[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.
rh
--
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University
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