# FOM: the categorical approach to logic

Stephen G Simpson simpson at math.psu.edu
Wed Apr 1 10:39:56 EST 1998

I thank Walter Felscher for his careful critique of the categorical
approach to universal algebra (four postings, March 20).  Now I want
to begin a critical discussion of the categorical approach to logic.
I'm hoping that the experts such as Makkai will chime in with their

Felscher writes:
> It appears astonishing to me that Mr Simpson ... chose the examples
> he did, because if anything, it is the situation of Boolean
> algebras (and propositional connectives) in which the advantages of
> the functorial approach to algebra may become convincing.

Why did I chose the example of Boolean algebras?  Well, let's recall
the context.  The context was my point that algebra is not the same as
logic, and that the algebraic or categorical approach to logic omits a
lot of important information that is relevant to f.o.m.  This point
came up because some category theorists wrongly claimed that topos
theory is "the same as" IHOL (=intuitionistic higher order logic).
They are not the same, because topos theory has a different signature;
it submerges the carefully chosen primitives of IHOL into a sea of
unanalyzed primitives, most of which make no sense from the viewpoint
of IHOL and f.o.m. generally.  See also my posting of 3 Feb 1998
14:24:10.

Unfortunately, the distinction between IHOL and topos theory was not a
good vehicle for me, because not all FOM subscribers are familiar with
IHOL and topos theory.  Therefore, in order to argue my point, I chose
the simpler and more familiar example of Boolean rings, because the
underlying logic and algebra (propositional connectives, Boolean
algebras, Boolean rings) are very well understood by everyone yet

Much to my surprise, even with this simple example, I ran into
difficulty because one of the category theorists denied the underlying
algebraic facts, e.g. the well known fact that Boolean algebras and
Boolean rings are not "the same", because they have different
signatures.  Perhaps we should learn to expect this kind of difficulty
whenever we attempt to communicate with category theorists.
Nevertheless, I still hold out hope that the attempt to communicate
with them will be worth while.

In any case, the outcome was that the discussion of the categorical
approach to logic bogged down in trivialities, and many people tuned
out.  I'm now hoping to reincarnate that discussion at a higher level.

Felscher's last posting of March 20 concludes with

> The inherent limitations of the functorial approach are that
> it captures aspects only which are expressible by equations
> (and here by no means all of them); hence it can never
> replace the traditional approach in the case of more general
> classes.
>
> Yet there is a further limitation, caused by our categorical
> brethren themselves: many of them do not even wish to know the
> methods of traditional universal algebra. ...  It seems that for
> many categorists the world only began when it were their brethren
> who started to name the concepts they employ ...

and I concur.

-- Steve