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Re: Matheology S 224
Posted:
Apr 14, 2013 11:19 AM
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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 ...]
--
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There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI
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