Proof Theoretic Foundations of Linguistic Structure


LING 548
Fall Term, 2005




Gerhard Gentzen


This course covers the fundamentals of proof theory and logic as they apply to linguistics. The notion of a well-formed derivation is fundamental to all flavors of formal linguistics and all sub-disciplines of linguistics. These foundations rest, ultimately, on axiomatic systems developed by logicians to encode the process of valid formal reasoning. We will place a particular emphasis on constructive methods and, where appropriate, develop connections with parsing theory, automatic theorem proving and computational semantics. We will consider some introductory topics in substructural logic, systems that encode some proper sub-part of first order logic. We will place particular emphasis on the Lambek Calculus which is a system for reasoning about the categories in a Categorial Grammar. We will, therefore, cover some of the basics of Categorial Grammar, type theory and the &lambda calculus.

The course is intended as a preparation for Linguistics 553 (Formal Semantics I). It includes a review of the propositional and predicate calculus before introducing tableaux and resolution systems, unification, axiomatic systems, natural deduction and sequent calculi. The latter two systems are particularly relevant for grammar formalisms like phrase structure grammars, TAGs and Categorial Grammar.

This year we will be doing things a bit differently. There will be no required textbook for the course, although I still recommend that students consult the excellent:
John Barwise and John Etchemendy (2002). Language, Proof and Logic. CSLI Publications, Stanford, CA.

Instead, I will develop the material on the board in class and make notes available. Thus, the course will have something of the flavor of a workshop. This makes attendance crucial.

Another book that I will refer to frequently is:
Marcus Tomalin (2006). Linguistics aqnd the Formal Sciences: The Origins of Generative Grammar. Cambridge University Press, Cambridge, UK.
This book places the development of generative grammar in the context of the development of logic and provides a lot of context to the philosophical and formal foundations of linguistic theory.

Grading will be based on homeworks. I will endeavor to give frequent (relatively small) homeworks. No late homeworks!

Topics
I hope to cover the following topics although the probability of coverage decreases as we go down the list.
  1. Set Theoretic Foundations
  2. The Propositional Calculus and Truth Functional Connectives
  3. The Predicate Calculus and Quantifiers.
  4. Game Semantics
  5. Tableaux Systems
  6. Natural Deduction
  7. Sequent Calculi
  8. The Lambek Calculus
  9. Elements of Substructural Logic
The above list does not correspond to the order of the course. The following are links to my lecture notes. Be advised that these are under constant revision. Check often.

Friends of game theoretic semantics will want to read Abelard's Historia Calamitatum for a first person account of his affair with Heloise and its distressing anatomical consequences!

Homework 1
Due: September 19, 2007

Homework 2
Due: September 26, 2007

Homework 3
Due: October 3, 2007

Homework 4
Due: October 17, 2007

Homework 5
Due: October 31, 2007

Homework 6
Due: November 19, 2007

Homework 7
Due: November 28, 2007

Homework 8
Due: December 7, 2007

[Linguistics Department] [Robin Clark's Homepage]