[FOM] Quine and the Principle of Abstraction

Chris Gray cpgray at library.uwaterloo.ca
Wed Sep 16 10:43:03 EDT 2009

The predicate substituted for 'F' must be 'is not an element of' or 'is 
not an element of itself' and for the very reason that you site.  A 
predicate may have more occurrences of the arguments than follow the 
schematic letter it replaces.
For instance, 'knows Jones and Smith plays squash with' giving:

(Ey) (x) (x is an element of y iff (x knows Jones and Smith plays squash 
with x))

Chris Gray

Alex Blum wrote:
> Quine seems to derive Russell's Paradox from:  
>                          (Ey)(x)(x is an element of y iff Fx)
> by substituting the sentence 'x is not an element of' for 'F', to get 
> for 'Fx', 'x is not an element of x'. Methods of Logic. (Revised edition 
> '64 p.249, 3rd edition '72 p.253). But doesn't this violate the 
> restriction: "Variables free in the predicate must not be such as to be 
> captured by quantifiers in the schema into which the predicate is 
> substituted." Quine. M of L. 3rd ed., p.148?

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