🚀 go-pugleaf

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

>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:

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

Then it befits me, a perpetually unemployable shirker, and certified
blowhard, to assert the obvious: there are just two truth values.

AHR:
>>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?

VRP:
>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.)

This is a point which comes up with a nauseating regularity; last time,
as I recall, it was Greg Restall who championed the many-valued path.
To recapitulate the substance of past battles, Alfred Tarski, in his
article "On Extensions of Incomplete Systems of Sentential Calculus",
appeals to what has come to be known as Lindenbaum's Lemma, proving that
there exists but one (absolutely) complete and consistent extension of
incomplete sentential calculi, which contain as theorems certain
formulae (namely, CpCqp, CpCCpqq, and CCqrCCpqCpr), expressing naturally
evident properties of material implication.  In other words, the
situation of these many-valued logics is quite unlike the case of
non-Euclidean geometries, in that the latter can be said to offer a
genuine alternative to the classical theory.

VRP:
>>>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!

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

VRP:
>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.

All fine texts; I would, however, add a recent comprehensive treatment
of the same topics from a logical point of view, in _A Study of Logics_,
by John Cleave, an obscenely expensive (though now sold at 20% off)
Oxford Logic Guide.  Furthermore, one should look into the subject of
"truth definitions" as given by forcing conditions, either in Cohen's
original exposition of forcing, or in Manin's course of mathematical
logic and Bell's account of Boolean-valued models of set theory; a fine
treatment of the connection between forcing and intuitionistic logic
(and, so by G\"odel's interpretation, the modal logic S4) will be found
in the course book by Bell and Machover.

However, none of the above is to be taken as in any way impugning the
incontrovertible fact that there are only two truth values, inasmuch as
the finely-grained models may (and ought to) be interpreted as applying
to the calculi of intensional entities, like propositions.

VRP:
>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.

However, see Cleave, and the references therein.  Also, please note that
the hoary old phallogocentric assumption of bivalence and excluded
middle underlies all of algebraic logic, at least as this subject is
conceived and developed in Tarski's classic paper on the concept of
truth in formalized languages.  If you are aware of any development of
model theory, which does not rely on the implicit assumption of
bivalence and excluded middle within its metalanguage, please let me
know.

Finally, the historical veracity of your disquisition on the subject of
truth-values in the preceding article is contradicted by most reasonable
interpretations of Aristotle, e.g. one found in Kneale & Kneale's _The
Development of Logic_, II.4.

>-- 
>Vaughan Pratt				There's safety in certain numbers.

cordially,
mikhail zeleny@husc.harvard.edu
"Les beaulx bastisseurs nouveaulx de pierres mortes ne sont escriptz
en mon livre de vie.  Je ne bastis que pierres vives: ce sont hommes."