HDTenEightyP/NetNews huggingface dataset:
From: loader@vax.oxford.ac.uk
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 13 Apr 92 18:48:58 GMT
Organization: Oxford University VAXcluster

In article <1992Apr5.213320.27618@morrow.stanford.edu>,
BR.MJC@forsythe.stanford.edu (Michael Carroll) writes:
  (Edited Highlights:)
> I am not aware of any attempt to construct a set theory with modal
> logic as its underlying logic.  I'd appreciate hearing about any
> research in this area.
> 
> Following are some speculations as to why this line of research
> might prove interesting.  Readers not interested in speculation are
> invited to hit <next> or <delete> now.
> 
> So closure is not definable absolutely.  To remedy this, I adopt as
> my underlying logic S4 or S5.  In these systems, as is well known
> (e.g. Sikorski, "Boolean Algebras", p.199), the possibility
> operator is a Kuratowski closure operator.  So in my set theory
> whose logic is S4 or S5 I may define: *the* closure of x is the set
> of all y such that it's *possible* that y is a member of x.
>
> So it would seem that using a modal logic of a certain sort as our
> underlying logic might permit us to follow Shoenfield's program of
> "introducing new operations on sets" in order to deal with
> "problems that cannot be settled from the axioms of ZFC."
> 
> No?
> --
> Michael Carroll                br.mjc@forsythe.stanford.edu
> Research Libraries Group
> 4/5/92
My first posting on this was terribly ambiguous - I was trying to be just too
concise - and seems to have caused lots of confusion, so I'll say it all
properly this time.
Define a "modal" logic Soo by adding the axiom A iff \box A to classical logic.
It's now really easy [exercise folks :-)] to see that:
(i) If B is a theorem of S5 (or S4, or any common modal logic), then B is
   a theorem of Soo.
(ii) Let C be any formula, and C' be C with all \box removed. Then:
   C is a theorem of Soo if and only if C' is a theorem of classical logic

Now, define theories ZFC(Sn) in the obvious way: ie the axioms of Sn plus
the axioms of ZFC (with comprehension / separation allowed with formulae
containing \box). Then easily (i), (ii) hold for these systems also:
(i) If  ZFC(S5) /- B  then  ZFC(Soo) /- B
(ii) ZFC(Soo) /- C  iff  ZFC /- C'

In particular, if D is a statement of classical set theory, then ZFC(S5) /- D
if and only if ZFC(S5) /- D.

Thus it seems that simply formulating set theory in a modal logic doesn't
introduce any new theorems that don't involve modal operators: Eg we can't
solve the continuum hypothesis in this way.

On the other hand, it may be that adding new axioms stating how modal operators
interact with sets might give us theories that prove things like the continuum
hypothesis. But I'll leave that to real set theorists.
--
R Loader, Oxford.
From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 12 Apr 92 23:33:30 GMT
Article-I.D.: husc3.1992Apr12.193331.11012
Organization: Dept. of Math, Harvard Univ.

In article <1992Apr11.130906.796@gacvx2.gac.edu> 
logic@gacvx2.gac.edu writes:

>In article <1992Apr10.161709.5397@vax.oxford.ac.uk>, 
>loader@vax.oxford.ac.uk (R Loader) writes:

>>In article <1992Apr8.100146.689@gacvx2.gac.edu>, 
>>logic@gacvx2.gac.edu writes:

>>>In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, 
>>>loader@vax.oxford.ac.uk (R Loader) writes:

RL:
>>>> S4 and S5 (and most other common modal logics), are subsystems of classical
>>>> (non-modal) logic extended with the axiom: A iff \box A.

>>>> [...]

>>>> Ralph Loader,
>>>> Mathematical Institute, Oxford.

AH:
>>> S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
>>> contain the axiom "A iff \box A", both contain the axiom "if \box A then A".
>>> 
>>> Al Hart

RL:
>> Sorry, My original posting was ambiguous: I meant ". . . are subsystems
>>   of the logic obtained by adding A iff \box A to classical logic"
>> --
>> Ralph Loader, Oxford.

This is exactly the way I originally interpreted your statement.

AH:
>I see the ambiguity now - missed it during my first post..  however, not having
>extracted your first post, i'm not sure i see the point.  extending classical
>logic with the axiom A iff \box A obviously makes \box completely superfluous.
>
>confused,
>al hart

Observe that Ralph says that S4 and S5 are *subsystems* of classical logic
extended with the axiom: A iff \box A. In that, he is absolutely correct: a
subsystem needn't contain the axiom in question, but only a deductively
closed proper subset of its consequences, e.g. those generated by \box A
\supset A in the presence of classical laws.  However, I am not sure
whether he could do without an additional rule of inference, viz. the Rule
of Necessitation:

		A
	       ___

	      \box A

which prima facie seems to be required for adequately characterizing S4 and
S5, even if all of their theorems can probably be obtained by the above
process.


`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'
: Qu'est-ce qui est bien?  Qu'est-ce qui est laid?         Harvard   :
: Qu'est-ce qui est grand, fort, faible...                 doesn't   :
: Connais pas! Connais pas!                                 think    :
:                                                             so     :
: Mikhail Zeleny                                                     :
: 872 Massachusetts Ave., Apt. 707                                   :
: Cambridge, Massachusetts 02139           (617) 661-8151            :
: email zeleny@zariski.harvard.edu or zeleny@HUMA1.BITNET            :
:                                                                    :
'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`
From: suchenek@nic.csu.net
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 9 Apr 92 00:10:08 PDT
Organization: Information Resources and Technology

In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
> In article <1992Apr5.213320.27618@morrow.stanford.edu>,
> BR.MJC@forsythe.stanford.edu (Michael Carroll) writes:
>   (Edited Highlights:)

[irrelevant stuff deleted]

> S4 and S5 (and most other common modal logics), are subsystems of classical
> (non-modal) logic extended with the axiom: A iff \box A.

Not true. S5 contains only \box A => A, and the rule of inference:

                      |- A
                      ---------
                      |- \box A

which is strictly weaker then an axiom A => \box A (i.e. it may happen that A
is true while \box A is not).


> Call this latter theory Soo. Obviously a theorem of set theory whose logic
> is Soo is just a theorem of usual set theory with a few \box inserted.

This is true under the above false assumption.

> Any modal logic which gives a set theory which is not extensional over usual
> set theory thus cannot be a subsystem of Soo.

Any prove of this?

> IMHO, extending the underlying logic of set theory is unlikely to give
> interesting set theorys. 

There is a lot of interesting results in the so called autoepistemic logic and
its relationships to set theoretic forcing by Marek and Truszczynski (see
Journal of the ACM vol 38 No 3 (June 1991) for details.)

Marek Suchenek
Cal State Dominguez Hills
From: loader@vax.oxford.ac.uk (R Loader)
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 7 Apr 92 11:49:48 GMT
Organization: Oxford University VAXcluster

In article <1992Apr5.213320.27618@morrow.stanford.edu>,
BR.MJC@forsythe.stanford.edu (Michael Carroll) writes:
  (Edited Highlights:)
> I am not aware of any attempt to construct a set theory with modal
> logic as its underlying logic.  I'd appreciate hearing about any
> research in this area.
> 
> Following are some speculations as to why this line of research
> might prove interesting.  Readers not interested in speculation are
> invited to hit <next> or <delete> now.
> 
> So closure is not definable absolutely.  To remedy this, I adopt as
> my underlying logic S4 or S5.  In these systems, as is well known
> (e.g. Sikorski, "Boolean Algebras", p.199), the possibility
> operator is a Kuratowski closure operator.  So in my set theory
> whose logic is S4 or S5 I may define: *the* closure of x is the set
> of all y such that it's *possible* that y is a member of x.
>
> So it would seem that using a modal logic of a certain sort as our
> underlying logic might permit us to follow Shoenfield's program of
> "introducing new operations on sets" in order to deal with
> "problems that cannot be settled from the axioms of ZFC."
> 
> No?
> --
> Michael Carroll                br.mjc@forsythe.stanford.edu
> Research Libraries Group
> 4/5/92
S4 and S5 (and most other common modal logics), are subsystems of classical
(non-modal) logic extended with the axiom: A iff \box A.
Call this latter theory Soo. Obviously a theorem of set theory whose logic
is Soo is just a theorem of usual set theory with a few \box inserted.
Any modal logic which gives a set theory which is not extensional over usual
set theory thus cannot be a subsystem of Soo.
IMHO, extending the underlying logic of set theory is unlikely to give
interesting set theorys. On the other hand, perhaps one could introduce new set
theoretic principles involving modal operators?
--
Ralph Loader,
Mathematical Institute, Oxford.
Newsgroups: sci.logic
From: BR.MJC@forsythe.stanford.edu (Michael Carroll)
Subject: Re: Set theory and modal logic
Organization: Stanford University, California, USA
Distribution: usa
Date: Fri, 10 Apr 1992 01:37:06 GMT

In article <1992Apr7.124948.5325@vax.oxford.ac.uk>,
loader@vax.oxford.ac.uk (R Loader) writes:
>IMHO, extending the underlying logic of set theory is unlikely to give
>interesting set theorys. On the other hand, perhaps one could introduce new set
>theoretic principles involving modal operators?

If one extends the underlying logic of set theory by adding the S4 modal
axioms, one obtains theorems such as:
    If necessarily (x is in y) then (x is in y);
    If (x is in y) then possibly (x is in y);
    If possibly possibly (x is in y) then possibly (x is in y);
and so on. If one adds the definitions
    The interior of x is the set of all y such that necessarily (y is in x);
    The closure of x is the set of all y such that possibly (y is in x);
then in virtue of the theorems it follows that
    x is a subset of the closure of x;
    The interior of x is a subset of x;
    The closure of the closure of x equals the closure of x;
and so forth.  That is, interior and closure defined in this way will have
many (all?) of the usual properties of interior and closure as defined in
topology. (And, in general, the interior and closure of x will be distinct
from each other and from x itself. We will have "more" sets than before,
I think, which might impact the independence proofs.)

It seems to me that this *does* introduce new set theoretic principles
involving modal operators, and does so by changing the underlying logic. So
I fail to grasp the contrast between the two approaches that R. Loader
describes.

I think the real problem lies elsewhere: it's not clear to me what we would
mean by *the* interior of a set. It would seem that there would have to be
some one topology that grew naturally out of set theory.  If there is one,
I don't know what it is.  Do you?

--
Michael Carroll             br.mjc@forsythe.stanford.edu
Research Libraries Group
From: logic@gacvx2.gac.edu
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 8 Apr 92 16:01:45 GMT
Article-I.D.: gacvx2.1992Apr8.100146.689
Organization: Gustavus Adolphus College, St. Peter, Minnesota

In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:

> S4 and S5 (and most other common modal logics), are subsystems of classical
> (non-modal) logic extended with the axiom: A iff \box A.

Other stuff deleted....

> Ralph Loader,
> Mathematical Institute, Oxford.

S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
contain the axiom "A iff \box A", both contain the axiom "if \box A then A".

Al Hart
From: rose@fsu1.cc.fsu.edu (Kermit Rose)
Newsgroups: sci.logic
Subject: power rings
Date: 2 Apr 92 01:52:06 GMT
Article-I.D.: mailer.1992Apr1.205206.4013
Reply-To: rose@fsu1.cc.fsu.edu
Organization: Florida State University
News-Software: VAX/VMS VNEWS 1.3-4

0^x is not well defined.  This is true, but is no problem for discussion of 
power rings.  
Define ordinary multiplication to be the "additive" operation.  Then the 
"zero" of this ring is 1, not 0.

0 can be eliminated as an element of the algebra, and the isomorphism will 
still hold. 
Kermit@fsu


This is the signature file of Kermit Rose.

Believing in dichotomies leads to hardening of your logic.

What determines the existence of Free Will?

Peace is the way to peace.  Hostility is the way to hostility.
From: versmiss@let.ruu.nl (Koen Versmissen)
Newsgroups: sci.logic
Subject: Interpreting modalities as subalgebras
Keywords: modal logic semantics categorial grammar
Date: 8 Apr 92 09:27:52 GMT
Organization: Faculty of Arts, Utrecht University, The Netherlands

In categorial grammar, the analogue of the S4 universal modality has
been introduced by several authors for linguistic reasons. 
Categorial grammars have a semantics in terms of semigroups. The
proposed interpretation for the box is in terms of a subalgebra: fix a
subalgebra S of the semigroup. Then, the interpretation of \Box A is the
intersection of S with the interpretation of A. This semantics is
said to be 'standard' in modal logic, but I haven't been able to find
this in the literature. Does anybody know a suitable reference? 

Thanks,
Koen.

-- 
Koen Versmissen, OTS, RU Utrecht, Trans 10, 3512 JK  Utrecht, NL
E-mail: Koen.Versmissen@let.ruu.nl...............Tel: 030-392057
__
"Listen, in order to maintain airspeed velocity, a swallow needs to flap
Newsgroups: sci.logic
From: hbe@sonia.math.ucla.edu (H. Enderton)
Subject: UCLA Logic Colloquium, April 17
Organization: UCLA Mathematics Department
Distribution: ca
Date: Thu, 9 Apr 92 15:54:12 GMT

                    UCLA LOGIC COLLOQUIUM
                    Friday, April 17, 1992
                           4:00 p.m.
                   Mathematical Sciences 6627
                             UCLA
                                                  
                       ALEXANDRA SOSKOVA
                      (Sofia University)

               "EFFECTIVE FIRST ORDER STRUCTURES"

		           Abstract

In our talk we shall discuss the notion of abstract data type from a
recursion theoretical point of view.  Our results are mainly
concentrated on the definition and on the characterization of the
effective abstract data types.

The main tool is a notion of computability over many-sorted first order
structure which puts together two non-deterministic notions of
computability -- search computability of Moschovakis and computability
of Friedman and Shepherdson.

The relationships between some special properties of the class of the
computable functions in an abstract structure and the existence of some
special enumerations of it will be presented.
From: suchenek@nic.csu.net
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 13 Apr 92 11:48:24 PDT
Organization: Information Resources and Technology

In article <1992Apr12.193331.11012@husc3.harvard.edu>, zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
[...]
> 
> Observe that Ralph says that S4 and S5 are *subsystems* of classical logic
> extended with the axiom: A iff \box A. In that, he is absolutely correct: a
> subsystem needn't contain the axiom in question, but only a deductively
> closed proper subset of its consequences, e.g. those generated by \box A
> \supset A in the presence of classical laws.  However, I am not sure
> whether he could do without an additional rule of inference, viz. the Rule
> of Necessitation:
> 
> 		A
> 	       ___
> 
> 	      \box A
> 
> which prima facie seems to be required for adequately characterizing S4 and
> S5, even if all of their theorems can probably be obtained by the above
> process.
> 
How, without the necessitation rule (and scheme K), can you get 

     p \supset \box (\box (p \supset q) \supset (\box p \supset \box q))

(which is a theorem of S4 and S5), without proving too much, e.g.

      p \supset \box p

(which is not a theorem of S4 and S5), where p is a propositional variable?

Greetings,

Marek Suchenek
Newsgroups: sci.logic
From: rar@csl.sri.com (Bob Riemenschneider)
Subject: Re: modal set theory
Organization: Computer Science Lab, SRI International, Menlo Park, CA.
Date: Mon, 13 Apr 1992 23:56:09 GMT

In article <1992Apr13.192451.26064@dcs.glasgow.ac.uk> jack@dcs.glasgow.ac.uk (Jack Campin) writes:

   Daniel Gallin's thesis (published as a book around 1976, I forget where) ...

It's Volume 19 of North-Holland's "Mathematical Studies" series, published
in 1975.

								-- rar
From: scharle@lukasiewicz.cc.nd.edu (Scharle)
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 8 Apr 92 17:26:14 GMT
Article-I.D.: news.1992Apr8.172614.27294
Reply-To: scharle@lukasiewicz.cc.nd.edu (Scharle)
Organization: Univ. of Notre Dame
Newsgroups: sci.logic
Distribution: world
Followup-To: 
From: scharle@lukasiewicz.cc.nd.edu (Scharle)
Reply-To: scharle@lukasiewicz.cc.nd.edu (Scharle)
Organization: Univ. of Notre Dame
Subject: Re: Set theory and modal logic
Keywords: 

In article <1992Apr8.100146.689@gacvx2.gac.edu>, logic@gacvx2.gac.edu writes:
|> In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
|> 
|> > S4 and S5 (and most other common modal logics), are subsystems of classical
|> > (non-modal) logic extended with the axiom: A iff \box A.
|> 
|> Other stuff deleted....
|> 
|> > Ralph Loader,
|> > Mathematical Institute, Oxford.
|> 
|> S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
|> contain the axiom "A iff \box A", both contain the axiom "if \box A then A".
|> 
|> Al Hart
    There is a problem with the terms "supersystem" and "subsystem".  
S4 and S5 are both subsystems and supersystems of classical logic, 
depending on the map one uses.
    Every statement true in classical logic is true in S4 or S5.
Hence classical logic is a subsystem.
    Every statement true in S4 or S5 is true in classical logic,
provided we interpret the box as the identity operator.  That, I
believe, is what R. Loader was saying.  Hence classical logic is a
supersystem.

-- 
Tom Scharle                |cm65n6@irishmvs(Bitnet)
Room G003 Computing Center |scharle@lukasiewicz.cc.nd.edu(Internet)
University of Notre Dame  Notre Dame, IN 46556-0539 USA



-- 
Tom Scharle                |cm65n6@irishmvs(Bitnet)
Room G003 Computing Center |scharle@lukasiewicz.cc.nd.edu(Internet)
University of Notre Dame  Notre Dame, IN 46556-0539 USA
From: logic@gacvx2.gac.edu
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 11 Apr 92 19:09:06 GMT
Organization: Gustavus Adolphus College, St. Peter, Minnesota

In article <1992Apr10.161709.5397@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
> In article <1992Apr8.100146.689@gacvx2.gac.edu>, logic@gacvx2.gac.edu writes:
>> In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
>> 
>>> S4 and S5 (and most other common modal logics), are subsystems of classical
>>> (non-modal) logic extended with the axiom: A iff \box A.
>> 
>> Other stuff deleted....
>> 
>>> Ralph Loader,
>>> Mathematical Institute, Oxford.
>> 
>> S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
>> contain the axiom "A iff \box A", both contain the axiom "if \box A then A".
>> 
>> Al Hart
> 
> Sorry, My original posting was ambiguous: I meant ". . . are subsystems
>   of the logic obtained by adding A iff \box A to classical logic"
> --
> Ralph Loader, Oxford.

I see the ambiguity now - missed it during my first post..  however, not having
extracted your first post, i'm not sure i see the point.  extending classical
logic with the axiom A iff \box A obviously makes \box completely superfluous.

confused,
al hart
Newsgroups: sci.logic
From: guttman@linus.mitre.org (Joshua D. Guttman)
Subject: Relevance logics 
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Date: Fri, 10 Apr 1992 21:13:50 GMT

A friend of mine asked me for a good reference on relevance logics.  I don't
know anything more recent than Anderson & Belnap, and I've forgotten the exact
title of that.  What would you recommend?  

His interest is in modeling systems with a state that consists of a set of
sentences.  However, this will occasionally become inconsistent.  He wants a
way of controlling the damage from the contradictions, so that the system can
be coaxed back to a consistent (and plausible) state.  Are there other kinds of
formalism you think he should also look at?  

Thanks so much.  

	Josh
From: loader@vax.oxford.ac.uk (R Loader)
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 10 Apr 92 15:17:09 GMT
Organization: Oxford University VAXcluster

In article <1992Apr8.100146.689@gacvx2.gac.edu>, logic@gacvx2.gac.edu writes:
> In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
> 
>> S4 and S5 (and most other common modal logics), are subsystems of classical
>> (non-modal) logic extended with the axiom: A iff \box A.
> 
> Other stuff deleted....
> 
>> Ralph Loader,
>> Mathematical Institute, Oxford.
> 
> S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
> contain the axiom "A iff \box A", both contain the axiom "if \box A then A".
> 
> Al Hart

Sorry, My original posting was ambiguous: I meant ". . . are subsystems
  of the logic obtained by adding A iff \box A to classical logic"
--
Ralph Loader, Oxford.
From: rapaport@acsu.buffalo.edu (William J. Rapaport)
Newsgroups: sci.logic
Subject: Computer Science position at SUNY Buffalo
Date: 6 Apr 92 17:02:10 GMT
Organization: State University of New York at Buffalo/Comp Sci


             STATE UNIVERSITY OF NEW YORK AT BUFFALO

                 Department of Computer Science
                   Temporary Faculty Position

We anticipate an opening for a one-year visiting faculty position
for  the  coming  year at a competitive salary for a new Ph.D. or
for a more senior person who wants  to  supplement  a  sabbatical
salary.  The successful candidate will teach one course or gradu-
ate seminar each semester. I would welcome you bringing  this  to
the  attention  of your finishing doctoral students or colleagues
who might be interested.  Please ask anyone interested to  submit
a curriculum vitae, and to arrange to have four letters of refer-
ence  sent  directly  from  their  referees,  to  me  by   e-mail
(selman@cs.buffalo.edu) or by mail to

                  Alan L. Selman
                  Department of Computer Science
                  226 Bell Hall
                  State University of New York at Buffalo
                  Buffalo, NY 14260

The Department currently  has  15  tenure-track  faculty, 12  ad-
ditional    faculty,   and   125   graduate   students.   Primary
research  areas  include:  artificial  intelligence,   complexity
theory,   computer vision, numerical linear algebra, parallel al-
gorithms, programming languages,  systems  and  VLSI.  Department
members   are   actively  engaged  in  interdisciplinary research
programs in the Advanced Scientific  Computing   Graduate  Group,
the  Cognitive Science Center, the Vision Graduate Group, and the
NSF National Center for  Geographic  Information  and   Analysis.
Departmental  computing  facilities include a network of worksta-
tions, hypercubes, Symbolics, an Encore   Multimax   and  several
image processing/graphics systems.
Newsgroups: sci.logic