[FOM] Odd Thought About Identity

[Paul Hollander]

The schema Fx stands for any open sentence containing x as a variable. 
So I see no problem with Rxy etc. so far as schemata go, because they 
are just open sentences containing x as a variable.

Since Rxy & ~Ryx entails ~(Rxy <--> Ryx), the inference from Rxy & ~Ryx 
to ~(x = y), if it proceeds by conditional proof, appeals to the 
non-identity of discernibles, i.e., to the contrapositive of the 
indiscernibility of identicals, that is, to (x)(y)(~(Fx <--> Fy) --> ~(x 
= y)).  Therefore, when constructed as a conditional proof, this is NOT 
an appeal to identity at all, but rather to non-identity.

But, if one constructs a proof by reductio ad absurdum, one appeals to 
the indiscernibility of identicals, because one proceeds by inference 
from x = y to Rxy <--> Ryx, which contradicts the premiss Rxy & ~Ryx.

So we have a nice duality here between conditional proof with appeal to 
non-identity, on one hand, and reductio ad absurdum with appeal to 
identity, on the other hand.  The logical relation between the two is 
contraposition between the non-identity of discernibles and the identity 
of indiscernibles.

-paul