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

[Jeffrey Ketland]

Dear Joe

Thanks for your message. (Sorry this reply's a bit long).

1. Some general points

In general, I want to give nominalists a fair crack of the whip, even though
I'm not a nominalist. If they argue that comprehension axioms, which are
claimed to be *logically* true, are not even true, then my feeling is that
they are half-right. The comprehension axioms (formulated in the right way,
so as not to succumb to Russell) are true, but not logically true. The
comprehension axioms are existence assertions - asserting the existence of
spooky abstract entities (properties, sets, etc.). The *property* of being a
chair is quite different from the physical chair itself. Of course, you know
all this. As a realist, I think that these spooky abstract entities do
exist, but I don't want to say that we know this by logic alone. We have to
work harder.

I'm saying that denying such things as the comprehension axioms is not
logically contradictory. In that sense, I'm a sceptic. C.f., Descartes
[Meditations] tells us that we can conceive that our bodies might not exist.
There's no solution to this. It would be bananas (clinically insane) to
actually believe that your body didn't exist, but it's not logically
contradictory. Similarly, Hume [Treatise] tells us that we can conceive that
the Laws of Nature might fail tomorrow. Or, to translate into modern
epistemology: that no amount of observation can verify (100%) a theory. As
Popper puts it: our theories are conjectural guesses about reality, no
matter how strongly supported by observation and pragmatic considerations.

2. Interpretation of Comprehension Axioms

Joe:
>  Anyway, in the context of second-order-logic rather than set theory,
>comprehension is not really problematic -- (3) becomes "there is a property
P
>such that for any y, y has property P iff y is an integer", which is
practically
>tautological.  And I was referring to comprehension schemes within SOL or
HOL,
>not set-theoretic comprehension.

By comprehension I *do* mean SOL (or HOL) comprehension. There are lots of
different interpretations of the SOL comprehension scheme.

    There is a property P such that ....
    There is a set X such that ....

(call the second one the "Quine reading" - "SOL is set theory in sheep's
clothing" [somewhere in Quine's Philosophy of Logic 1970]).
The nominalists argue that *none* of these are true (and certainly not
tautological). They argue that properties/sets/collections/etc., don't
exist. I want to take this argument seriously.

So, I do not think that a statement like,

    (3)* There is a property P such that for all y, y has P iff y is an
integer

is tautological or even logically true. What if there just aren't any such
things as properties? (E.g., Quine doesn't believe in properties, but does
believe that there are sets). Properties are quite spooky entities. When we
see a physical object like a chair, we interact with the physical object
itself (and then the sentence "That's a chair" pops into our minds). We
don't causally interact with the *property* of being a chair. Nominalists
deny their existence. Properties, universals, etc., are in Ockham's phrase
"flatus vocis" (the blowing of voice).
Then there's the "individuation problem". Take Quine's example: consider the
property "being a creature with a kidney" and the property "being a creature
with a heart". Let's assume that these are co-extensive. Are these then the
same property? One wants to say that they're not, but why? Well: we
philosophers say:

    property P1 is the same as property P2 just in case, at any possible
world w, P1 and P2 are  co-extensive.

Now we up to our neck in *modality*, *possible worlds*, ... Ouch! Out of the
frying pan into the fire!
Modern nominalists are just as sceptical about sets.

3. Monadic SOL, Plural Quantification and nonfirstorderizability

The late George Boolos devised a fascinating interpretation of the monadic
second-order variables called "Plural Quantification". He reads

    EX Ay  (X(y) iff phi(y))

as

    There are *some things* such that an object y is *one of them* iff y is
phi

and he argues that this doesn't commit you to a collection/property/etc.,
but only commits you to certain "plural way" of referring to the individuals
(values of y).
See: Boolos, G 1998: "Logic, Logic and Logic" (Harvard). Especially, the
papers
[1] "To be is to be a value of a variable (or to be some values of some
variables)"
[2] "Nominalist platonism".

Actually, this whole book is great. Boolos uses this apparatus to discuss
"nonfirstorderizability", which sounds like something from Mary Poppins.
Sentences like the famous "Geach-Kaplan" sentence,

    [4]    Some critics admire only one another

which, prima facie, look like they go into FOL, but seem to require SOL,
namely

    [4]*    EX(Ex X(x) & AxAy[ (X(x) & Admire(x,y)) --> [~x=y & X(y)])])

Boolos's book is fully of fascinating stuff about these things.

4. Logical truth and verificationism

Joe:
>This seems to make "logically true" dependent on the power of our brains.
I
>think "logically true" is closely related to "necessarily true", or "true
in all
>possible worlds", or "true in all possible interpretations", none of which
>notions contain any such dependency.  In the particular case of FOL, the
set of
>validities is r.e., so you can identify "logically true" with "recognizably
true
>in a finite time", but before Godel discovered this in 1929 I don't think
it
>would have been regarded as a necessary property.
[snip]
> ... the truth value of S should be non-contingent, but that is different
from
>"determining the truth value of S" which is a humanistic process.

I *do* want to argue that logic is intimately connected to epistemic
matters, and thus inseparably connected to what (an idealization of) human
minds can recognize to be true. Logic means reason. A truth of logic is a
"truth of reason" (in Leibniz's terminology).

Actually, I do think that mathematical truths are synthetic (in Kant's
sense). We don't logically contradict ourselves if we assert that there are
no prime numbers. I don't think that nominalists like Field and Chihara (and
so on) are contradicting themselves in their nominalism.

Joe:
>If you're going to insist that "logic" entails computability, then
obviously
>there must be second-order validities which are not "logical", but again
this is
>just a dispute about the definition of "logical".

Perhaps this is partly terminological, but I think it's important. If
something is logically true, then I think that the human mind should be able
to determine this. I'm not a verificationist. I do not think that, in
general, truth (even mathematical truth) should always be epistemically
determinable. But I'm a sort of verificationist about *logic*!
Using Godel's results, we can argue that true mathematical propositions in
general transcend logical truth (massively), and that we may never come to
know the truth values of many of these propositions. Many of these
propositions will not be self-evident in any immediate sense, and we may
have to adopt a thorough-going pragmatism in deciding how to extend our
mathematical knoweldge.... territory explored by Quine and Putnam (and more
recently by Harvey Friedman and John Steel here on FOM).

5. Axioms of infinity

(I said that we need axioms of infinity to do maths)

Joe:
>  This is wrong. See work by Simpson and Friedman -- a huge amount of
>mathematics can be developed in a system which is conservative over Peano
>Arithmetic and requires no ontological commitment to infinite objects.

Sorry. I didn't mean the axiom: "There is a set X containing 0 and all its
successors". I'm thinking of axioms of infinity in the sense discussed here
on FOM a couple of months ago.
I think of PA as an infinitistic axiom system (since it only has infinite
models). In fact, I think of even successor arithmetic ("s(x) = s(y) -> x =
y" and "~[s(x) = 0]") as an axiom of infinity.
I agree: you can do lots of maths in ACA_0 which conservatively extends PA
(or WKL_0 which conservatively extends PRA for Pi^0_2 sentences). But PA
implies the existence of 0, 1, 2, etc. (and their distinctness) and ACA_0
trivially implies the existence of the set N. It's a delicate debate, I
know.

>Let T be an arithmetical theorem which requires an axiom of Infinity to
prove
>(for definiteness, Con(PA)).  T can be stated without any ontic commitments
>because it is arithmetical, but (unlike, say, the Prime Number Theorem) it
>cannot be regarded as a logical truth unless you accept some axiom of
infinity
>as "logical".  On the the other hand, S-->T where S is an axiom stating
that the
>set of integers exists is indeed a logical truth, and the ontic commitment
>accepting the set of integers is sufficiently fundamental that I think it
is
>reasonable to call "maths=logic+infinity" a form of logicism.

I do count the axioms of PA as having ontic commitments: to the numbers 0,
1, 2, ...!
I count an (unbounded) Pi^0_1 sentence as making an infinite assertion
(about all the numbers). Perhaps we might count bounded statements like Ax<t
R(x) or Ex<t R(x) as properly finitistic. I don't know - I haven't followed
all the literature on this sort of thing. (In your example, S would
presumably be the following comprehension axiom of ACA_0:
  EX  An  (n is in X iff n = n).

6. Logical foundations versus set-theoretic foundations

Joe:
>It seems strange to argue in favor of a set-theoretical foundation and
against a
>logical foundation on the grounds that we have to account for people who
don't
>believe in sets!

I agree that this is a bit strange. I defend realism but not logicism. I
think Frege-Russell logicism was a great idea, but it has just failed, and
won't be recuscitated.
(There is the St Andrews school - Crispin Wright and Bob Hale - who are
trying to recuscitate Fregean neo-logicism, based on Hume's Principle. I'm
not persuaded that Hume's Principle is logically true).

I think that the mere (empirical) existence of coherent nominalist arguments
(e.g., by people like Hartry Field, Charles Chihara) shows that the
comprehension axioms cannot be *logically* true.
Still, I disagree with nominalism, and argue that the comprehension axioms
are true, but not logically true.

(The only axiom of set theory that I think might be accounted as logically
true in some sense is extensionality. Even the null set can be doubted - I
seem to recall that one of Tarski's teachers, S. Les'niewski, in developing
mereology, was motivated by his repugnance for the concept of an empty
collection).

7. Another test for logical truth - conservativeness

Here's another proposal that I've toyed with. We have a consistent formal
system S in a language L. Suppose we add a new axiom A, perhaps in an
extended language L+. Obviously, the result S+A could be a conservative or a
non-conservative extension (w.r.t L-sentences).
But if S+A is a *non-conservative* extension of S, then we might argue that
A is not a logical truth.
For example, I think there's a result due to Levy, that adding a function
symbol c(x) and cardinality axiom
        c(x) = c(y) <--> x and y are equipollent
to certain first-order set theories can yield non-conservative extension.

(Technical question: if S is an incomplete theory in classical propositional
logic, then does adding any first-order logical truth A ever yield a
non-conservative extension of S? If the answer is yes, then this proposed
criterion is wrong)

(A few years ago I proved that adding the Tarski T-sentences Tr(#A) <--> A
(with A arithmetic) to Peano arithmetic is a conservative extension. That
provides a sort of argument that the T-scheme is logically true or perhaps
"analytic", or that this minimalist notion of truth is a logical notion).

8. Back to Quine/Putnam and pragmatic realism

Like Quine and Putnam, I think that the best overall arguments for the truth
of mathematical axioms are not based on a priori considerations, or
self-evidence, or intuition, or anything like that. They are based on
pragmatic considerations concerning their indispensability for doing
science. (If we could contrive a way of doing Quantum field theory or GR
without assuming mathematics, I'd be a happy formalist).

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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~