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). > -- *PS: A number of people ask me about the attachment to my email, which is of type "application/pgp-signature". Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics 0425 253119 (") UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 ----------------------------------------------------------------------------
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