[FOM] Odd Thought About Identity

Alex Blum blumal at mail.biu.ac.il
Thu May 21 03:56:01 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 (if y does not become bound where x was free) 
and any free occurences of y in the formula may be replaced by x (if x does not become bound where y was free).

This would immediately yield from 'x=y and Rxy', 'Ryx'. But '~Ryx'. Hence '~(x=y) v ~Rxy'. And 'Rxy', hence '~(x=y)'. Thanks to John Corcoran
for prompting a correction in the deduction.
Alex Blum

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