Tuesday,  5 March 2002
   2:30 - 4:00  Mihaly Makkai

	On syntax: an exhortation

Abstract: The recent discussion on Syntax on the Categories list exhibited
a surprising disregard for the achievements of Mathematical Logic in the
subject of Syntax, and thereby for the only existing system of thought
that can be called a science of Syntax. (This statement is subject to
possible errors insofar I may have missed some relevant contributions to
the discussion.) This disregard is particularly relevant in the context of
Categorical Logic.

My papers "Generalized sketches as a framework for completeness theorems",
JPAA 115 (1997), pp. 49-79, 179-221 and 241-274, are intended as a
principled approach to the syntax of Categorical Logic in general, one
that does take Mathematical Logic into account (I am unaware of other
relevant work that would address the same issues).

In the talk, I will discuss, among others, why the concern with
completeness theorems is natural. As a particular instance, the General
Completeness Theorem of the work is used to give a uniform specification
of the syntax of well-defined programs in any one of the programming
languages based on categorical doctrines such as (1) the product/coproduct
doctrine (D-sketches), (2) the finite-limit doctrine, (3) the
Cartesian-closed doctrine, and many possible others. Other known
specifications in these areas are of interest, but they are unsatisfactory
insofar they do not remain in the original categorical framework, but
translate it into a traditionalist linguistic framework in an ad hoc,
case-by-case, manner.