[FOM] Logicomix and intuitionistic truth
pratt at cs.stanford.edu
Sun Apr 18 10:32:14 EDT 2010
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
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
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
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
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
The same information is found in the internal logic of a topos, see e.g.
or my October 19 FOM post
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.
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