> Peter Percival wrote: >>Nam Nguyen wrote: >>> >>> I certainly meant "odd(x) can _NOT_ be defined as a >>> positive formula ...". >> >>Prove it. > > With Nam's new definition of positive/negative, I think > it's immediately provable (subject to some clarification as > to what a formula is) that odd(x) is a negative formula. > > Let even(x) <-> Ey(x=2*y). > > Assuming Nam's definition of "formula" supports the claim > that even(x) is a positive formula, then odd(x) must be > a negative formula since odd(x) is equivalent to ~even(x).
That does not match my reading of the new definition. It states that a formula is positive if it can be written in a particular form. That odd(x) can be written in at least one form that does not match the requirements for positivity does not mean that it can't be.
My first counter example, odd(x) <-> Ey[Sx=2*y] seems to me to be to as positive as Nam's version of even, but he's forgotten about that as far as I can tell.