Predicate Modal Logics and Non-standard Set Theories
A weekly seminar devoted to exploring the application of non-standard logics to philosophical problems.
January 16, 2009: Grigori Mints, Stanford University.
Review of Chapter 1 of Fixing Frege
Russell's paradox shows inconsistency of Frege's original system.This books surveys methods put forth for fixing Frege's system after it was discovered how much can be consistently done “almost” Frege's way. Chapter 1. Frege's system. Developing arithmetic. Cantor and Russel's paradox. Russel's solution: type theory. Interpretations. Reverse mathematics. Type theories vs Set Theories.
January 23, 2009: Solomon Feferman, Stanford University.
Review of Chapter 2 (“Predicative Theories”) of Fixing Frege
January 30, 2009: John Burgess, Princeton University.
Frege: Impredicative Fixes
Highlights of chapter 3 of Fixing Frege: “Frege's Theorem”, the “bad company objection”, Kit Fine's general theory of abstraction, and a sketch of Fregean and quasi-Fregean set theories.
February 6, 2009: Jesse Alama, Stanford University.
Part 1 (of 2) of a Discussion of Types, Tableaus, and Gödel's God
Melvin Fitting's Types, Tableaus, and Gödel's God aims to formalize Kurt Gödel's proof for the existence of God. Fitting develops a logical system needed to make Gödel's argument precise: a higher-order intensional and extensional modal logic. This presentation—part 1 of a two-part discussion of Fitting's book—will focus on that logical machinery. The presentation and evaluation of Gödel's argument (and related arguments for the existence of God), formally and informally, will take place in the meeting of the workshop after this one.
February 13, 2009: Eric Pacuit, Stanford University.
Part 2 (of 2) of a Discussion of Types, Tableaus, and Gödel's God
We will discuss part III of Fitting's book Types, Tableaus and Gödels's God. In particular, we will look at Fitting's formalization of Gödel's proof of the existence of God in the system of higher order modal logic developed in the text.
February 27, 2009: Wilfried Sieg, Carnegie Mellon University.
Uncovering Aspects of the Mathematical Mind
What is it that shapes mathematical arguments into proofs that are intelligible to us, and what is it that allows us to find proofs efficiently? — This is the informal question I intend to address by investigating, on the one hand, the abstract ways of the axiomatic method in modern mathematics and, on the other hand, the concrete ways of proof construction suggested by modern proof theory. These theoretical investigations are complemented by experimentation with the proof search algorithm AProS. It searches for natural deduction proofs in pure logic; it can be extended directly to cover elementary parts of set theory and to find abstract proofs of Gödel’s incompleteness theorems. The subtle interaction between understanding and reasoning, i.e., between introducing concepts and proving theorems, is crucial. It suggests principles for structuring proofs conceptually and brings out the dynamic role of leading ideas. Hilbert’s work weaves these strands into a fascinating intellectual fabric and connects, in novel and surprising ways, classical themes with deep contemporary problems. They reach from proof theory through computer science and cognitive psychology to the philosophy of mathematics and all the way back.