[FOM] Odd Thought About Identity

[Harvey Friedman]

> On May 12, 2009, at 2:18 PM, Richard Heck wrote:
>
> 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?

This is merely a simple comment.

In, e.g., Enderton's book, Intro to Math Logic, there is the following  
formulation of the axioms of identity:

x = x
x = y implies (A implies A'), where A is atomic and A' is obtained  
from A by replacing some occurrences of x by y.

So this is in this direction. Obviously, we can use

x = x
x = y implies (A implies A*), where A is atomic and A* is obtained  
from A by simultaneously replacing all of the occurrences of x,y in A  
by either x or y.

Harvey Friedman