On Tue, Jul 05, 2005 at 12:09:24PM +0200, Bruno Marchal wrote:
> >
> >How does it give the logic of "temporal knowledge"? I understand from
> >your points below, that the necessitation rule is necessary for Kripke
> >semantics, and its is clear to me that necessitation follows from
> >Thaetetus 1 & 3, whereas it doesn't follow from consistency alone (one
> >could consistently prove false things, I guess).
> Right. But then I guess you mean Theaetetus 0 and 1. We loose 
> necessitation once we just add the consistency ~B~P requirement (in 
> Theaetetus 2 and 3). For example from the truth t we can deduce BP, but 
> we cannot deduce Bt  & ~B~t nor Bt  & ~B~t & t.
> I recall:
> BP   (Theaetetus 0)
> BP & P  (Theaetetus 1)
> BP  & ~B~P  (Theaetetus 2)
> BP & ~B~P & P  (Theaetetus 3) ?

If D'P = BP & ~B~P & P, then D'P => P (ie necessitation). So it seems
it is the conjunction of truth of P that gives rise to necessitation, no?

> >
> >I still haven't figured out how to get temporality from a modal
> >logic. Sure I can _interpret_ a logic as having Kripke semantics, and
> >I can interpret the Kripke semantics as a network of observer moments,
> >with the accessibility relation connecting an observer moment to its
> >successor. However, what I don't know is why I should make this 
> >interpretation.
> Why not? It is a "natural" interpretation of S4 type of logic, 
> especially if you accept to interpret the accessibility relation as 
> relation between OMs. It is the case for any interpretation of any 
> theory. Perhaps I miss something here. Of course we could feel even 
> more entitled to take the temporal interpretation once we accept 
> Brouwer "temporal" analysis of intuitionist logic.
> Beth and Grzegorczyk have defend similar interpretations. I will come 
> back on the question of interpreting Kripke structure once I will 
> translate a theory by Papaioannou in those terms next week (after a 
> brief explanation of what Kripke structures are for the 
> non-mathematician).
> Bruno

Fair enough. It is very similar to the situation in my ontology of
bitstrings, asking how bitstrings can observe themselves.

The way I would probably phrase things is to appeal to something like
my TIME axiom as implying a relationship between observer
moments. These in turn naturally map into a Kripke structure defining
a modal logic for knowlegde contained in each observer moment. Then we
can do your Thaetetus move and so on. This is in the reverse order to
the way it is presented in your thesis,  but it makes more sense to
me. Is there some error of logic in thsi process?


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