Philosophy 3340
Symbolic Logic II

The Project
Spring, 2008

This page contains information on the final project. This will most likely be a paper, but there are a variety of different approaches you could take to the paper, which could explore the philosophical implications of a metalogical result; explain and interpret a metalogical theorem other than the ones we are covering in class; or do additional research on a theorem we will cover, and give a different proof or an explanation from a different angle. I am also open to the idea of a "literate programming" project, integrating a program with a paper describing its motivations and significance.

Most of the suggestions below are in the "philosophical implications" category.

Format:

10-15 pages typed double-spaced (ten- or twelve-point font, one-inch margins, no really weird fonts, and preferably also not space-eating fonts like Courier New).

The paper should strive for the following virtues: clarity at every level (sentence, paragraph, section, paper as a whole); argumentative rigor.

The paper must be on a topic closely related to our class, and should make use of class readings in addition to outside research.

Due:

A proposal is due no later than Friday, March 14 (the Friday before Spring Break). The project itself is due Monday, April 28.

Possible Topics:

1. Does Gödel's incompleteness theorem show that Artificial Intelligence cannot match human reasoning abilities? You would want to read selections from Penrose, Shadows of the Mind, and an online discussion of this book by various philosophers with a reply by Penrose. Lots of other possible sources that I can provide.

2. Does the Löwenheim-Skolem theorem show that metaphysical realism is false, and that words in a natural language cannot refer to metaphysically real entities? You would want to read Putnam's essay "Models and Reality," and perhaps his less technical presentation of similar ideas in his book Reason, Truth, and History along with responses to Putnam's arguments, e.g. David Lewis's excellent "Putnam's Paradox" (Australasian Journal of Philosophy 62 (1984), pp 221-36). Again, lots of other possible sources; I can give a more complete list to anyone who's interested.

3. Logicians like to start proofs by saying things like "Let a be any arbitrary element of set A . . ." (sound familiar?). This is usually interpreted along the lines of boxing a constant for use in Universal Introduction in Barwise and Etchemendy's system (or flagging a constant in Klenk's). But Kit Fine has suggested that it would be better to simply accept the idea that there are strange objects that are "arbitrary" -- for instance an object that does have the property of being an element of A but does not have the property of being identical with any particular element of A. Might be interesting to explain and evaluate this idea. See Fine's book Reasoning with Arbitrary Objects (Oxford: Basil Blackwell, 1985). There is a shorter article that preceded the book: "Symposium: a defence of arbitrary objects: I," Proceedings of the Aristotelian Society Supplementary volume 57: 55-77 (1983).

4. We will discuss Turing Machines as an abstract model of computation. It is an interesting and surprisingly difficult issue exactly how these abstract models relate to the real world: when should we say that a physical process is an implementation of an abstractly defined computation? John Searle (in The Rediscovery of the Mind) and Hilary Putnam (in Representation and Reality) have suggested that any sufficiently complex physical system implements pretty much every abstractly defined computation. (Searle's famous example is that the wall behind him implements the Wordstar program.) David Chalmers has proposed an interesting account of implementation that seems to avoid this conclusion. (See the first two papers at http://www.u.arizona.edu/~chalmers/ai-papers.html, or the discussion in his book The Conscious Mind.) You could defend or criticize an account of implementation. (I have lots more references if you'd like them.)

5. Some possible sources of ideas for other topics:  (1) Torkel Franzen, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Very nice (and highly accessible) book which includes both good discussion of the theorem and a critique of a lot of loose and fuzzy (or just plain wrong) attempts to apply it to various areas. (2) John Etchemendy, The Concept of Logical Consequence. Etchemendy's a very clear writer, and this book raises interesting philosophical issues about the idea of logical consequence. (Yes, this is the Etchemendy from Barwise and Etchemendy.) (3) Mark Sainsbury, Logical Forms: An Introduction to Philosophical Logic. Mostly involves philosophical issues about the attempt to use logical machinery to understand natural language. (4) Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics: Selected Readings. Includes articles on the ontology of mathematical objects, the concept of set, and the idea of mathematical truth. 1983, so it's not exactly up to the minute. (5) Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. The giants in their own words: papers by Frege, Peano, Hilbert, Skolem, Gödel, and others.