> Here is where my complexity when interpreting > Aristotle's remarks arise: > > Something like > > x=y > > is called informative identity. > > But, within a formal proof, one can never > write that as an assumption. In a formal > proof, one begins with true sentences and > ends with true sentences. So all the > metalogical analysis of meaning outside > of a deduction is "armchair quarterbacking" > whereas what happens in a proof is > "regulation time".
I'm afraid I have no idea what you mean here. Natural deduction proofs involve assumptions all the darned time. For instance, one may begin the proof that sqrt(2) is irrational by:
Assume that a and b are natural numbers such that (a/b)^2 = 2.
Now, given what you say below (snipped, since I don't really understand what you're getting at and have no comment), it may be that you think such a statement involves quantifiers and so is an exception to the rule you state above. But, honestly, there is *NOTHING* in the usual presentation of natural deduction that entails an equation cannot be taken as an assumption.
Do you think that these comments of yours are widely accepted or original?
-- "Often times, when [...] looking for a sense of adventure, I'd doodle with math equations. Often too, when pressures of regular life were really bothering me, I'd go for the adventure of fiddling with math." -- James S. Harris, a man of many mathematical adventures!