[FOM] Odd Thought About Identity
Richard Heck
rgheck at brown.edu
Tue May 12 14:18:48 EDT 2009
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?
Richard
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