[FOM] Odd Thought About Identity

rgheck rgheck at brown.edu
Sat May 16 12:46:37 EDT 2009

Neil Tennant wrote:
> The usual formulation of substitutivity of identicals in natural 
> deduction is
>          P          t=u
>          ________
>                 Q
> where P and Q become the same sentence upon uniformly replacing 
> occurrences of t by occurrences of u.
> This allows the instance
>         Rab     a=b
>         _________
>                Rba
The way I usually see it is that Q arises by uniformly replacing (some)
occurrences of t by occurrences of u. So, well, I'm not sure if  Neil's 
statement of the rule permits the instance he mentions or not. But it 
would not be permitted under my form of the rule.

Perhaps one thing worth saying here is that my form of the rule is the 
one that corresponds to what you get in second-order logic, if you take 
Leibniz's Law in quantified form:
	(F)(x)(y)(x = y --> Fx <--> Fy)
and then get instances via comprehension. So perhaps the reason for 
taking the schema in its usual form, which would not permit what Neil 
wants (directly), is that it's what corresponds to what we really mean, 
which is the quantified form.


Richard G Heck Jr
Professor of Philosophy
Brown University

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