[FOM] Odd Thought About Identity

[Allen Hazen]

The semantics for modal logic derived from David Lewis's Counterpart Theory
(cf. Lewis, Counterpart Theory and Quantified Modal Logic, in Journal of
Philosophy, 1968; repr. with addendum in his "Philosophical Papers, vol. I")
does not validate full standard substitution of identicals in modal
contexts, but does validate something that would give Heck's students what
they thought they had: from x=y and A(xy) to infer A(yx), where the latter
formula has free occurrences of y in all and only the positions where A(xy)
has free occurrences of x and vice versa.  Kit Fine discusses this
phenomenon somewhere.

Allen Hazen
Philosophy Department (PASI)
University of Melbourne


On 13/5/09 4:18 AM, "Richard Heck" <rgheck at brown.edu> 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?