From: "Elwood Wilkins" <elwood@essex.ac.uk>
To: "Categories" <categories@mta.ca>
Subject: Re: Topos logic arises naturally
Date: Fri, 30 May 2003 12:35:06 +0100 [thread overview]
Message-ID: <002801c326a1$05dc0000$f112fea9@essex.ac.uk> (raw)
In-Reply-To: <3ED34194.9050005@informatik.uni-bremen.de>
Isabelle's internal logic is not constructive. The existential clause of the
BHK-interpretation is violated so that a consquence of "all unicorns have
horns" is that "some unicorn has a horn". The moral (biased towards
theorists): software engineering considerations are not enough, a coherant
philosophy of mathematics is also needed.
Elwood
----- Original Message -----
From: Till Mossakowski <till@Informatik.Uni-Bremen.DE>
To: Categories <categories@mta.ca>
Sent: Tuesday, May 27, 2003 11:44 AM
Subject: categories: Topos logic arises naturally
> 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.
>
>
>
>
next prev parent reply other threads:[~2003-05-30 11:35 UTC|newest]
Thread overview: 8+ messages / expand[flat|nested] mbox.gz Atom feed top
2003-05-27 10:44 Till Mossakowski
2003-05-30 11:35 ` Elwood Wilkins [this message]
2003-05-30 14:37 ` Till Mossakowski
2003-06-01 16:05 ` Toby Bartels
2003-06-02 18:11 Valeria.dePaiva
2003-06-03 9:24 Lawrence Paulson
2003-06-03 16:04 ` Todd Wilson
2003-06-04 9:25 ` Lawrence Paulson
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