[FOM] Quine and the Principle of Abstraction

rgheck rgheck at brown.edu
Mon Sep 21 14:47:03 EDT 2009

On 09/20/2009 03:15 PM, Alex Blum wrote:
> Thank you. Two questions:
> Is Quine deriving Russell's paradox from
>    "(Ey)(x)(x \in y<-->  ~(x \in x))"
> rather than from the naive schema with a monadic predicate?
The sentence you display is one instance of the naive schema. In this 
instance, what replaces "F(1)" itself contains a binary predicate, plus 
negation. But, in "asserting" or "accepting" the schema, one thereby 
"asserts" or "accepts" all of its instances, including these. Subject, 
as you have of course been pointing out, to the restriction that no 
variable that occurs in what replaces "F(1)" can contain any variable 
that will become bound on substitution.

One could, of course, limit the schema in some way. We do precisely this 
in weak theories of arithmetic, where we accept induction but, say, only 
for quantifier-free formulae. It is obvious that if the schema is 
limited to formulae in which \in does not occur, then the result is 
consistent (though probably very weak---how weak?). I wonder if there's 
something else that is true, such as: if the schema is limited to 
formulae in which \in "occurs positively" in the usual sense, then the 
result is consistent. Or something like that. (Hey, this is 
stream-of-consciousness, so don't fry me if that was silly.)

> And, how does "~[(1) \in x]" differ from "~[(1) \in (1)]" or " ~(x\in x)"
> when it substitutes for 'Fx'?
Let me clarify something: We are substituting "~[(1) \in (1)]" not for 
"Fx" but for "F(1)", where the argument places marked by (1) are filled 
by whatever fills the argument place of "F(1)", in this case "x". If we 
tried substituting it for "Fx", then we would end up with:
(Ey)(x)(x \in y <--> ~((1) \in (1)))
which is nonsense. It can seem pedantic to insist upon this kind of 
thing, but we are in a neighborhood where pedantry is crucial.

Now, in the case we are discussing, the *result* of the substitution is 
of course the same, but the formulae are themselves different. In 
particular, only the middle one can properly be substituted for F(1) in 
the schema, and this is so even though the result happens to be the same 
in the other cases. There is no rule to the effect that substituting in 
violation of the capturing restriction must lead to nonsense. Only that 
respecting the capturing restriction won't.

If we had something like "(x)(Fx) --> (y)(Fy)", then the results would 
be different, and the violation of capturing would lead to invalidity. 
Your three cases would give:
(x)~(x \in x) --> (y)~(y \in x)
(x)~(x \in x) --> (y)~(y \in y)
(x)~(x \in x) --> (y)~(x \in x)
and only the middle one is OK.

To ride my own hobby horse a bit, one of the amazing things about Frege 
is how clear he is about this kind of thing, already in 1893. It's 
another thirty years before anyone approaches his level of clarity about 
syntax again.


Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

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