Re: Belief Knowledge
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
Re: Belief Knowledge
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
Re: Belief Knowledge
Robert W. wrote: Let's be clear on the meaning of *logic*. Logic as a reasoning power existed before the discipline of formalized logic. One can have a very powerful logical reasoning facility without having attented a course on the subject. I agree. I guess some misunderstandings came from that double sens. I do use logic as a branch of math. To be sure I don't believe that logic is a special branch capable to provide foundation for the other part of math. This is the philosophical logicist thesis which has been abandonned since Godel's result. Quite the contrary, big part of math are used in logic. Having said all this, I must say that I believe logic can have important application in biology, psychology, theology, philosophy, artificial intelligence, computer science, etc. Bruno
Re: Belief Knowledge
Robert W. wrote in part: One must realize in the attempt to constrain common everyday experience to a finite conceptual space, that something will be lost in the translation. Brent Meeker answered in part (to Robert W., in the invisible post): Of course, ineffable mystical experiences will be left out. Is not consciousness the first ineffable experience? At least it seems to me that consciousness in unprovable, uncommunicable. You can just hope people understand what you say by having had similar experiences. This is one of my motivation for thinking that consciousness and consistency share the formula: -[]c Saying just that c (consistency, consciousness) is not provable, not finitely communicable, ... ineffable? Goedel's second theorem: c - -[]c(c = TRUE, or -[]FALSE). Bruno
Re: Belief Knowledge
Robert W. wrote: [...] Logic is a powerful tool for analysis. Some use it intuitively, you people seem to have mastered formalized, symbolic logic. That's great. Logic is just a branch of mathematics which studies discourse and their interpretation. My point has to do with the way you folks seem to be trying to understand *everything*. Logic will always play a powerful role in understanding and analysis. I am basically trying to say, there are ways of seeing and understanding that transcend sequential thinking. Most discourses are sequential but the thinking behind does not need to be sequential. The semantics are in general not sequential. Maticulously wondering a search space, with logic or any other method, only reveals what's in that space. It does not help one see outside of the space. Recall I have until here give only an informal (although persuasive IMO) argument showing that if we can survive with a digitale brain then physics transform itself into a branch of machine psychology. UDA = UDA http://www.escribe.com/science/theory/m1726.html And currently, I have just proposed to some to explain the technical part of my thesis where I will explicitely show how quantum logic in the discourse of the introspective UTM. I' dont feel myself searching in a space. I'd love to be expert enough in logic and mathmatics to demonstrate all the brick-walls I see intuitively, unfortunately, I am not now, or likely to be good enough in formalized systems of logic to do so. It is true that logic is poorly taught. But then that is the reason why I propose to take it at the beginning. Of course that demand some works. I was hoping to call attention to other facilities we all posses to expand understanding in areas that seem to defy understanding. Thank you. Such facilities are welcome, although they are in general much more difficult to communicate. Read the UDA. It is IMO such a facililities. The logic is only for those who have grasped UDA and want to see the emergence of the quantum in machine's mind. To be honest I think you were also rather hardhearted with Brent Meeker's post. That post was indeed indirectly pointing to the very reasonable critics which can be adress to Thaetetus definition of knowledge (true and justify) which entails we can know think for bad or reason. To introduce causality in the definition of knowledge is a move which will be forbidden by comp and UDA, we must on the contrary extract the squeletton on causal proposition from our definition of knowledge or better observation. In fact that remarks is important and we will be obliged to say more on that, soon or later. Bruno
Re: Belief Knowledge
rwas rwas wrote: I guess the virtue of that depends on what you want to achieve. Sequential thinking is nice for applying formalized methods of analysis on something. Lateral thinking is far better for creative problem solving. Ahh - now I see that your sequential thinking means the same thing as analytic or deductive thought. As opposed to synthetic or integrative thought (which is in essence what lateral thinking is all about). My frustration with this group has been to observe (apparently) no creative ideas in transcending the darkness which our ignorance of consciousness and the nature of existence creates. Robert W. You must have a particular problem in mind to have made this comment. In fact this group has come up with many novel ideas that have been discussed. Some were preconceived ideas brought to the discussion, other arose in the heat of debate. Some have been found to be wanting, and so have been left on the wayside. Other ideas have better stood the test of criticism, but are still contentious. Consciousness per se, is a philosophical quagmire, that is unlikely to be solved anytime soon. I believe it is somewhat on the periphery of the sort of ensemble theories we discuss here, as it appears through the various types of anthropic principles, and that more progress will be made by postulating certain properties of consciousness and working with that. I know that several members of this list are also interested in the AI problem, that is how to generate a conscious being in an artificial machine - as much as this would be a significant philosophical milestone in many areas, I don't believe it is needed to make progress. Cheers Dr. Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967 UNSW SYDNEY 2052 Fax 9385 6965 Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks
Re: Belief Knowledge
You must have a particular problem in mind to have made this comment. I do, a set of problems: 1. Machine cooperation in fabrication robots and other agents that can cooperate in complex tasks such as manufacturing non-robot-friendly components. 2. Synthetic Autonomous Agent Engineering such as circuit board design (this would operate completely in software) 3. Autonomous Agents for mission and vehicle operation. AA's used in space, and undersea exploration 4. General Robotic Autonomous Agents ie., androids, specialized autonomous robots for augmentation or replace of human presence in any number of environments. Obviously these are biggies no matter how you look at it and are common interests to many researchers. I can invision methods of crude consciousness but consciousness in terms of what is required to solve a particular problem In fact this group has come up with many novel ideas that have been discussed. Some were preconceived ideas brought to the discussion, other arose in the heat of debate. Some have been found to be wanting, and so have been left on the wayside. Other ideas have better stood the test of criticism, but are still contentious. I guess I missed those. Consciousness per se, is a philosophical quagmire, that is unlikely to be solved anytime soon. Based on my explorations I must conclude consciousness of *any* kind is a thread that intersects the physical realm (from somewhere else). Any tool or machine construct that lends itself to observation of identity seperate from other identities provides a scratch and snif kind of intersection between here and *there* and necessarily forms a thread of consciousness. Some might describe this as an emergent construct. Dr. Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967 UNSW SYDNEY 2052 Fax 9385 6965 Australia [EMAIL PROTECTED] Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks Robert W. __ Do You Yahoo!? Yahoo! Auctions - buy the things you want at great prices http://auctions.yahoo.com/
Re: Belief Knowledge
--- Marchal [EMAIL PROTECTED] wrote: Robert W. wrote: [...] Logic is a powerful tool for analysis. Some use it intuitively, you people seem to have mastered formalized, symbolic logic. That's great. Logic is just a branch of mathematics which studies discourse and their interpretation. Let's be clear on the meaning of *logic*. Logic as a reasoning power existed before the discipline of formalized logic. One can have a very powerful logical reasoning facility without having attented a course on the subject. My point has to do with the way you folks seem to be trying to understand *everything*. Logic will always play a powerful role in understanding and analysis. I am basically trying to say, there are ways of seeing and understanding that transcend sequential thinking. Most discourses are sequential but the thinking behind does not need to be sequential. The semantics are in general not sequential. Maticulously wondering a search space, with logic should be wandering. or any other method, only reveals what's in that space. It does not help one see outside of the space. Recall I have until here give only an informal (although persuasive IMO) argument showing that if we can survive with a digitale brain then physics transform itself This assumes a great deal... into a branch of machine psychology. UDA = UDA http://www.escribe.com/science/theory/m1726.html And currently, I have just proposed to some to explain the technical part of my thesis where I will explicitely show how quantum logic in the discourse of the introspective UTM. I' dont feel myself searching in a space. How can it be anything but a space to be searched? One can invision a possibility space where one dimension is time and others possible expression types. One might attempt to write a program to demonstrate consciousness by generating a random program within some set of constraints and then generating many until he stumbles upon what he's looking for. When all the possibilities within the constraints are exhausted, he's searched the possibility space. Obviously there are faster ways to search this space for meaningful results, but it's still a search. I'd love to be expert enough in logic and mathmatics to demonstrate all the brick-walls I see intuitively, unfortunately, I am not now, or likely to be good enough in formalized systems of logic to do so. It is true that logic is poorly taught. But then that is the reason why I propose to take it at the beginning. Of course that demand some works. I have not taken a course on it. I have read a book on Symbolic Logic but found the discipline of little utility and so not worth my time to incorporate as a persistent knowledge. My logical reasoning power is reasonable developed however and find it more than adequate for most endeavours. I have noticed that when it comes to exhausting a possibility space with logic and finding an answer using intuitive methods then verifty the results with logic is far more productive. On the other hand, I can see the temptation to use one's already developed formalized method to explore a subject. I however, do not enjoy considering the possiblity of having my mind placed in a mathmatical bottle by you people. I was hoping to call attention to other facilities we all posses to expand understanding in areas that seem to defy understanding. Thank you. Such facilities are welcome, although they are in general much more difficult to communicate. Read the UDA. It is IMO such a facililities. The logic is only for those who have grasped UDA and want to see the emergence of the quantum in machine's mind. To be honest I think you were also rather hardhearted with Brent Meeker's post. I was having a bad day. My appologies for being harsh. Bruno Robert W. __ Do You Yahoo!? Yahoo! Auctions - buy the things you want at great prices http://auctions.yahoo.com/
Re: Belief Knowledge
Robert W. wrote: I also mention it because it seems that much of the dicussion here is forcing understanding through symbolic logic. There is no way to force understanding. You know there was a time when people believed that the 5th postulate of Euclide geometry was a consequence of the four others. Until Lobachevski shows a mathematical model obeying to the four first axioms and not the 5th. This shows the independance (the relative consistency of the negation of the 5th postulate). This is exactly what modern logician does. Building structures for viewing the relative independance or dependance of statements. In the same way, we know through Kripke models that the formula K, T, 4, 5, B are mutually independant (and we can prove semantically that KT45 - B, because a relation which is reflexive transitive and euclidian is necessarily symmetrical, or that KT4B - 5 because a relation which is reflexive, transitive and symmetrical is automatically euclidian, etc. Modern logic help us to forget syntactical symbolic derivations. This is liberating the mind. IMO logic is just a polite way for helping others (including oneself) to realise they have prejudices. The french poet and novelist Paul Valery said it in a rather forceful manner: You have just one choice in life, the choice between war and logic. Robert wrote also A lot will be left out. There are many things physically observable that defy excluded-middle analysis, like much of the phenomenon found in quantum mechanics. Intuitionist logic (classical logic minus the excluded-middle principle p v -p) and quantum logic (classical logic minus distributivity axioms) will find their proper place in the dialog with the UTM. Bruno
Re: Belief Knowledge
On Wed, 2 May 2001, Brent Meeker wrote: A true belief that has a casual connection with the fact that makes it true. Knowledge is when predicted.
