[FOM] Quine and the Principle of Abstraction

Alex Blum blumal at mail.biu.ac.il
Thu Sep 17 03:42:47 EDT 2009

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)
Alex Blum

Chris Gray wrote:

>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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>FOM at cs.nyu.edu

Alex Blum,Department of Philosophy,Bar-Ilan University   
Ramat-Gan 52900,Israel 
Home: tel 972-3-5352386
12 Rotem St.,Ramat-Gan 52644,Israel


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