FOM: Re: The logical, the set-theoretical, and the mathematical

[Jeffrey Ketland]

Replying to Joe Shipman (<shipman at savera.com> Date: 12 September 2000 01:15
Subject: FOM: The logical, the set-theoretical, and the mathematical)

1. The Logical

Joe said:
>My position is as follows.  Comprehension axioms are "logical".

I think that can't be right. Suppose we are interested in whether an
interpreted sentence S is logically true. Then I would propose these
necessary conditions:

    (i) S should be recognizably true (in a finite time) by any reasonable
person, perhaps after careful deliberation, and possibly re-checking.
    (ii) S should be devoid of ontic commitments
    (iii) determining the truth value of S should (somehow) be independent
of contingent knowledge or empirical information.

These conditions are met for simple FO logical truths like,

    (1) All wise Greeks are Greeks
    (2) For any object x, if x is a wise Greek then there is a y such that y
is a Greek
    etc.

(more precisely, these English sentences have a FOL logical form, which is a
theorem of any standard system).

It seems that our brains are built to see the truth of sentences like (1),
(2), etc. In some sense, our human brains contain FOL machines (and if
they're not damaged or intoxicated, then they work properly). FOL exhibits
the basic structure of human mental reasoning.
Perhaps our mind/brains also contain the rules for modal logic, tense logic
and provability/epistemic logic and the (lowest level) truth predicate, as
well. It would be interesting to see empirical evidence concerning how
logical rules like these are psychologically implemented in the human
mind/brain.

But the conditions I stated above don't seem to hold for the comprehension
axiom:

    (3) There is a set X such that for any y, y is in X iff y is an integer

There are philosophers who, after many years of deliberation, think that the
sentence (3) is *false* (because sets don't exist). E.g., contemporary
nominalists, like Hartry Field and Charles Chihara. For that reason---the
existence of human minds who do not find (3) to be self-evidently logically
true---I cannot count (3) as a truth of logic. A truth of mathematics maybe,
but not a truth of logic.

2. Logical validity and consequence

If V is a set of *logical* validities, then I do not see how V can be non
r.e. And if R is a *logical* consequence relation, then I do not see how it
cannot be matched up with a deductive system which verifies all and only the
valid consequences.
(Sometime in the 1960s, Hilary Putnam suggested that it is probably
impossible for the human mind to come to know, a priori, a set of sentences
which is not r.e.. How could the set of self-evident logical truths not be
r.e.?)

I would argue (like Quine - see his Philosophy of Logic 1970) that the very
notions of "logic" and "complete proof procedure" are intimately tied
together. The reason is that "logos" means "reason": and thus the very
notion of logic is inseparable from what human minds can (in principle)
recognize to be the case by actual reasoning. In particular, human reason
seems to be bounded by certain finiteness constraints. Perhaps there are
other creatures in the universe whose physical structure means that they can
perform mental supertasks (e.g., determining the truth values of any Pi^0_1
sentence by running through all the positive integers). But we're not them.

The set of (full) second-order validities is highly non-computable (as we
have seen, with the messages from Solovay) and, by Godel's 1st
Incompleteness Theorem, we all know that there is no complete proof
procedure (deductive system) for the full second-order consequence relation.

I can only conclude that the full second-order consequence relation is not a
*logical* consequence relation, but is an intrinsically mathematical notion.
It makes perfect mathematical sense to talk about this relation (as it does
to talk about many other computationally intractable relations).

3. Axiom of Infinity

Joe:
>(Mossakowski insists on an axiom of Infinity as well, but Jones argues
>that this axiom is purely ontological and is only necessary in a
>metaphysical sense.   However, I am willing to accept a modified
>logicist thesis that (ordinary) mathematics = logic plus Infinity.)

I agree. I cannot see how anything like an axiom of infinity could count as
logically true. Plenty of people don't believe that there are infinitely
many objects, but they understand logical reasoning. One cannot develop
(much) mathematics without an axiom of infinity.
So, the thesis that maths = logic + infinity cannot count as a logicist
thesis.

Better to say,
    maths = logic + set-theoretic comprehension + infinity
(where the latter 2 are extra-logical).
But this brings us back to set-theoretic foundations - i.e., anti-logicism!

(I seem to recall that Hintikka has recently argued that AxC is logically
true. Is it "Principles of Mathematics Revisited" (1997, I think)?).

4. "Logical" revisited

My proposal is vague. In general, if a sentence S is considered *false* by
well-informed rational people, who have thought about the question for many
years, then I cannot count S as logically true. And the comprehension axioms
fit into this category. The comprehension axioms say that sets exist. The
comprehension axioms are false if there are no sets (or properties, etc.),
and this is what nominalists say. (I think they're wrong: but the mere
empirical fact that they can argue this way strongly suggests that
comprehension axioms are not *logically* true).


Regards - Jeff

~~~~~~~~~~~ Jeffrey Ketland ~~~~~~~~~
Dept of Philosophy, University of Nottingham
Nottingham NG7 2RD United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~