Wolfgang Schwarz

What does Russell's Paradox Teach in Semantics?

On Friday, I wrote:

Conclusion 2: If we want to avoid Bradley's regress, there is no reasonable way to defend the principle that every meaningful expression of our language has a semantic value. (Russell's paradox is an independent argument for the same conclusion.)

Today, I was trying to prove the statement in brackets. This is more difficult than I had thought.

Semantic paradoxes usually (always?) arise out of an unrestricted application of schemas like

'p is true' iff p;

'F' is true of x iff F(x);

't' denotes x iff t=x.

The paradoxes prove that these schemas have false instances and therefore aren't generally correct. (Maybe they are correct only for a certain part of our language, the relevant 'object language'; or maybe they are correct only when "iff" is replaced with some non-standard operator; Anyway, the important thing is that, as they are, they are not generally correct.) So they can't be used to define the concepts "true", "true of", "denotes", etc.

Now if, as I have claimed, Russell's paradox teaches a general lesson about semantic values, the paradox has to arise out of certain assumptions in semantics without application of any provably incorrect schema. And this is what I found to be difficult. Clearly,

1) Every predicate expression has a semantic value

in itself doesn't give rise to a contradiction. There's nothing wrong with assigning the moon to every predicate and calling it the predicate's "semantic value". We get closer to a contradiction by strengthening (1) to a semantic version of Axiom 5:

2) If F is a predicate expression and t a singular term, then the truth of F(t) depends only on the semantic values of F and t.

Now a contradiction can be derived, but only with the help of heavy second-order logical machinery. Given the controversial nature of second-order logic, this is still not very interesting as a general lesson. So (2) must be strengthened further by including some kind of substitute for the comprehension principle:

3) If F is a predicate expression and t a singular term, then the truth value of 'F(t)' is determined by a general rule of the form: 'F(t) is true iff the value of F blah the value of t'.

Using an irritating amount of corner quotes, I think a contradiction can now be derived, provided the further assumption that

4) Any sentence p is true iff 'not p' is not true.

Here is the derivation:

Reading (3) from right to left, we get:

i) 'x blah y' is true iff x is the value of a predicate expression F such that 'F(y)' is true.

Let R be the value of the expression 'not: x blah x', which must exist by (1) and (3). So by (i),

ii) 'R blah R' is true iff R is the value of a predicate expression F such that 'F(R)' is true.

Since by definition, R is the value of 'not: x blah x', we get by (2),

iii) For every expression F such that R is the value of F, 'F(R)' is true iff 'not: R blah R' is true.

From (ii), (iii) and the fact that R is the value of 'not: x blah x' it follows that

iv) 'R blah R' is true iff 'not: R blah R' is true.

By (4), this is impossible.

I'm not fully convinced that this is valid. Even if it is, (3) is already much stronger than I thought would be necessary. (And (4) of course stands and falls with the assumption that having sex for non-procreative reasons is naughty.)

[Update 04-09: I now believe that the derivation is really invalid. I think it uses both the principle that the value of 'the value of x' = the value of x and the principle that if p, then 'p' is true. Since these are exactly the kinds of principles I didn't want to use, the derivation is probably useless.]