Le 05-juil.-05, à 09:39, Russell Standish a écrit :

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) aboutthe logic of knowledge "CP". This gives a logic of "temporalknowledge"of a "knower" verifying the philosophers' most agreed upondefinition.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) ?

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 itssuccessor. However, what I don't know is why I should make thisinterpretation.

`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

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 theKripke semantics, but get interesting quasi-topological spacesinstead.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 theatomic P to DU accessible truth, the Theaetetus 1, 2 and 3 all leadstodifferent "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).

http://iridia.ulb.ac.be/~marchal/