Re: Kripke semantics

From: George Levy <>
Date: Sun, 08 Apr 2001 14:41:24 -0700

Marchal wrote:

> Hi George,
> I finish my post yesterday a little too quickly.
> I said:
> >I hope your binder will not explode.
> >My next post will anticipate toward the end of my proof to glimpse
> >the "quasi-appearance of Hilbert Space" when we explain the UDA TE
> >to the guardian angel of the sound machine.
> >
> >And, just because you promise me a prize for deriving SE from
> >the "psychology of machine" I tell you that I have decided
> >to call the modal formula (the one for the symmetrical frame):
> >
> > p->[]<>p,
> >
> > the little abstract Schroedinger Equation (LASE),
> >
> >as I have called before the (godel-like) formula
> >
> > <>p -> -[]<>p
> >
> > the first theorem of machine's psychology. (FTMP)
> >
> >And our goal is to find a natural bridge from FTMP to LASE.
> >
> >I must say that I have believe for a quite long time that this was
> >impossible. But then when you take the definition of knowledge
> >belief, observation/perception in Plato's thaetetus, then the
> >aritmetical translation of the UDA TE will lead us directly toward
> >the solution.
> >And so you will be obliged to give me the prize (at least
> >a little abstract price!).
> Of course this is not entirely correct. You should "give me
> the prize" only:
> 1) when you will see what I mean when I say that
> <>p -> -[]<>p is a fundamental theorem in machine's psychology.
> + (above all):
> 2) when you understand why I dare to call the formula p->[]<>p the
> little abstract schroedinger equation.
> + (of course):
> 3) when it will be clear how I translate
> the UDA TE and how that translation isolates a derivation of
> LASE from FTMP.

As I said in my earlier post, 1) seems inconsistent, 2) seems obvious.
3) I do not recall the acronyms.

Please refresh my memory with regards those acronyms UDA and TE

> I explain informally a little bit.
> ABOUT 1):
> When the box []p is interpreted in english as provable(p),

You are saying that "p is true in all worlds" is identical to "p is
provable." You have lost me....

> i.e.
> provable by a honest UTM, the <>p, which is -[]-p, can be read as
> p is consistant. This follows from the fact that if -p is not provable
> then you can add p to the set of axioms used by the machine without
> being lead to a contradiction.

Great adding an unprovable statement to a set of axiom is OK. But how do
you know it is not provable until you actually attempt to prove it... and
how long will you attempt to do so?

> (Indeed if the machine derives the
> false from -p, then the machine derives (-p -> FALSE), which is --p,
> which is p). More directly <>TRUE is the same as -[]-TRUE = -[]FALSE,
> = I do not prove the FALSE = I am consistent.
> So <>p -> -[]<>p is just a modal form of a generalisation of Godel's
> second incompleteness theorem, and in English you can read that
> formula in the following way. if p is consistent then I cannot prove
> that p is consistent.
> When the machine proves the particular case (<>TRUE -> -[]<>TRUE),
> that is Godel's second theorem, it is as if the machine was telling
> us: if I am consistent I cannot prove it.

Consistency ("logical thinking") is certainly a necessary condition for
consciousness ("I think" a la Descartes)

> A more psychological reading of that formula is, by identifying
> (audaciously perhaps) consistency with consciousness (or awakeness)
> you get "if I am conscious then I cannot prove it.

but consistency is not a sufficient condition for consciousness. Or is it?
What else would be required?

> Note that any formula with the form <>p can be read there is an
> accessible observer-moment (world) with p true at it. So when the
> talk about <>p, the machine talk about a consistent extension of
> herself.
> To sum up, the 1) above is linked to Godel's theorem, seen as a
> psychological limitation of machine. Later I will be hopefully
> a little more rigorous about the link consciousness/consistency.
> ABOUT 2):
> Why do I consider the formula p->[]<>p as an abstract form
> of SE ?
> Surely that deserves some words of explanation. Unfortunately
> to explain this, we must leave the cocooning logic of plato
> heaven (classical logic) for the jungle of what is called
> The theorems of a weak logic makes a subset of the theorems
> of classical logic.
> We will meet essentially two principal weak logics:
> intuitionistic logic and quantum logic.
> For exemple a typical classical tautology which is not a theorem
> for intuitionnistic logic is the principle of excluded middle
> p v -p
> And a classical tautology which is not a theorem of quantum logic
> is
> p & (q v r) <-> (p & q) v (p & r)
> That is, in quantum logic there is a failure of the distributivity
> axioms.
> It has been proved that both intuitionist and quantum logic
> doesn't have truth table semantics. But modal logics can help
> us to handle those weak logics without leaving plato heaven.
> In particular the modal logic, known as B, with the axioms
> []p ->p, p->[]<>p and with the rule of modus ponens and necessitation
> has a non trivial relationship with quantum logic.
> In fact we have a representation theorem (by Goldblatt) which
> says that when B proves []<>A, QL proves A. (I simplify a
> little bit). Sometimes []<>A is called the quantization of A.
> Isolating the formula p->[]<>p is also a good step toward
> defining a measure on the relative set of consistant extension.
> ABOUT 3). (later).
> I stop here. I hope you are not overwhelmed by the information.

Yes I am.... but as the saying goes "l'appetit vient en mangeant." We
shall see.
Forgive me for not being too prompt....Responding to your posts is not
simply a matter of making a few cute remarks....We are certainly going in
the right direction....I think that the basic ideas are probably very
simple... the hard part is to communicate them.

> Here I have just try to anticipate a little. I must say that I
> not to busy now, but unfortunately I will be quickly more and more
> busy. So I give you something to eat for a long period.
> (except that I will try to say a little more about 3) still
> this week).
> Nevertheless, don't hesitate to tell me "enough!" or one of your
> "hmmmmmmmmmmmmmmmmmm". I would understand. Of course it is not
> so easy to understand my work which is not only transdisciplinar
> but also goes accross very different kind of logics.
> Bruno

Received on Sun Apr 08 2001 - 14:44:42 PDT

This archive was generated by hypermail 2.3.0 : Fri Feb 16 2018 - 13:20:07 PST