At 09:38 AM 11/30/2004, you wrote:
At 13:40 26/11/04 -0500, Hal Ruhl wrote:What does "logically possible" mean?
In the above I meant in the context of the larger phrase of: "logically possible worlds".
In the following call an individual [Ai,Dj] pair logic system Ln where "i", "j", and "n" can go from 1 to an uncountable infinity and all possible [Ai,D,j] pairings are considered.
A proposition P is logically possible, relatively to 1) a consistent set of beliefs A 2) the choice of a deduction system D (and then consistent means "does not derive 0=1).
if the negation of P is not deductible (in D) from A.
So in the larger phrase rather than dealing with a proposition P in relation to Ln I am exploring the range of [Ai,Dj] pairs that would be valid descriptions of "worlds". Call this sort after ensemble "W".
The further issue is induction and whether or not it fails for a particular Ln.
Now suppose that "belief" set Ai includes the "belief" that Ai, and Dj for j over some range are both subject to random input from outside the system.
I see no reason to exclude the Ln which have such an Ai from being a valid description of a World. It is just an explicit expression of incompleteness rather than an implicit one. Thus there could be two subsets of Ai in W.
Is there any reason why the ensemble W can not for reasons of its own structure include Ai from both subsets and also insist that the incompletenesses both implicit and explicit be progressively resolved? I know of none and to avoid a "selection" within the W it would seem that this arrangement is unavoidable.
Thus induction would fail for all worlds in W because the logical foundation for all worlds would be constantly shifting from one Ln to another.
Concerning many theories, to say that a proposition (or a set of propositions) A is logically possible is the same as saying that A is consistent (i.e you cannot derive 0 = 1 from it),
When talking of descriptions of worlds - in such a venue consistency would only be applicable to individual states [if at all] and not to successions of states. The question then is can the All [which contains W] contain self inconsistent states such as one with a correctly and completely assembled two wheeled tricycle or a cat that is both alive and dead or the same thing having two valid sets of coordinates? Now the All is complete so it is internally inconsistent so I see no way to argue against the presence of such states founded on inconsistent Ai.
or saying that A has a model (a reality, a mathematical structure) satisfying it.
It seems that the idea that mathematical structures are actually consistent is nice but lacks any basis.
To help place my model in context with the above:
A core idea is the definitional pair relationship. The [All,Nothing] pair is unique in being inherently unavoidable but still summing to no information. Thus it has no initiation and no end.
Another core idea is: Is there a meaningful question the Nothing must resolve? The answer to this is: Yes there is: The Nothing either continues [persists], or it does not. The answer must be inherent in the information within the Nothing but there is none in there by definition. Therefore the Nothing is incomplete - it can not resolve any meaningful question. But in this case it must do so. The only reservoir of information is the All. Therefore it must breach the barrier between itself and the All. In doing so it losses contact with what it was [an Ln shift] and becomes an evolving [including successive Ln shifts] - a multiverse - within the All. Since the [All,Nothing] is as above an unavoidable definitional pair a "new" Nothing simultaneously replaces the old one. The cycle repeats. The cycle always was and always will be and the All contains an infinite number of these Somethings all evolving towards completeness. This produces waves of "physical reality" passing through a random sequence of states [including Ln shifts as per above]. The Somethings evolve because of their own incompleteness and the need for no selection no net information within the All. The evolution must be random because of no selection and the All is internally inconsistent since it is complete.