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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Received on Tue Jul 05 2005 - 04:56:44 PDT