[FOM] Quine and the Principle of Abstraction

rgheck rgheck at brown.edu
Fri Sep 18 22:47:40 EDT 2009

On 09/17/2009 03:42 AM, Alex Blum wrote:
> There is no problem with the number of occurrences of the variable in
> the substituted predicate,but there is a problem  in bringing in a
> variable free in the substituted predicate which will be bound in the
> substituted schema. An example from Quine, M o L. 3rd ed. pp152-3:
> Substitution of 'Gx' for 'F' is improper in :
>   if Fy then (Ex)Fx  (valid)
> for it would yield:
>   If Gxy then (Ex)Gxx (invalid)
This is correct, but it is not relevant.

In the way you are thinking of it, the scheme of naive abstraction is:
(Ey)(x)(x \in y <--> Fx)
Quine is suggesting that the predicate F(1) be replaced by: ~[(1) \in 
(1)], to yield:
(Ey)(x)(x \in y <--> ~(x \in x))
Here, I am using Quine's device of "placeholders" to indicate the 
argument-places of F(1). You will note that the (complex) predicate that 
replaces F(1) does not contain any occurrences of "x", hence the bar on 
capturing does not apply. If F(1) had been "~[(1) \in x]", then it would.

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

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