[FOM] Odd Thought About Identity
rgheck at brown.edu
Wed May 13 19:04:59 EDT 2009
It seems maybe my previous post wasn't clear.
The usual formulation of the Leibnizian law of identity takes it to be:
(x)(y)[x = y --> A(x) <--> A(y)]
where this is a schema, and A(x) is any formula you like, subject to the
usual sorts of restrictions, viz, x is free for y. (Not that it matters
here, but of course "<-->" can be weakened to "-->".)
The observation was that this
(1) (x)(y)[x = y --> Rxy <--> Ryx]
is not an instance of that schema. Of course, it can be derived from the
(2) (x)(y)[x = y --> Rxx <--> Rxy]
(3) (x)(y)[x = y --> Rxx <--> Ryx]
but what struck me was that, intuitively, there really isn't any
difference between (1), on the one hand, and (2) and (3) on the other.
My students, almost all of whom appealed to (1) in their proofs, were
going on instinct, and their instincts are right. Moreover, the sort of
thing Leibniz (and he's hardly alone) says in justification of the
schema (not that Leibniz knew about schemata....) seems quite naturally
to justify (1) just as well as it justifies (2) and (3). And it's easy
enough to state an axiomatization of the laws of identity that would
permit (1). However, such an axiomatization is not "schematic" in what I
take to be the standard sense of that term, viz, the sense used by Vaught.
Obviously, this is of limited significance, but it does seem to me
possibly to point to an oddity, maybe even a shortcoming, in our usual
notion of a schema.
Richard G Heck Jr
Professor of Philosophy
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