| Winter Term
2008 PHIL 650: Symbolic Logic
Lecture/seminar
University Hall, Room 353
TR 12:30-2:18 p.m.
Aims of this course
This course aims to provide a comprehensive coverage of the syntax and
semantics of first-order languages, and the positive results
concerning them. First-order languages contain the expressions "for
some x", "for every x" and "x is identical to
y", in addition to the connectives "not", "and", "or" and "if
... then ..." of propositional logic (which will have been studied in
PHIL 250: Introduction to Symbolic Logic).
Topics
We address various philosophical problems concerning reference,
definite descriptions, predication, identity and existence; and cover
the rudiments of informal set theory that are needed for a
rigorous discussion of syntactic and semantic matters. We give a
precise compositional grammar for the generation of well-formed
expressions (both terms and formulae) of first-order languages. We
give the famous Tarskian definition of truth (in a model) for sentences
of first-order languages as these are standardly interpreted, and the
definition of classical logical consequence that is based on that
definition of truth, classically conceived. (The Tarskian definition
of truth is important for an understanding of the work of influential
philosophers of language such as Davidson.) We also define the
important semantic concepts of isomorphism of models,
categoricity-in-power and elementary embedding.
We provide rules of inference for the connectives, quantifiers, and
the identity predicate. Combinations of these rules make up the
various well-known systems of natural deduction The classical
system of natural deduction is proved (1) sound, and (2)
complete, with respect to this notion of classical
consequence. That is, we show that (1) every classical proof preserves
classical truth from its premisses to its conclusion; and (2) every
argument that preserves classical truth from its premisses to its
conclusion can be given a proof in the system of classical natural
deduction. The method we use for proving completeness is that of
Henkin. It is economical and elegant, and affords also a proof of the
compactness theorem and of the countable models theorem,
whose philosophical consequences are important. Important
examples of first-order theories will be given, in fully
explicit logical form, which will be accessible even to the
non-mathematician. These theories are: Peano-Dedekind arithmetic;
various theories of orderings; the theory of Boolean algebras; the
theory of groups; the theory of real closed fields; and set theory.
Both the theory of descriptions and set theory
motivate the philosophically more sophisticated and technically
(slightly) more elaborate system of free logic, which we shall
treat in the necessary detail. Free logic provides a
satisfactory resolution of Russell's Paradox in a language in
which abstractive terms such as "the set of all Fs" are taken as genuine singular terms. If (and only if) time
permits: We shall prove Craig's interpolation theorem and
Beth's definability theorem. (These are important for
technically sophisticated discussion of the philosophical problems of
reductionism and supervenience in the special sciences.) We shall
establish the decidability of classical monadic first-order
logic. We shall isolate intuitionistic logic as a
subsystem of classical logic, and examine the relationship between the
two systems, as given by the Gentzen-Gödel-Glivenko
theorem. (This theorem is important for an understanding of how
the weaker subsystems of classical logic can suffice for the
hypothetico-deductive method in natural science.) Finally, we shall
cover the normalization theorems for these systems of
proof. (These theorems are of growing importance in the field of
automated deduction, where one needs to exploit the strongest
constraints that one can find on the search-space, in order to ensure
the most efficient proof-search possible.)
Textbook: Neil Tennant, Natural Logic, Edinburgh
University Press, 2nd edn., 1990. Second-hand copies might be
available at SBX. You can also
purchase a printout of a digitized photocopy, available from Copez at
Tuttle (near Oxley's coffee shop).
Useful supplementary reading (on reserve in the Main Library):
- G. Boolos and R. Jeffrey,
Computability and Logic, Cambridge University Press, 1974 [1989].
- J. Etchemendy, The Concept of Logical Consequence, Harvard
University Press, 1990.
- M. Dummett, Elements of Intuitionism, Clarendon Press,
Oxford, 1977.
- J. Bell and A. Slomson, Models and Ultraproducts: an introduction,
North-Holland, Amsterdam, 1974.
- D. Prawitz, Natural Deduction: A Proof-Theoretical Study,
Almqvist and Wiksell, Stockholm, 1965.
Handouts (downloadable Word files or .pdf files):
Handouts in Winter 2008
|
  
|
NEIL TENNANT