[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)
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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