FOM: preamble to a Dummettian account of intuitionistic logic

[Neil Tennant]

This posting is by way of preliminary preparation for the longer
disquisition (on foundations of intuitionism) invited by Harvey and Steve.

Harvey wrote:

 > You seem to suggest that there is a coherent view of intuitionism -
 > different than Godel, Heyting, and Brouwer. I would like to see you
 > explain *carefully* what this is without sending us to papers and
 > books. If it is good, it can explained on the fom, at least enough
 > so that one can get a feel for it.

Harvey more than anyone else knows just about everything worth knowing
about the technical interrelationships between classical and
intuitionsitic formal systems---so I hesitate to undertake the
challenge with anything like a hope that anything I say will strike
him as new or useful. On the other hand, he appears to be issuing a
genuine invitation to make a philosophical case of which he professes
an honest innocence. This places a rather onerous responsibility on
one rash enough to forego royalties... but I'll try to do my
best. That is, I shall try [in a SUBSEQUENT mailing] to state as
succinctly as I can what exactly the Dummettian case is for revision
of logic to (something in the neighbourhood of) intuitionistic
logic. Note that I shall confine myself to *logic*. I shall not
attempt to go further by arguing for any intuitionistic principles of,
say, real analysis.

By way of further preamble (to which this posting is wholly confined),
let me react to two observations by Steve Simpson.  Steve wrote:

> it's difficult in principle to
> separate f.o.m. research from philosophical motivation.  If Dummett's
> philosophical position is truly different from Brouwer's, then this
> should be reflected in the mathematics.

My first reaction is: why? Why should it not be possible for a
recommended practice, initiated by someone with a certain set of
quirky inspirations, to be amenable to a better rationalization later
on from wholly different, and more appealing, principles? 

The picture to be defended against Steve's dictum is this. Brouwer had
a quirkily subjectivist `justification' for intuitionistic principles
in logic and mathematics. He believed that mathematics consisted of
mental considerations, which were essentially private, and which would
be corrupted by having to be put into symbolic form for communication
to other minds. This led Brouwer to recommend a different mathematical
practice, or set of norms: roughly, the eschewal of all uses of the
classical law of excluded middle Pv-P, except when the proposition P
in question is decidable. Naturally, there was a backlash from
classical mathematicians unwilling to have their inferential hands
tied in this fashion. These defenders of classicism stand to make
valuable debating points by casting aspersions on the apparently
whacky philosophical ideas that *appear* to form the basis of
Brouwer's deviant recommendations---at least, given the historical
record thus far.  Enter the Dummettian, after the later
Wittgenstein. The Dummettian argues that exactly the same set of
normative recommendations (at least, concerning the logic used in
mathematical discourse) can be recovered from philosophical
considerations of a wholly different flavour, considerations
emphasizing the *publicity* of communication, and the
*indispensability* of language for the expression of thought. The new
philosophical basis of intuitionism turns out, ironically, to be
diametrically opposed to Brouwer's on major counts.

How, one might ask, could it possibly be that one and the same set of
norms could be argued for in such different ways? Well, consider
ethical norms. Originally, they were taken to be of divine
inspiration, given from on high, and backed by divine reward and
retribution in the afterlife. Nowadays, however, exactly the same
ethical norms might be grounded by a humanist in evolutionary theories
about human psychology, particularly as it relates to sexuality,
family structure and reciprocity. 

I don't need to argue the adequacy of any attempted non-theistic
recapitulation of ethics. The weak point I need is only that it
wouldn't be held against the would-be naturalizing humanist that
he/she was seeking to give a different kind of backing for a set of
norms that had already, as a matter of (pre-)history, been provided
with *a* backing (of whatever dubious merits).

So: if it can happen with ethical norms, then why shouldn't it also be
possible in the case of logical norms?

To end this preamble, let me note that Steve replied to my claim

 > the Dummettian can claim that once one *does* take the physical
 > reality of actual communication into account, one is led along a
 > justificatory path to intuitionism.

by saying "this confusion of physical reality with language does indeed
strike me as subjectivist." 

On the contrary, I was not confusing language with physical
reality. Rather, I was emphasizing that our use of language is a
process that takes place in the physical world. We make sounds and/or
inscribe symbols. These are usually heard or seen by others. (This is
the central case, which motivates the argument against the possibility
of a private language.) Our competence as language users consists in
our ability to produce and to process finite audible sequences of
sounds and finite visible sequences of written symbols. Moreover, any
semantic competence that we may wish to attribute to ourselves or to
other speakers must be able to be made manifest in our productions of,
and/or in our behavioural responses to, finite sequences of spoken or
written symbols. The latter responses *might* be linguistic, such as
an utterance of a simple "Yes" or "No", or of a more complicated
sentence such as "As far as I know, no-one has ever proved that claim,
nor has anyone ever refuted it"; or the responses might be
non-linguistic, such as obedient retrieval of a red ball, or the
raising of a hand. Either way, the responses in question will allow
the behaver to manifest his/her linguistic understanding of whatever
is at issue.

It is the consequences of this "manifestation requirement", in
conjunction with certain other basic principles concerning the
language of mathematics, that the Dummettian pursues in order to
arrive at intuitionistic logic as the logic justified by the best
available account of the meanings of the logical operators. The most
important among the latter principles are the semantic principles of
molecularity, separability and harmony [to be explained in the next
promised posting]. Dummett's argument is meaning-theoretic, and may
therefore be held (by Quineans, for example) to beg certain important
questions about the analytic/synthetic distinction etc. I shall let
those chips fall where they may. The point to be emphasized at this
stage is that the Dummettian's case is based on the (alleged/argued
for) achievement of a certain `reflective equilibrium' between
linguistic practices, on the one hand, and, on the other, a semantic
theory that (so the Dummettian maintains) best accounts for our
abilities in learning the underlying norms and then participating in
that practice. The idea is that such reflective equilibrium can
ultimately *enjoin* reform of certain aspects of the practice (such as
illicit uses of excluded middle). Rather than saying of such uses
"They have to be justified, because they are part of the practice",
the Dummettian says "They cannot be justified, because they are not
consequences of our best theory explaining our acquisition and
manifestation of linguistic competence". Anyone hostile to the
possibility in principle of reform-via-reflective-equilibrium will of
course not be moved to hew to intuitionistic logic.

In addition to the semantic principles just mentioned is the
anti-realist's principle of knowability (that all truths are in
principle knowable). Note that the principle of knowability if *not*
the same as the principle of in-principle-decidability, according to
which every declarative sentence will admit in principle of proof or
of refutation (in some acceptable system of mathematical proof,
perhaps yet to be invented). An anti-realist can hold that all
*truths* are in principle knowable (via proof in some future system)
*without* holding that all conjectures are in-principle-decidable.

For those who are not prepared to *give up* classical forms of
reasoning as a result of the Dummettian considerations, there is still
the prospect of shedding some foundational light on exactly *what they
are doing*, and *why they take themselves to be justified*, when using
those classical forms of reasoning. This is where various relative
consistency proofs, and results about conservative extensions by
classical rules, might have a role to play.

This was, after all, intended to be only a preamble; so I shall end it
here. I can't promise when the next installment will be
ready. *Condensing* books already written is almost as hard as writing
them!


Neil Tennantd