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) 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).


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 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


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).


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


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