George Levy wrote:

>Would Descartes' statement be written as :
>
>                (c -> -[]c) -> c
>
>How would you prove it? As it stands it appears to be a third person
>statement. How would you make it a first person statement with Kripke's
>logic?

About your formula "( c -> -[]c) -> c", We can say (with [] = Godel's 
Bew):

1) It is true.

2) It is trivially provable ... by the Guardian Angel (GA), because the GA
   can prove c, and so, by pure elementary propositional calculus, the GA
   can prove   # -> c  (# = any proposition). Remember the GA, alias G*, 
will
   play the role of the "truth theory" for the sound UTM's discourse.
   See below.

3) My feeling is that your intuition is correct. There is a link between 
the
   Descartes' cogito and diagonalisation (or Godel's proof). See Slezak
   1983 (ref in my thesis). Slezak idea is that we cannot doubt 
everything, 
   because by doing so we will be trapped on a indubitable fixed point. 
   I have proposed myself a refinement of Slezak idea (not in my thesis, 
   but in the technical reports).

4) Now, you are doing the same "error" as Descartes! Because, although
   (c -> -[]c) is true and provable (provable by both G* and G)
   c and (c -> -[]c) -> c are both true but unprovable (provable by G* 
only).
   So Descartes reasoning show only that c is bettable (if that is 
english).
   c does remain "ineffable".

5) As you say (c -> -[]c) and (c -> -[]c) -> c are both third person 
statement.
   I am not sure we can translate it into a first person statement. But I 
hope
   you will be convinced that some good approximation exists. 

George, I am thinking about a way to explain Godel's theorem (and Lob, 
and 
Solovay) without going to much into heavy details and without losing to 
much
rigor. Unfortunately I have not much time to really think about it now. 
I propose you read or reread, in the meantime, my old post:

          http://www.escribe.com/science/theory/m1417.html

where I present G and G* (and S4Grz). The notation are different:
Bp = []p, -B-p = <>p, T = TRUE, etc.
The difficult point is to explain how Bew(p) (modaly captured by []p) can 
be
correctly defined in the language of a sound machine or in the language
of arithmetic. 
In the (same) meantime, perhaps you can begin to study quantum logic which
is well explained by Ziegler at

           http://lagrange.uni-paderborn.de/~ziegler/qlogic.html

We will not need all details of Quantum Logic. Note that quantum logic is
a sublogic of classical logic. Traditional semantics of weak logic are
algebraic. The algebraic semantics of classical propositional is given by
the (boolean) algebra of subset of a set. A proposition is interpreted
by a subset. Negation by the complement of the subset, "&" by 
intersection,
"v" by union. TRUE by the whole set. FALSE by the empty set. You can 
verify
that all tautology are interpreted by the whole set (TRUE). For exemple
p v -p = P U cP = the whole set.
Intuitionistic logic admits a semantics given by topological spaces. 
Propositions are interpreted by open sets. The complement of an open set 
is
not necessarily open so that p v -p is not necessarily interpreted by the
whole space. So p v -p is not intuitionistically  valid.

Well quantum logic admits a semantics given by Hilbert Space (or simply
linear or vectorial space + a notion of orthogonality). A proposition
is interpreted by a subspace of dimension one. "&" is interpreted by
intersection, "v" is interpreted by the linear sum of the subspaces (the 
minimal subspace generated by the two subspaces). The negation is 
interpreted by the orthogonal subspaces. You can verify (on euclidian R3)
that p v -p is a quantum tautology. In quantum logic we lose the
distibutivity axioms. This is explained and illustrated page 92 in my 
thesis.

Intuitionist logic and quantum logic have also Kripke semantics. This
will help us to recognize their apparition in our dialog with the machine
and its guardian of truth ...

Hoping this help ...

Bruno

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