🚀 go-pugleaf

A radical critique of language which essentially denys its ability to
characterize reality is one of the foundations of Buddhist thought
(or more accurately, of Mahayanic Buddhist thought).  Mapped onto
ordinary bimodal logic, this is something like asserting that all
statements are false, and there is even a system of "systematic
denial" invented by a major figure in the formation of Mahayanic
"theory".  Of course, this quickly becomes self-referential
(the statement that all statements are false is itself false) and
thus paradoxical from a two-valued point of view.  However, it is
seen as an advantage (a feature not a bug) in the Buddhist context.

Taoism contains a similar anti-linguistic theme (independently derived?)
expressed by aphorisms such as "Those who speak do not understand; those
who understand do not speak" (a suitable slogan for the net?) and
"The Tao which can spoken is not the true Tao."

It is, thus, unsurprising that the branch of Mahayanic Buddhism
(Ch'an or zen) formed under the influence of Chinese Taoism rejects
language most forcefully and defines itself as a means of transmission
of the truth "by direct pointing of the mind" without reliance on any
words or scriptures.

Systems of Buddhist logic which are sometimes described as multi-valued
are more understandable in this framework.  In some sense, the various
possible values which can be assigned to a statement are intended to
indicated how much or in what general way a statement errs (or, in more
Buddhist language, the degree and nature of the delusion it expresses).

If this clears anything up, I'll be surprised.


--
Ed Turner			"There ain't no answer.
elt@astro.princeton.edu		 There ain't going to be any answer.
				 There never has been an answer.
				 That's the answer."
                                             - Gertrude Stein

In article <1993Jan9.191854.10303@dcs.qmw.ac.uk> arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:

   In <1993Jan9.185347.9555@CSD-NewsHost.Stanford.EDU>
   pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

   >Any culture with a weak work ethic is vulnerable to the
   >oversimplification that there are just two truth values.

   That's a beautiful-looking generalisation, but I don't understand
   quite what it means or how you arrive at it. Can you expand on it
   a bit?

Yeah, me too, me too.
--
		Charles R. Martin/(Charlie)/martinc@cs.unc.edu
     Dept. of Computer Science/CB #3175 UNC-CH/Chapel Hill, NC 27599-3175
	    3611 University Dr #13M/Durham, NC 27707/(919) 419 1754
     "Oh God, please help me be civil in tongue, pure in thought, and able
      to resist the temptation to laugh uncontrollably.  Amen." -- Rob T

In article <1993Jan8.223915.17370@cs.sfu.ca> jamie@cs.sfu.ca (Jamie Andrews) writes:
>(Note cross-posting; also posted to soc.religion.eastern
> and soc.religion.bahai, which are moderated.)
>
>     There has been some talk lately on sci.logic about cultures
>and religions which have multiple truth values.  In particular,
>it has been asserted that the Jains have (or had at some time)
>seven truth values.
>
>     In traditional Western thought, in contrast, there are just
>two truth values (true and false).

Ouch.  This is an extreme misrepresentation of "Western thought."  Few
if any Western logicians or philosophers limited themselves to two
truth values prior to Peirce (c. 1870) in *any* context.  Before Boole,
people thought in terms of whether sentences were true or false, not in
terms of whether they had values like arithmetic expressions do.  And
no philosopher in his right mind, Western or otherwise, and before or
after Boole, would make the bald statement that sentences were either
true or false.  The diversity of the logical taxonomy of sentences has
long been a staple of the philosophical diet.

Boole was the first to provide an algebraically tractable truth value
semantics.  However he did not provide a two-truth-value semantics for
his logic, this was left to Peirce to supply, more than two decades
latter.  Boole interpreted his formulas as classes, i.e. as
predicates.  For Boole a proposition was not a Boolean formula in the
sense we understand it today but an equation between classes.  Boole
did not propose a truth-valued interpretation of equations, two-valued
or otherwise. And even if he had, he would presumably have acknowledged
the distinction between those equations that were tautologically true,
those that were true by definition of their terms, those that were
empirically true universally, those that were true here and now, those
that were satisfiable, etc.

Peirce invented the truth-table decision method for recognizing
tautologies.  But not even Peirce insisted that the two truth values of
his method constituted the scope of possible truth values.  This was a
particular mathematical abstraction for Peirce, not a hypothesis about
the possible truth values of real propositions.

The impression that "Western thought limits itself to two truth values"
appears to have obtained its real impetus from the binary nature of
digital computing and its connections with logic, and only then when
that became a commonplace within the past two or three decades.  People
have discovered that identifying truth values with bits makes logic
much easier to understand.  This is true, and two-valued logic is the
best place to start learning *mathematical* logic.  But it is no more
representative of "Western thought"'s taxonomy of truth values than the
natural numbers are representative of mathematics' diversity of
datatypes.

This is yet another example of how little time it takes for a
perspective to be perceived as a tradition.

Any culture with a weak work ethic is vulnerable to the
oversimplification that there are just two truth values.  So if the
Japanese are right about the Western work ethic then you may indeed be
right about traditional Western thought, provided "traditional" is
taken to mean something on the order of "dating back to the 1950's".

>Lately, logic researchers
>have been playing around with the "undefined" truth value and
>with a "both true and false" truth value.

True enough, provided you will also allow that lately mathematicians
have been playing around with negative numbers.  The algebra of truth
values is a rich subject that is not adequately represented by the
passage from {0,1} to its power set, to say the least!

>I think we sci.logic
>people would be interested in hearing about philosophically- or
>religiously-motivated theories of truth.

I think so too, although the context of your preceding sentence makes
this rather a non-sequitur.  Can we start over with just this sentence,
stripped of its misleading introduction?
-- 
Vaughan Pratt				There's safety in certain numbers.

In <1993Jan9.185347.9555@CSD-NewsHost.Stanford.EDU>
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>Any culture with a weak work ethic is vulnerable to the
>oversimplification that there are just two truth values.

That's a beautiful-looking generalisation, but I don't understand
quite what it means or how you arrive at it. Can you expand on it
a bit?

>The algebra of truth
>values is a rich subject that is not adequately represented by the
>passage from {0,1} to its power set, to say the least!

Do you have any references to more adequate studies of the algebra(s)
of truth values?
--
Gus Rodgers,  Dept. of Computer Science, | Cinema's worst romantic dialogue? --
Queen Mary & Westfield College, Mile End | "Mechagodzilla's brain is  installed
Road, London, England.   +44 71 975 5241 | in my stomach!" "I don't care if you
E-mail (JANET):   arodgers@dcs.qmw.ac.uk | are a cyborg  --  I still love you!"

In article <1993Jan9.191854.10303@dcs.qmw.ac.uk> arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:
>In <1993Jan9.185347.9555@CSD-NewsHost.Stanford.EDU>
>pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>
>>Any culture with a weak work ethic is vulnerable to the
>>oversimplification that there are just two truth values.
>
>That's a beautiful-looking generalisation, but I don't understand
>quite what it means or how you arrive at it. Can you expand on it
>a bit?

Meaning only that it is all too easy to relax after passing an
introductory course on two-valued Boolean logic and assume you now
understand truth values.  I phrased it that way just in case it had
really become true that Western thought had settled for 2-valued logic,
so that I could then suggest an explanation concomitant with Japanese
speculation.  (I hope you can see my tongue sticking well into my
cheek.)

>>The algebra of truth
>>values is a rich subject that is not adequately represented by the
>>passage from {0,1} to its power set, to say the least!

>Do you have any references to more adequate studies of the algebra(s)
>of truth values?

Let me strongly recommend Birkhoff's Lattice Theory, if you don't mind
the pace.  Although it is not exactly offered as such by Birkhoff, I
think it fair to say that even today it remains the definitive text on
the algebra of truth values for those interested in algebraic logic.
It has long been a source book for algebraic logicians (well, not all,
HMT (see below) don't even cite it), and will continue to supply them
with material they will discover they need for quite a while.

Texts on universal algebra and on category theory are also quite
relevant, although you have to know what you are looking for in them,
they aren't aimed at logicians per se unfortunately.  There are Henkin,
Monk, and Tarski (HMT)'s 2-volume set on Cylindric Algebras, and
Halmos' book on Algebraic Logic, but neither has adequately broad
coverage (though sect. 5.6 of Vol. 2 of HMT surveys a number of other
logically motivated algebras besides cylindric algebras).  The books on
universal algebra by Cohen; Gratzer; Burris and Sankappanevar; and
McKenzie, McNulty, and Taylor, all convey many insights into equational
and universal Horn logic, again if you know what to look for.

If you know Stone's representation theorem for Boolean algebras, namely
that every Boolean algebra is isomorphic to a "field" of sets (one
closed under finite unions and intersections as well as complement with
respect to the union of all the sets in the field), and the more
general one for distributive lattices, that every DL is isomorphic to a
"ring" of sets (drop "complement" from the definition of "field"), you
are ahead of the game.  If furthermore you know a few of the basic
facts about Heyting algebras (Birkhoff calls these Brouwerian lattices)
then you will be able to hold your own in discussions of intuitionistic
vs. classical propositional logic.  For example you will be able to
contradict people like Roger Penrose who don't even understand the
basics, witness the middle of p.114 of "The Emperor's New Mind", where
he repeats the common mistake of interpreting the absence of excluded
middle as an upper bound on the expressiveness of intuitionistic logic
instead of a lower bound.  If he only understood that double negation
served to retract a Heyting algebra onto a Boolean subalgebra of it
thereby erasing information he would find it much easier to see that he
had things backwards.  (I mentioned this to him briefly when he visited
Stanford last month, hope it took.)

Unfortunately there is no one place to look for the bulk of the
worthwhile material.  Despite its antiquity Birkhoff's Lattice Theory
remains an encyclopedic starting point (rather like a volume of Knuth),
but does not reflect the last three decades of development, whose
impact can be felt at all levels, even the most elementary.  The
algebra of truth values is a very rich subject today, from bottom to
top, that is not adequately served by its literature.
-- 
Vaughan Pratt				There's safety in certain numbers.

In <1993Jan10.005215.13278@CSD-NewsHost.Stanford.EDU>
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>(I hope you can see my tongue sticking well into my
>cheek.)

Thanks to the telescope which you have just provided, yes. :-)

[Much interesting stuff deleted; thanks again.]

>[...] double negation
>served to retract a Heyting algebra onto a Boolean subalgebra of it

Is this by any chance related to the construction of the regular open
algebra of a topological space?
--
Gus Rodgers,  Dept. of Computer Science, | Cinema's worst romantic dialogue? --
Queen Mary & Westfield College, Mile End | "Mechagodzilla's brain is  installed
Road, London, England.   +44 71 975 5241 | in my stomach!" "I don't care if you
E-mail (JANET):   arodgers@dcs.qmw.ac.uk | are a cyborg  --  I still love you!"

In article <1993Jan10.005215.13278@CSD-NewsHost.Stanford.EDU>
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
 
>Unfortunately there is no one place to look for the bulk of the
>worthwhile material.  Despite its antiquity Birkhoff's Lattice Theory
>remains an encyclopedic starting point (rather like a volume of Knuth),
>but does not reflect the last three decades of development, whose
>impact can be felt at all levels, even the most elementary.  The
>algebra of truth values is a very rich subject today, from bottom to
>top, that is not adequately served by its literature.
 
However, there are now - or will be shortly - some comprehensive
texts reflecting the state of the art of the subject:
 
1. The proceedings of the conference on algebraic logic, Budapest 1988,
Coll Math Soc Janos Bolyai 54
 
2. The proceedings of the logic semester at the Banach Center, Warsaw,
Banach Center Publ 28, to appear 1993, as well as special issue of
Fund Inf
 
3. Istvan Nemeti's survey article "Algebraizations of quantifier logics,
an introductory overview", now in its 11th version,
to appear in Studia Logica.
 
And don't forget Rasiowa & Sikorski "The mathematics of metamathematics",
quite old but still rather useful and good to read (though
others may think differently).
 
Ivo Duentsch
Univ. Osnabrueck

In article <1993Jan12.224350.10516@dcs.qmw.ac.uk> arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:

   Are there only two truth values, or aren't there?

Maybe yes, maybe no.
--
		Charles R. Martin/(Charlie)/martinc@cs.unc.edu
     Dept. of Computer Science/CB #3175 UNC-CH/Chapel Hill, NC 27599-3175
	    3611 University Dr #13M/Durham, NC 27707/(919) 419 1754
     "Oh God, please help me be civil in tongue, pure in thought, and able
      to resist the temptation to laugh uncontrollably.  Amen." -- Rob T

Are there only two truth values, or aren't there?

(And if it's false that there aren't only two, are there only two?)
--
Gus Rodgers,  Dept. of Computer Science, |
Queen Mary & Westfield College, Mile End |                       This .signatur
Road, London, England.   +44 71 975 5241 |                       British Rail a
E-mail (JANET):   arodgers@dcs.qmw.ac.uk |                       Normal service

In article <1993Jan12.224350.10516@dcs.qmw.ac.uk> arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:
>Are there only two truth values, or aren't there?

Depends on who you ask.

>(And if it's false that there aren't only two, are there only two?)

Are you asking me or Zeleny?
-- 
Vaughan Pratt			There's safety in large condition numbers.