> On 23/08/2013 7:49 AM, Ben Bacarisse wrote: >> quasi <email@example.com> 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. >> >> My first counter example, odd(x) <-> Ey[Sx=2*y] seems to me to be to as >> positive as Nam's version of even, > > Right. But remember that odd(x) is a _non logical_ expression, hence > it does matter (on it being positive or negative) whether or not, say, > 'S' is part of a language. > > In the language L1(S,*), both even(x) <-> Ey(x=2*y) and odd(x) <-> > Ey[Sx=2*y] are positive, while in L2(*), only even(x) would be.
How is 2 defined in L2(*)? What are the axioms for *? Don't both use S?