FOM: Semantics and the problem of reference

[Jeffrey Ketland]

Semantics and the Problem of Reference

Just some comments on recent discussions, about 2nd-order logic, semantics,
truth value of CH, etc.

1. What is set theory about?

Sets, obviously. ZFC is a theory (a load of axioms and schemes that can be
written down) and is about sets---not models of set theory.
I agree with Friedman and Steel: ZFC is about sets.

(It's hard to see why models have anything to do with what language, or a
theory in a language, is about. Consider the inconsistent statement "Quine
is a human and the taller than relation is irreflexive and Quine is taller
than all humans". This statement has 0 models, but surely it's about Quine).

2. What exactly is the (intended) semantics for set theory?

This question seems to be bothering Roger Jones. But surely it's easy - it's
the disquotational Tarskian semantics.
Define a valuation v as a function from variables to (all!!) sets. Then

(i) "x_n is an element of x_m" is satisfied by a valuation v iff v(x_n) is
an element of v(x_m).
(ii) "x_n = x_m" is satisfied by a valuation v iff v(x_n) = v(x_m).

(iii) ~A is satisfied by v iff A is not satisfied by v
(iv) A & B is satisfied by v iff A is satisfied by v and B is satisfied by v
(v) (Ex_n) A is satisfied by v iff there is an valuation v* such that v and
v* differ at most at the nth value, and A is satisfied by v*

(v) If A is a sentence of the first-order language and has no free
variables, then A is TRUE iff A is satisfied by every valuation v.

(NB: in these axioms, "satisfies" has to be treated as new primitive
predicate. It is not definable in ZFC).

3. What is the language of mathematics about?

Presumably it's about numbers, functions, graphs, sets, vector spaces,
groups, manifolds, etc. Isn't that obvious? It's not about "intuitions",
"memories", "Julius Caesar", or anything like that.

4. What does it mean to say that a sentence of mathematics is true?

This question is bothering Kanovei. As Tait (following Frege, Ramsey,
Tarski, Quine, etc.) pointed out, "CH is true" means the same as CH. That
is, CH is true just in case any infinite subset of |R is either denumerable
or equipollent to |R. It's hard to see how anyone could be confused about
this, given that Tarski's work is well-known.
In particular, the following T-sentence can be *proved* from the definition
(v) and axioms (i)-(iv):

 CH is true iff CH

5. Does mathematical truth mean "having a mathematical proof"? (Kanovei)

Obviously NOT. The truth set for (|N, 0, s, +, x) is not axiomatizable. This
was discovered by Kurt Goedel in 1930. N.B. The definition (v) of truth (for
sentences of the lang of set theory) above makes no mention of *proof*.

6. How do we *know* that a mathematical statement/axiom is true?

Question postponed!
(Intuition? Self-evident? Follows from logic alone? Indispensable to
empirical science? Most pragmatically coherent explanation of our basic
intuitions?).
NB. Epistemology is HARD, and there are no Nobel Prizes for us poor
epistemologists.

7. The problem of reference (Quine, Putnam, the Skolem paradox, etc.):
The predicate "set" refers to sets (just as "chicken" refers to chickens).
But what is it about the word "set" that makes it refer to sets, and not,
say, to chickens? Is it something about our *use* of the word "set"? Is it
our best overall theory containing this word?
So: what is that makes the word "set" refer to sets?

7(a) Causal theory
A popular theory of reference (i.e., a theory of what makes certain WORDS
refer to a certain CLASS of THINGS) is the causal theory of reference.
Consider "chicken". Once upon a time, someone used the word "chicken" with a
pointing gesture, towards a real chicken, and said "That's a chicken". In
this way, there was a certain kind of causal connection set up between the
word "chicken" and that particular chicken pointed to. But the class of
chickens forms a NATURAL grouping in the world. So, the word "chicken"
referred not just to that particular chicken, but also to all members of
that particular group. That's how "chicken" got to refer to (the class of)
chickens.
(Implausible isn't it?)
Anyway, can we do the same with "set" and sets? Probably not, because we
cannot point at sets.

7(b) Use theory (Wittgenstein, Quine?)
Another popular theory is the use theory of meaning/reference. The use of
language fixes its meaning. In particular, there is a part of the use of our
language which concerns which sentences we accept (as true) and utter in our
conversations, etc. Then, the meaning of words in our language fixes their
reference. So, it is because we use the language of set theory in a certain
way that the word "set" refers to sets (and not, say, to chickens). But what
way?

7(c) Implicit Definition (several philosophers of science, Ramsey? Lewis?,
Shapiro?)
Another theory of reference, popular in the philosophy of science, is the
IMPLICIT DEFINITION theory of reference. Consider "quark". We cannot point
at quarks. So, how do we refer to them? Well, we have some big scientific
theory using the word "quark". This theory implicitly defines the word
"quark". (We hope that there will be only one model up to isomorphism). This
partly explains why some philosophers want to use second-order logic -
because the categoricity results suggest a way of uniquely fixing reference
(up to isomorphism).

You can try and combine 7(a) with 7(c). Causal theory for observational
terms and implicit definition for theoretical terms. Very popular move in
philosophy of science.

7(d) Disquotational theory (Quine?)
Another theory of reference is the disquotational theory (associated with
Quine, although there is no locus classicus. "Ontological Relativity" 1969,
maybe).
The disquotation statements,

 "chicken" refers to chickens
 "set" refers to sets
 etc.

are all somehow analytically or tautologously true. To understand the word
"refers" is to understand these statements. So, there is no question of
EXPLAINING why "chicken" refers to chickens or why "set" refers to sets.
That's just like asking why everything is identical to itself.
The disquotational theory is very good, for at least 5 reasons:

(1) we can refer to anything we like - there doesn't have to be a causal
connection.
(2) it seems that it's impossible to explain in precise detail what this
alleged causal link is (Putnam has written a lot about this).
(3) we all agree that "unicorn" refers to unicorns, even though there aren't
any!
(4) the Skolem, Putnam, etc., paradoxes all disappear. "Uncountable set"
refers to uncountable sets; "constructible set" refers to constructible
sets, and so on.
(5) the disquotational theory makes "true" and "refers" more like *logical*
notions rather than physical or causal notions (so "true", "refers",
"satisfies" are analogous to "not", "there is", "for all", "and", etc.).

(As you can see, I'm rather keen on the disquotation theory of reference.
That's partly why I was interested to show a couple of years ago that when
you add all the disquotation statements to (almost) any theory T, the result
is a conservative extension of T. That sort of explains why the disquotation
statements are "analytic" or "tautologous").

As always - 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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~