Carnegie Mellon University

Ph.D. Program in Pure and Applied Logic

The Ph.D. Program in Pure and Applied Logic is an interdisciplinary program designed to support students seeking a career in Mathematics, but interested in working in an area of logic supported by the Department of Philosophy. As part of the program, students earn an M.S. degree from the Department of Mathematical Sciences.

The program's course requirements are designed to provide students with a shared introduction to basic tools of philosophical analysis, a shared background of philosophical issues, significant interdisciplinary competence and an introduction to research topics in the department.

Core Philosophy requirements (4 courses)

  • 80-600 Philosophy Core Seminar: Survey of crucial research in philosophy, logic, and related areas
  • 80-601 Philosophy Core Seminar II: Continued survey of crucial research in philosophy, logic, and related areas
  • 80-616 Formal Methods (1.5 semester): An introduction to contemporary formal frameworks, including Bayes Nets, Decision Theory, Game Theory, and Formal Learning Theory
  • 80-618 Topics in Logic II (half semester): The theory of computability, and Gödel's incompleteness theorems
  • Professional development seminar: Students must enroll in the professional development seminar in the spring semester of each of their first three years

Mathematics requirements (8 courses, sufficient to earn an MS in Mathematical Sciences)

  • One course in algebra (such as 21-610 Algebra I, 21-611 Topics in Algebra, or 80-713 Category Theory)
  • One course in topology (such as 21-651 General Topology)
  • One course in analysis (such as 21-720 Measure and Integration or 21-721 Probability)
  • Three courses in logic (such as 21-602 Set Theory, 21-603 Model Theory, 21-604 Recursion Theory, or 80-711 Proof Theory)
  • Two electives in mathematics or computer science

At most two of the eight courses can be taken outside of Mathematics. Students should contact the Director of Graduate Studies of the Mathematics Department to determine if courses taught outside of Mathematics will satisfy this requirement.

Breadth (2 courses)

  • One course in the analytic tradition (such as 80-605 Rational Choice, 80-612 Philosophy of Mathematics, or  80-680 Philosophy of Language)
  • One course in the history of philosophy (such as 80-254 Analytic Philosophy, or 80-255 Pragmatism)

Electives (4 courses)

  • Four unconstrained electives (including directed reading and dissertation research)

These requirements can be filled in three years by taking three courses each semester. Coursework must be completed by the end of the fourth year, at the latest. Electives should be chosen in consultation with the Director of Graduate Studies and the student's advisor, to ensure that the courses chosen will support the student's career goals.

Research areas include:

  • automated deduction and automated reasoning
  • category theory and categorical logic
  • computability and computable analysis
  • constructive logic and type theories
  • homotopy type theory
  • proof theory

Students are advised to do supervised reading and research with a member of the faculty in the spring of their first year, to explore possible research topics for the MS thesis. In the spring of the second year. The MS thesis itself should be completed by the end of the second year.

In the third year, students choose a dissertation topic and committee. Students must present and defend a prospectus by the end of the fourth year, at the latest, and are expected to complete their doctoral dissertation by the end of the fifth year.

Teaching experience is an essential component of the Ph.D. program. It provides fundamental professional skills for those pursuing academic careers and broadens the focused expertise of pure researchers. Students are required to serve as an assistant or instructor each semester, with the following exceptions: students are exempt from teaching one semester in their first year, and one semester later on, while they are carrying out research on their thesis.

Students serving as principal instructors are mentored by a faculty member. Each foreign student whose native language is not English must take a proficiency exam with the ESL Center prior to teaching.

The department's interdisciplinary research thrust affords an unusually broad range of career possibilities. Graduates of the program have been offered positions in Philosophy, Mathematics, Psychology, Computer Science, and Statistics, as well as research positions in industry. This wide range of interesting career opportunities reflects the department's unique dedication to serious, interdisciplinary research ties.

For a complete listing of our graduates and placement record, see our Ph.D. alumni page.