to propositional modal logic. The aim of this paper is to show how the topological interpretation can be extended in a very natural way to first-order modal logic. 1. Topological Semantics for Propositional Modal Logic Let us review the topological semantics for propositional S4. 1.1. The System S4 of Propositional Modal Logic. Modal logic is the study of logic in which the words “necessary” and “possible” appear in statements such as • It is necessary that the square of an integer is not negative. • It is possible that there are more than 8 planets. The history of modal logic is as old as that of the study of logic in general, and can be traced back to the time of Aristotle. The contemporary study of modal logic typically treats modal expressions as sentential operators, in the same way as ¬ is treated. That is, for each formula φ of propositional logic, the following are again A grateful acknowledgment goes to inspiring discussions with and helpful comments by Horacio Arlo-Costa, Nuel Belnap, Johan van Benthem, Mark Hinchliff, Paul Hovda, Ken Manders, Eric Pacuit, Rohit Parikh, Dana Scott, and especially Guram Bezhanishvili, Silvio Ghilardi and Rob Goldblatt as well as an anonymous referee for their accurate suggestions which improved Section ??. We also thank the organizers, Aldo Antonelli, Alasdair Urquhart, and Richard Zach, of the Banff Workshop “Mathematical Methods in Philosophy” for the opportunity to present this research. Philosophy Department, Carnegie Mellon University; awodey@cmu.edu. Philosophy Department, University of Pittsburgh; kok6@pitt.edu.

Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-first-Century Semantics

This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics so as to shed new light on the relevant strengths and limits of higher-order logic.

Natural models of homotopy type theory

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums, dependent products, and intensional identity types, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: they should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.

From Wittgenstein's prison to the boundless ocean : Carnap's dream of logical syntax

The Logical Syntax is a revolutionary book. How did the author of the Aufbau, whose viewpoint is so very different, come to write such a book? It was a drama in two acts, comprising not one but two major breakthroughs within less than two years. The first of these, in January 1931, was the one Carnap describes vividly in his autobiography: After thinking about these problems for several years, the whole theory of language structure and its possible applications in philosophy came to me like a vision during a sleepless night in January 1931, when I was ill. On the following day, still in bed with a fever, I wrote down my ideas on forty-four pages under the title ‘Attempt at a Metalogic’. These shorthand notes were the first version of my book Logical Syntax of Language.(Carnap 1963a, p. 53)

A cubical model of homotopy type theory

We construct an algebraic weak factorization system

Sheaf representation for topoi

(L, R)

Continuity and logical completeness: an application of sheaf theory and topoi

on the cartesian cubical sets, in which the canonical path object factorization