Contra Equivalence?
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From: MoeBlee on 13 Aug 2010 19:36 On Aug 13, 5:59 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Assuming there's a defined unary predicate P(x), the statement > "There are infinitely many examples of P" would be written as > the formula (*)P(x): > > (*)P(x) <-> ( Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))] ) Okay, if that's the formulation you want. > Also, the only 2 relevant formal systems are {GC} and {cGC}, NOT PA. Okay, but please state these things first as part of your question or formulation. But you're not concerned at this point with the theory PA but only the language for the theory PA, right? So you want to consider the formal systems S and cS in the language of PA, where S = {F | F is provable from GC} i.e., S = {F | GC |- F} cS = {F | F is provalbe from cGc} i.e. cS = {F cGC |- F} Right? > On the issue of 2 FOL formulas being "logically equivalent (or > reducible)", the context I intend is the following. If A1 and A2 > are atomic formulas You didn't say anything about atomic formulas before. Please include what you want to include all at once so a reader doesn't have to keep getting additional updates. > each with a different non-logical symbol, > then > though in an inconsistent T, from A1 we could infer A2 and vice versa, > but we can not logically reduce say A1 to A2, the way to reduce say > ~~A to A. Yes, in an inconsistent theory T (now we're talking about theories other than the aforementioned S and cS, right?), we can prove any formula in the language of the theory. So, where A1, A2, and B are formulas in the language of inconsistent T, we have: A1 |-_T B A2 |-_T B also |-_T B. and A1 and A2 might not be logically equivalent. Okay, but what is your question or point about this? MoeBlee |