Carnegie Mellon University

Summer School in Logic and Formal Epistemology

There is a long tradition of fruitful interaction between philosophy and the sciences.

Logic and statistics emerged, historically, from combined philosophical and scientific inquiry into the nature of mathematical and scientific inference; and the modern conceptions of psychology, linguistics, and computer science are the results of sustained reflection on the nature of mind, language, and computation. In today's climate of disciplinary specialization, however, foundational reflection is becoming increasingly rare. As a result, developments in the sciences are often conceptually ill-founded, and philosophical debates often lack scientific substance.

The Department of Philosophy at Carnegie Mellon University holds a three-week summer school in logic and formal epistemology for promising undergraduates in philosophy, mathematics, computer science, linguistics, economics, and other sciences. The goals are to:

  • introduce promising students to cross-disciplinary fields of research at an early stage in their career; and
  • forge lasting links between the various disciplines.

The summer school is usually held the first half of June. There will be morning and afternoon lectures and daily problem sessions, as well as planned outings and social events.

The summer school is free. That is, we will provide:

  • full tuition
  • dormitory accommodations on the Carnegie Mellon campus

So students need only pay for round trip travel to Pittsburgh and living expenses while here. We expect to be able to accept about 25 students in 2018. There are no grades, and the courses do not provide formal course credit.

The summer school is open to undergraduates, as well as to students who will have just completed their first year of graduate school. Applicants need not be US citizens. There is a $30 nonrefundable application fee.

Information about applications for 2018 will be available in January.

2017 Summer School Topics

Week #1a
(June 5-9)

Instructor: Adam Bjorndahl

Title: Epistemic Logic and Topology 

Description: In this course we introduce epistemic logic and topology and explore the relationship between the two. No background in modal logic or topology is assumed.

We begin by motivating logics of knowledge and belief, and develop the formal tools that are typically used to study them; we then survey some classic results in this field. Next we turn to topology, building intuitions using a variety of metaphors and intuitions. In essence, by connecting the formal definition of a set's interior to the abstract concept of "robustness", we are able to co-opt the spatial notion of "nearness" as a means of representing uncertainty. To make this precise, we introduce topological semantics for the basic modal language, investigate its connection to the more standard relational semantics, and establish the foundational result that S4 is "the logic of space" (i.e., sound and complete with respect to the class of all topological spaces).

Viewing epistemology through the lens of topology highlights the distinction between the known and the knowable, between fact and measurement. To more fully incorporate this conceptual framework into our analysis, we introduce topological subset space semantics, which allows us to manipulate separately the state of the world and the epistemic state of the agent. We close with a look at some recent work that uses topology to improve our understanding of the dynamics of knowledge.

Week #1b
(June 9-10)

Center for Formal Epistemology Workshop

Title: Modality and Method

Franz Huber (University of Toronto)
Melissa Fusco (Columbia University)
Alexandru Baltag (Oxford)
Tamar Lando (Columbia)
Harvey Lederman (Pitt)
Dmitri Gallow (Pitt)

Week #2
(June 12-16)

Instructor: K.T. Kelly
Title: The Topology of Inquiry

Description: The standard mathematical frameworks for understanding reasoning are logic and computability for mathematical reasoning and probability theory for empirical reasoning. In this summer school session, we examine an alternative, topological viewpoint according to which computational and empirical undecidability can both be viewed as reflections of topological complexity. That may sound a bit odd, since topology is usually understood to be "rubber geometry", or the study geometrical relationships preserved under stretching operations that neither cut nor paste pieces together. In fact, topology is better understood as studying the mathematical structure of epistemic verifiability. Topological concepts and results will be applied to provide a unified, explanatory perspective on undecidability, on empirical underdetermination, on bounded rationality, and on the elusive connection between simplicity and empirical truth.

Background Reading:
1) "The Logic of Success"
British Journal for the Philosophy of Science, special millennium issue, 51, 2001, 639-666. 

2) Several overview papers are available on my web-site:
(with O. Schulte) "Church's Thesis and Hume's Problem," in Logic and Scientific Methods, M. L. Dalla Chiara, et al., eds. Dordrecht: Kluwer, 1997, pp. 383-398.

3) "Justification as Truth-finding Efficiency: How Ockham's Razor Works"Minds and Machines 14: 2004, pp. 485-505.

Week #3
(June 19 -23)

Instructor: Kevin Zollman
Title: The Economics of Science

Description: Science is a unique institution.  In most fields, people are rewarded for hard work with more money and promotions.  Scientists on the other hand are primarily paid in terms of credit for discoveries.  A scientists strives to be known as the person who discovered this thing or invented that theory.  What effect does this desire for credit have on the progress of science as a whole?  Is science benefited by this motivation or is it harmed?  In this course we will look at a number of mathematical and computer models of scientific behavior which strive to answer this question.