Nam Nguyen wrote: > On 23/08/2013 7:51 AM, Peter Percival wrote: >> quasi wrote: >>> 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). >>> >>> So, conceding that, where does he go with it? >>> >>> quasi >> >> But who's to say that there isn't a positive formula phi with one free >> variable x such that phi(x) <-> odd(x)? > > >> phi may be taken as a definition of odd. > > Sure. If in the language of arithmetic - or L(*), you defined: > > odd(x) <-> Ey(x=2*y) > > then _this_ odd(x) would be positive as expressed in _both_ languages.
Then why did you write: > I certainly meant "odd(x) can _NOT_ be defined as a > positive formula ...".
> That still doesn't "invalidate" the _definitions_ . >
-- Sorrow in all lands, and grievous omens. Great anger in the dragon of the hills, And silent now the earth's green oracles That will not speak again of innocence. David Sutton -- Geomancies