[FOM] Odd Thought About Identity]

Alex Blum blumal at mail.biu.ac.il
Fri May 15 06:07:49 EDT 2009

Richard Heck wrote:

>This came up in my logic final. There was a deduction in which one got 
>to here:
>    Rxy . ~Ryx
>and needed to get to here:
>    ~(x = y)
>What a lot of students did was this:
>    (x)(y)(x = y --> Rxy <--> Ryx)
>This does not, of course, accord with the usual way we state the laws of 
>identity, but it struck me that it is, in fact, every bit as intuitive 
>as the usual statement. Which, of course, is why they did it that way.
>It wouldn't be difficult to formulate a version of the law of identity 
>that allowed this sort of thing. But I take it that it would not be 
>"schematic", in the usual sense, or in the strict sense that Vaught 
>uses. I wonder, therefore, if a logic that had a collection of axioms of 
>this sort might not yield an interesting example somewhere. Or if there 
>isn't a similar phenomenon somewhere else.
>Anyone have any thoughts about this?

If x=y, then any free occurences of x in a wff F may be replaced by y 
and any free occurences of y  in the formula may be replaced by x.

This would immediately yield from 'x=y' and 'Rxy' , 'Ryx', but ' 'x=y' 
and 'Rxy' also yield '~Ryx'. Hence '~(x=y) v ~Rxy' But 'Rxy' Hence '~(x=y).
 Alex Blum


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