# [FOM] Odd Thought About Identity

Neil Tennant neilpmb at yahoo.com
Wed May 13 20:31:15 EDT 2009

```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

So the relevant fragment of the natural deduction sought is

:        ____(1)

Rab      a=b
:                    _________

~Rba                    Rba
__________________

#
______(1)

~a=b

Neil Tennant

________________________________
From: Richard Heck <rgheck at brown.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Tuesday, May 12, 2009 2:18:48 PM
Subject: [FOM] Odd Thought About Identity

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)
(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.

Richard

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