> Ben Bacarisse wrote: >>quasi writes: >>> 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. > > But Nam has already agreed to the properties: > > (1) Every formula is either positive or negative, but not both. > > (4) If P is positive, ~P is negative. > > (4') If P is negative, ~P is positive. > > Hence, since even(x) is positive, it's automatic that odd(x) must > be negative.
The example I gave (odd(x) <-> Ey[Sx=2*y]) is clearly positive according to the definition he gave. It may well be that ~even(x) is equally clearly negative, but that just means that my odd(x) and ~even(x) are not logically equivalent. Given that we've not seen the system in full, and there even seems to be some doubt about basic permitted equivalences (see the recent exchange with Peter), I don't think one can say more at this point.
I do, however, think it's premature to agree that odd(x) must /always/ be a negative formula and more on from there.
>>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. > > Provided Nam allows the use of 'S' in formulas.
He uses 2 in his "acceptable" formula for even and what is 2 if not SS0?