Topos logic arises naturally

[Till Mossakowski <till@Informatik.Uni-Bremen.DE>]

You see: even with just aesthetic and software engineering
considerations you are eventually lead to topos logic...

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

Message-Id: <200305270616.h4R6G3w8008134@plxc2089.pdx.intel.com>
To: Randy Pollack <rap@inf.ed.ac.uk>
cc: John Harrison <johnh@ichips.intel.com>, isabelle-users@cl.cam.ac.uk
Subject: Re: HOL without description operators
Date: Mon, 26 May 2003 23:16:03 -0700
From: John R Harrison <johnh@ichips.intel.com>
Sender: Larry Paulson <Larry.Paulson@cl.cam.ac.uk>

Hi Randy,

| Perhaps more usefully, how do you suggest I do this task of developing HOL
| without description operators.

You might find it interesting to look at HOL Light. This gives a
different axiomatization of the HOL logic, developed precisely because I
was dissatisfied with the one in the original HOL system, on which I
believe the Isabelle/HOL logic is based.

Although I do eventually introduce the description operator, quite a lot
of the basic logic --- including the automation of inductive definitions
--- is developed without it (and indeed without excluded middle or
extensionality). You may find it a more congenial starting-point.

As Konrad Slind later pointed out, what I settled on is remarkably close
to the internal logic of a topos, as presented for example in Lambek and
Scott's book. This was a surprise to me since at that time I knew next to
nothing about toposes and was motivated by a mixture of aesthetic and
software engineering considerations.

Cheers,

John.