[FOM] consistency and completeness in natural language
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Fri Mar 28 22:09:02 EST 2003
Neil Tennant has recently presented the fine details of just how it
is that we humans can prove what PM, and the like, cannot, in the
case of Goedel's Theorems (Mind, July 2002). As a consequence, we
are in a far better position to evaluate just what it is that
distinguishes our natural notion of proof. Tennant has argued that
we have external means of judging the consistency of the formal
systems Goedel was concerned with, and so know that reflection
principles, which might be written in the form
(|- p) -> p,
hold for them. Strictly, the left-hand side of such conditionals
should be meta-linguistic, a fact which will have considerable
significance later, but the common expression for the objects of
proof in formal systems represents them as propositions, as above,
not mentioned sentences, as with "|- 'p'". Keeping to the common
convention, for reasons which will become apparent in a moment, we
then have that, given, for a specific 'q',
q <-> ~ |- q,
we can deduce that ~ |- q, since the alternative, |- q, would entail
q, by the reflection principle, and so ~ |- q, by its definition.
But our proof of this truth cannot be internalised in Goedel's own
systems of proof, i.e. they are incomplete: we cannot deduce
|- ~ |- q.
Tennant's proof reminds us that previous claims about PM
providing a foundation for Mathematics are invalid, since the needed,
extra proof of consistency would use many of the arithmetical
principles PM was supposed to establish. But the most immediate
philosophical question his analysis gives rise to is: what happens if
we now start to talk about our own systems of proof? Is natural
language shown to be inconsistent, if the turnstile relates to our
own proof processes? This latter question has been faced by a number
of thinkers, notably Graham Priest, and recently J.C.Beall has
elaborated its large bearing on other major questions in the
Philosophy of Mathematics. Beall saw little hope of denying that a
sentence like 'q' above could be found in our informal language, and
hence was led, explicitly, to conclude that informal mathematics is
inconsistent (Philosophia Mathematica 1999, p324). Beall's sentence
g was defined to be 'g is (informally) unprovable', and in Priest's
previous previous discussions this kind of sentence has been said to
generate 'Goedel's Paradox'.
There is, of course, no difficulty in finding such a sentence
as Priest and Beall suppose, but,what is questionable is whether it
is relevant to anything to do with truth and provability in informal
Mathematics. What is (informally) provable, as Tennant's analysis
also reveals, is some fact, not some sentence. Forgetting the
difference is presumably why, in common representations of the
objects of formal proofs, a proposition is indicated, as before,
rather than a mentioned sentence. For if '|- p' is used to represent
our informal notion of provability, then it certainly is 'p' that is
involved, and not "'p'", as is evidenced by the right-hand side of
the reflection principle before. The proved fact in question is then
certainly what some sentence is standardly taken to express, i.e.
what is true in its standard model. But even then it is not the
sentence which is what is true in that standard model: what is true
is that 1+1=2, not '1+1=2', and 'that 1+1=2' is not a sentence, it is
a noun phrase. The relevant locutions are therefore operator
locutions like 'it is provable that p', and it is trivial that there
can be no self-reference with such locutions. Certainly we can say
such things as 'It is provable that this fact does not obtain', but
'this fact' cannot refer to what is expressed by the whole sentence.
Thus while there can be a sentence g which consists in the words 'g
is unprovable', there is no sentence 'q' such that 'q' = 'it is
unprovable that q' - that is ruled out by mereology, since nothing
can be a proper part of itself. In addition, there can be no 'q'
such that it is necessary that
q <-> it is unprovable that q,
since an argument somewhat similar to the one at the start shows this
leads to a contradiction. Specifically, since it is provability of
facts, i.e. truths, which is involved, 'it is provable that' at least
obeys the rules for the modal operator in the system T. With 'it is
provable that' as 'L', therefore, Lq would entail q, and so ~Lq,
giving ~Lq absolutely, and so q likewise, and hence, by the Rule of
Necessitation, contradictorily Lq. In fact '~L(p <-> ~Lp)' is a
theorem of T.
The residue in the natural language version of Tennant's
analysis of Goedel's Theorems is therefore just the reflection
principle, which becomes the characteristic T-axiom 'Lp -> p'. One
can have q <-> ~ |- 'q' (where '|-' is a provability predicate of
mentioned sentences), but not q <-> ~ |- q (where '|-' is a
provability operator on used sentences). So there is no proof of
incompleteness. Moreover, no induction to the first epsilon number,
or anything remotely like it, is needed to previously show that the
natural notion of proof is consistent. Because of the T-axiom again,
there is a simple one line proof of this fact: one can no more have
Lp.L~p than one can have p.~p.
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html
More information about the FOM
mailing list