From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Multiple Truth Values
Organization: Computer Science Department,  Stanford University.
Date: Sun, 10 Jan 1993 00:52:15 GMT

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.