[FOM] Odd Thought About Identity

[Neil Tennant]

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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