On 14/04/2013 9:19 AM, Nam Nguyen wrote: > On 14/04/2013 12:44 AM, Nam Nguyen wrote: >> On 13/04/2013 7:10 PM, Jesse F. Hughes wrote: > >> >> Now that that has been spelled out, however unnecessarily, what's next? >> >> Can you or they give me a straightforward statement of understanding >> or not understanding of Def-1, Def-2, F, F' I've requested? >> > > I don't remember if I asked Chris Menzel directly or he might have just > been in the post, but once (iirc) I wondered if there is a way to > express something like "There are infinitely many individuals" _without_ > any non-logical symbols. > > I did define the "Mx (Many quantifier) and 0x (Null quantifier)" in: > > https://groups.google.com/group/sci.logic/msg/8fd316ddcfc09e5c?hl=en > > <quote> > > (1) Mx[P(x)] df= There exist more than one x such that P(x). > (2) 0x[P(x)] df= There exists no x such that P(x). > > </quote> > > And in the post: > > http://groups.google.com/group/comp.ai.philosophy/msg/58615203416c4d7e?hl=en > > > I did define: > > - The "I-form (Inductive) of infinity expression": > > (I)P(*) <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ Ez(y = x + Sz))] > > - The "aI-form (anti-Inductive) of infinity expression": > > (aI)P(*) <-> Ex[P(x)] /\ AxEy[P(x) -> (P(y) /\ (x < y))] > > The long and short of it I've been frustrated that the Many Quantifier > Mx doesn't make a lot of logical sense: how many should be logically > considered as "many"? But now I see in Mx and 0x (The Null quantifier) > a quite relevancy to the relativity of the truth values of cGC and its > negation ~cGC. > > The difficulty in the Mx quantifier is actually a reflection on the > need of introducing to FOL new logical quantifiers: > > - Ix (There are infinitely many x's) > - Fx (There are finitely many x's) > > Where some of the _traditional_ rules of inference on these two new > quantifiers are: > > - Ix <-> ~Fx /\ Fx <-> ~Ix > - Ix -> Ex. > > And of one of the new "Anti-Inference" rules is: > > - From Fx one shall _not_ infer Ex. > > More properties and rules might be forwarded, but these definitions > will bring more crisp the reasons why the there exists the relativity > of the truth values of cGC and its negation ~cGC > > [To be continued ...]
Apropos out of nothing, the caveat here is that the issue of the relativity of the truth value of cGC in the naturals is an _independent_ issue from the suggested new FOL with the 2 new quantifiers Ix and Fx.
And one doesn't have to discuss about these 2 new quantifier in discussing the issue of cGC.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.