On Sun, Jun 26, 2005 at 05:30:08PM +0200, Bruno Marchal wrote:
> >  
> > This reminds me of something I wanted to ask you Bruno. In your 
> > work 
> > you axiomatise knowledge and end up with various logical systems 
> > that 
> > describe variously 1st person knowledge, 1st person communicable 
> > knowledge, 3rd person knowledge etc. In some of these, the Deontic 
> > axiom comes up, which if translated into Kripke semantics reads 
> > "all 
> > worlds have a successor word" (or "no worlds are terminal").  
> > 
> I recall that for knowledge CP, philosopher asks for both CP -> P, and 
> the closure for the necessitation rule. 
> But then this means we can define "knowledge of P", CP, by BP & P. 
> And then we can interview the machine (through an infinite 
> conversation, ok, but finitely summarized thanks to Solovay's G) about 
> the logic of knowledge "CP". This gives a logic of "temporal knowledge" 
>  of a "knower" verifying the philosophers' most agreed upon definition. 

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).

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.

>  I take it as the simplest first person notion "definable" in the 
> language of the machine. 
> [Careful here: CP will appear to be only very indirectly definable by 
> the machine: no machine can give a third person description of its "CP" 
> logic! 
> The logic of CP is the system known as S4Grz. The subjective 
> temporality aspect come from the fact that on finite transitive frames 
> respecting the Grz formula the Kripke accessibility relation is 
> antisymmetric and reflexive, like in Bergson/Brouwer conception of 
> time. See perhaps: 
>  van Stigt, W.?P. (1990). Brouwer's Intuitionism, volume?2 of Studies 
> in the  history and philosophy of Mathematics. North Holland, 
> Amsterdam. 
>  Boolos, G. (1980b). Provability in Arithmetic and a Schema of 
> Grzegorczyk. Fundamenta Mathematicae, 96:41-45 
>  Goldblatt, R.?I. (1978). Arithmetical Necessity, Provability and 
> Intuitionistic Logic. Theoria, 44:38-46. (also in Goldblatt, R.?I. 
> (1993). Mathematics of Modality. CSLI Lectures Notes, Stanford 
> California). 
> See also http://homepages.inf.ed.ac.uk/v1phanc1/dummet.html 
> Note that BP -> P is equivalent to ~P -> ~B~ ~P, and if that is 
> true/provable for any P, then it is equivalent to P -> ~B~p, so BP -> 
> P, as axioms, entails BP -> ~B~P (the deontic formula). But, by 
> incompleteness the reverse is false. 
> Now you were just pointing on tis little less simple definition of 
> first person based on the deontic transformation. This one has been 
> studied in my thesis, so I have only my papers in my url for 
> references). Here a new logic is defined by DP = BP & ~B~P. It is not 
> used to define a first person knower, but more a first person plural 
> gambler. The logic of DP loses the necessitation rule and loses the 
> Kripke semantics, but get interesting quasi-topological spaces instead. 
> A "immediate time" notion (re)appear though the combination of the two 
> ideas: define D'P by BP & ~B~P & P. 
> Do you you grasp the nuance between 
> BP   (Theaetetus 0) 
> BP & P  (Theaetetus 1) 
> BP  & ~B~P  (Theaetetus 2) 
> BP & ~B~P & P  (Theaetetus 3) ? 
> Only Theaetetus 1 gives rise to a "temporal subjectivity". 
> (Now if you interview the machine on *comp* itself, by limiting the 
> atomic P to DU accessible truth, the Theaetetus 1, 2 and 3 all leads to 
> different "quantum logics". In my thesis of Brussels and Lille I have 
> been wrong, I thought wrongly that the pure (given by Theaetetus 1) 
> first person collapse with comp). 

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A/Prof Russell Standish                  Phone 8308 3119 (mobile)
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