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.
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.
A proposal is due no later than Friday, March 14 (the Friday before Spring Break). The project itself is due Monday, April 28.
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.
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Last update: March 3, 2008. |