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
Received on Tue May 08 2001 - 07:30:48 PDT