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I have problems understanding the argument against unrestricted quantification based on Russell’s paradox, which does not work with sets, but instead with semantic interpretations (as objects).

The general schema of Russell’s paradox is:

∃y ∀x (R(x, y) ↔ φ(x))

with the contradiction resulting when setting φ(x) to be ¬R(x, x).

It seems the goal is to show that when attempting unrestricted quantification, objects are involved that were not originally part of “everything”.

In one version (p. 8) it is stated as:

Say that P applies to an object x according to a semantic interpretation i if and only if x is in the extension assigned to P by the interpretation i. In symbols:

i ⊨ P(v) [a(v/x)],

where a is any variable assignment suitable for i, and a(v/x) is just like a except possibly for the fact that the singular variable v denotes x. The naive principle of interpretations is obtained by instantiating R(x,y) with ‘P applies to x according to y’.

The principle tells us that for any formula φ(x), there is an interpretation according to which the extension of P is given by φ(x):

(I) ∃i ∀x (P applies to x according to i ↔ φ(x)).

The problematic instance results from taking φ(x) to be ‘P does not apply to x according to x’:

(I*) ∃i ∀x (P applies to x according to i ↔ P does not apply to x according to x).

If the range of the universal quantifier is all inclusive, or at least contains every interpretation, then we obtain a contradiction:

(I**) P applies to i* according to i* ↔ P does not apply to i* according to i*, for some interpretation i*.

Is there any way to make this a bit more accessible?

I find the part “P does not apply to x according to x” especially odd. In most cases x would not be an interpretation – what does “apply … according to x” then even mean?

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  • You're on the right track. Williamson essentially used quantification of semantic interpretations (as objects) as a form of proof by contradiction. He assumes first we can do this kind of unrestricted quantification and follows Russell's line, arrives at a similar logic contradiction. If you view set theory as a theory of everything (a semantic interpretation is nothing but an extensional set (function) with certain features), then this proof is nothing but a simple application of R's paradox. So Williamson conflates naive comprehension with absolutism while absolutists may reject like Zermelo
    – Double Knot
    Commented Nov 28, 2021 at 3:52
  • Links are not an acceptable way to pose a question. You need to quote or put into text your question.
    – Swami Vishwananda
    Commented Nov 28, 2021 at 5:56
  • ...that's why later Zermelo proposed axiom scheme of specification replacing the naive axiom of comprehension as ways to define or construct all sets, then we can safely quantify over the domain of all sets in an unrestricted manner, and note that the domain of all sets by no means a set itself thus free of any Russell like logical paradox. Some metaphysics need unrestricted quantification's expressivity, for most practical problems it's almost useless except maybe providing some psychological confidence with one's theory about a particular practical field...
    – Double Knot
    Commented Dec 17, 2021 at 23:54

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The author is quantifying over interpretations, that's the whole point. So when he says P applies to x according to i he is saying just that : Predicate P is true of the interpretation "x" according to interpretation "i".

Naturally, then, if quantification is unrestricted then we can say P is true of "i*" according to "i*" if and only if P is not true "i*" according to "i*".

To be technical you can say P applies to x (as opposed to is true of x) because we are quantifying over interpretations.

"I find the bit P does not apply..."

Do you find x ∉ x odd? What about all those people that do not shave themselves?

It's the same maneuver, P does not hold for interpretation "i" according to "i".

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  • Do you find x ∉ x odd I do not, but just because we’re talking only about sets in which case “∉” always has a meaning.
    – viuser
    Commented Dec 18, 2021 at 0:29
  • @viuser "i" is a just a meaningless formula at this point which means we can interpret it however we want. Like, AxPx, can be interpreted as: some bikes have 8 wheels. It's Nonsensical, but it's a semantic interpretation nonetheless.
    – Bertrand Wittgenstein's Ghost
    Commented Dec 18, 2021 at 17:08

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