1st Semester 2009/10: Philosophical Aspects of Proof Theory

Instructors

Theodora Achourioti.
If you are interested in this project, please contact the instructor by e-mail.

ECTS

6

Description

This project is an introduction to some of the main philosophical ideas surrounding proof theory. Among the themes to be covered are the following: the origins of proof-theory in the foundations of mathematics, proof-theoretic semantics as a species of verificationist theories of meaning, harmony, the rejection of classical logic on proof-theoretic grounds, structural proof theory, what proofs look like after normalization or cut-elimination. The project is primarily of philosophical interest and technically non-trivial. For technical precision and a thorough understanding of the main notions we will refer to the textbooks below. A selection of exercises will be part of the preparation, and they will be discussed in class, but they will not form part of the grading. The project will be more like a seminar than a series of lectures.

Prerequisites

Basic Logic or equivalent course.

Assessment

Assessment will be based on:

1. preparation and participation in the discussions.
2. two presentations of papers in class.
3. submission of a short final essay based on an interesting question developed by the student throughout the course.

Working in groups of two is an option.

References

• M. Dummett. The Philosophical Basis of Intuitionistic Logic. In Truth and Other Enigmas, pages 215-247, 1978.
• D. Hilbert. On the Infinite. In J. van Heijenoort, editor, From Frege to Gödel: A Scource Book in Mathematical Logic, pages 1879-1931. Harvard University Press, 1967.
• D. Prawitz. Meaning and Proofs: on the Conflict between Classical and Intuitionistic Logic. Theoria, 43:2-40, 1977.
• D. Prawitz. Some Remarks on Verificationistic Theories of Meaning. Synthese, 73:471-477, 1987.
• D. Prawitz. Logical Consequence from a Constructivist Point of View. In S. Shapiro, editor, The Oxford Handbook of Philosophy of Mathematics and Logic, pages 671-695. Oxford University Press, 2005.
• P. Schroeder-Heister. On the Notion of Assumption in Logical Systems. In R. Bluhm & C. Nimtz, editors, Selected Papers Contributed to the Sections of GAP 5, pages 27-48, Paderborn: mentis 2004.
• P. Schroeder-Heister. Validity Concepts in Proof-Theoretic Semantics. Synthese 148:525-571, 2006.

Textbooks:

• D. van Dalen. Logic and Structure. Springer Verlag, Berlin, fourth edition, 2004.
• A. S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge University Press, second edition, 2000.