Marchal wrote: > 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 ... > Thanks Bruno.... this is much more than I bargained for... I can barely keep afloat...I have a lot of homework to do. I am also very busy these days with my regular work so I don't mind going slow. Just a couple of questions: 1) What is Godel's "Bew". Probably not something he drank. 2) You say : >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". I am not sure if I understand. You are saying that c and (c -> -[]c) -> c are both true but unprovable? So the statements "I am" and "I think therefore I am" is not provable? You say: the GA can prove # -> c (# = any proposition). Interestingly the reverse c -> # means that if there is a consciousness, then the plenitude exists, I think, or maybe more appropriately from the relativistic point of view, if "I am" then "the plenitude is." As you recall, a few months ago, I gave a "short" proof to derive the existence of the plenitude using Descartes "cogito", the Anthropic principle (I am therefore the world is) and the principle of sufficient reason ( if there is no reason for this world, W, then all the worlds, #, are). Basically what I conjectured was (c -> -[]c) -> c -> W -> #. You have shown that the first step is trivial. It would be interesting to see if we can derive the Anthropic principle. c -> W and the principle of sufficient reason W -> # from logic. The question is how to define W, "that specific world that sustains the existence of c." What is the meaning of "sustain the existence of." Surely, W, will have a "thickness" or uncertainty associated with it. And now we go full circle back to the Schroedinger Equation. Take your time.... George

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