[FOM] Quine and the Principle of Abstraction

[Alex Blum]

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?
And,how does "~[(1) \in x]" differ from "~[(1) \in (1)]" or " ~(x\in x)" 
when it substitutes for 'Fx'?

Alex Blum



rgheck wrote:

>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
>
>  
>