[FOM] Logicomix and intuitionistic truth

[Vaughan Pratt]

On 4/17/2010 10:25 AM, Timothy Y. Chow wrote:
> A search of the FOM archives turned up
> only a passing reference to [Logicomix] by Vaughan Pratt, so I figure that
> not everyone has heard of it.

Tim is referring to my post of October 19 last at

   http://cs.nyu.edu/pipermail/fom/2009-October/014146.html

which I bring up here for its relevance not so much to Logicomix (which 
despite a three-digit Amazon ranking at one point was narrowly edged out 
by Hilary Mantel's Wolf Hall in David Gutowski's self-styled 
"boxing/wrestling match" at

   http://www.themorningnews.org/tob/wolf-hall-v-logicomix.php )

as to the recent exchange between Nik Weaver and Panu Raatikainen 
concerning the former's intuitionistic account of the liar paradox.

My post was about intuitionistic truth, a concept which Raatikainen 
flatly denies.  And not just with a simple "Ain't no such thing" but 
with a detailed rejection at

 
http://www.mv.helsinki.fi/home/praatika/conceptions%20of%20truth%20in%20intuitionism.pdf

Having recently engaged some quite determined AGW (anthropogenic global 
warming) deniers, I have a strong sense of deja vu here.  On the climate 
question I thought at first that I was dealing with paid distributors of 
misinformation, but after a few exchanges I came round to the view that 
most of them were simply finding the AGW idea implausible (though I bet 
a few were on a payroll).  The picture gradually developed and refined 
by de Saussure (1760-1790), Fourier (1807-1827), Tyndall (1850-1860), 
Arrhenius (1895-1910), etc. meant nothing to them.  Evidently those not 
on the payroll had little intuition for the underlying physics (no idea 
about those on it).

There is a striking difference between FOM and the categories at mta.ca 
discussion group moderated by Bob Rosebrugh for the past 18 years or so. 
  Whereas the former largely denies intuitionistic truth (albeit not 
quite as vehemently as in the late 1990s when Sol Feferman and I 
simultaneously unsubsubscribed from FOM, Sol because of its intolerance 
of non-classical viewpoints---recall Sol's frustrated cry of 
"Basta!"---and I because of other increasing demands on my time just 
then), the latter largely denies classical truth.  The intersection of 
followers of both groups is distressingly small: Fred Linton is one I'm 
aware of, but who else?

There are two approaches to deniers of AGW.  One is to give a short list 
of the most compelling literature.  The other is to appeal to the 
experimentally inclined by suggesting they place two sheets of glass in 
front of respectively a white and black sheet of paper, leaving a 
half-inch gap to rule out conduction, and leaving them out in the sun. 
After ten minutes the glass in front of the black sheet is appreciably 
warmer (alternating the two sheets between your hands amplifies the 
difference).  Despite what many claim (even some climatologists!) the 
mechanism is the same as for AGW.

I wish I could come up with as compelling a demonstration of 
intuitionistic truth, but alas truth is not the sort of phenomenon of 
natural philosophy that lends itself to direct experiment, and I 
therefore have only the literature to point to.

The most fundamental semantics of intuitionistic logic is that of Kripke 
as per

http://en.wikipedia.org/wiki/Kripke_semantics#Semantics_of_intuitionistic_logic

This can be boiled down to replacing the Stone duality of sets and 
Boolean algebras with the Birkhoff duality of posets and distributive 
lattices (with additional qualifiers for infinite structures that Stone 
was more in command of than Birkhoff in the 1930s but the finite case 
suffices).

The same information is found in the internal logic of a topos, see e.g.

http://en.wikipedia.org/wiki/Topos#Explanation

or my October 19 FOM post

   http://cs.nyu.edu/pipermail/fom/2009-October/014146.html

that Tim cited (albeit in connection with Logicomix rather than 
intuitionism).  A more complete account can be found in Peter 
Johnstone's three-volume update of his 1977 book on Toposes, of which 
the first two volumes are now in print.  (Johnstone calls it his 
Elephant, I would characterize it more simply as unnecessarily 
slow-paced.  A Readers' Digest version is urgently needed, for now look 
for short explanations couched in terms of subobjects such as the above 
Wikipedia *explanation* of toposes.)

In support of classical logic is the fact that there is only one 
subdirectly irreducible Boolean algebra, namely the two-element one that 
most philosophers and many mathematicians identify with the concept of 
truth.   This gives classical logic a huge advantage over intuitionistic 
logic, for which the counterpart to the two-element Boolean algebra 
amounts to those Heyting algebras having the property that that the top 
two elements form a chain, these being precisely the subdirectly 
irreducible Heyting algebras.  (An algebraic structure is said to be 
subdirectly irreducible when it is isomorphic to some factor in every 
subdirect factorization of it, where a subdirect factorization of A is a 
subalgebra of a direct power of A such that for every factor the 
projection of A to that factor is surjective.  The subdirect 
irreducibles suffice to determine the equational theory.)

Bottom line: the nature of intuitionistic truth, as understood in terms 
of possible worlds and subobjects of algebraic structures, is a tad 
subtle, very much like anthropogenic global warming.  In neither case is 
it fair to infer nonexistence from subtlety.

Vaughan