Re: The role of logic, planning ...
Russell Standish wrote: If treasures were not hidden in the mud, they would not be treasures :-) I suspect your words came out wrong here - do you mean A treasure hidden by mud is still a treasure? Which, of course I agree with. Thanks for correcting my words. I would be ecstatic if someone developed a working theory of consciousness. You know that the UDA explains that matter emerge from consciousness (to say the things roughly). If by working theory of consciousness you mean a theory giving explicitely results on physical matter, then you will be ecstatic. If you mean a theory useful in AI, you will be disappointed. Actually my theory is rather negative for AI, it seems at first that machine will be intelligent *despite* humans. I don't think the time here is too mysterious. All one needs is a one dimensional parameter t, so one can write down an evolution equation. A differential equation? It could be a lot for me. This could lead us toward a new thread. I propose to come back later to the time problem. A sort of time will appear in my machine psychology through the Grzergorczyk formula. --- Russell Standish wrote also: Thanks for this extended discussion. It does help a lot, and makes even more sense if one assumes COMP (which actually I don't, but for the sake of argument, wil do). Thanks. My practical philophy is to accept comp as a working hypothesis and to push it to its ultimate limits. I do take the MWI aspect of realist QM as a sort of confirmation, though. Just one further question. Is it possible for one machine to know p and another machine to know -p? Soon p will be interpreted as arithmetical propositions, in which case if p is true it will be true for all machines. Bruno
Re: The role of logic, planning ...
Marchal wrote: Hi Russell, I am glad you borrowed Booloses from a library and that you spent a while poring over my thesis. I want just made precise that I have never try to modelise knowledge by Bew(|p|). This is, actually, a rather sensible point. Most philosopher agree that S4 is a good *axiomatic* of knowledge. Precisely S4 is KT4 + MP,NEC or, explicitely (added to the Hilbert Ackerman axioms) : AXIOMS [](A - B) - ([]A -[]B) K []A - AT []A - [][]B4 RULES A/[]A(A (A-B)) / BNEC MP. That is, most philosopher (since Plato, but I remember having seen a Buddhist similar writing) agree that: -if A-B is knowable and if A is knowable, then B is knowable. (K) - if A is knowable then A is true. (T) - if A is knowable than that very fact (that A is knowable) is knowable (4) Would you agree with that? 4 makes that knowledge somehow introspective. Now we will see that if []A represent the formal provability of A, or (provability by a sound machine), i.e. Bew(|A|), although 4 and K are verified, we don't have T, that is, we don't have []A - A provable for all sentence A. Bew(|A|) - A is not always provable. This entails that formal provability cannot and should not be used for the formalisation of knowledge. Thanks for this extended discussion. It does help a lot, and makes even more sense if one assumes COMP (which actually I don't, but for the sake of argument, wil do). Just one further question. Is it possible for one machine to know p and another machine to know -p? It seems from the above discussion, you are only considering consistent machines, which of course cannot know p and -p simultaneously without being inconsistent. However, you're not ruling out a society of such machines who argue over what statements they know to be true (just like my ardent theists and atheists in Australia - actually this last example is largely hypothetical - when it comes to religions, Australians are amongst the most apathetic in the world - an important fact in us enjoying peace and prosperity). 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: The role of logic, planning ...
Russell Standish wrote: Ah! You mean the problem of consciousness (or more exactly, the problem of having a theory of conscsiousness). Yes - I'm well aware of this problem, and unlike some, I don't believe it is a non-problem. OK. I prefer to call it the mind-body problem. That reminds us that there is both a mind problem *and* a body problem. Now I do have a theory of mind with comp: mainly given by the discourse of the self-referential sound UTMs, and its guardian angel: G and G* The body is less obvious, but translating the UDA in arithmetic, this gives Z1 and Z1* for physical experiment and experiences. Open problem are: is Z1* completely axiomatisable? Does Z1* give the entire Birkhof von Neuman quantum logic. Does Z1* violates Bell inequality, is there a unique representation of Z1* in term of subspaces of Hilbert space, etc. In Occam's razor, I don't just ignore this problem, I sweep it under the rug. At some stage I have said Theories of Consciousness have bogged down in a quagmire. I think I must have just said it, as I cannot find it written in any publications. Perhaps I want to distance myself somewhat from the mud-slinging going on. If treasures were not hidden in the mud, they would not be treasures :-) Without providing a theory, or even a definition of what consciousness is, in the paper I assume 3 propositions to be true about consciousness: i) It is capable of universal computation (in order to interpret) ii) It experiences time (in order to compute) iii) It projects out actual events from the set of potential events. i) is ambiguous. And doubtful for most crisp interpretation I try. For exemple I believe cats are conscious. Cats are capable of universal computation only in a very large sense. Still I appreciate, but dangerous because most people will derive that animal are not conscious. ii) needs a definition of time. You will see how time emerges in my approach. It is linked to consciousness, and basicaly I would agree, although the subject of experience is a person and not consciousness, which is a qualitative state of that person experiencing time. iii) is a correct description of the appearances. [...] The reason I say this is that while homo sapiens is capable of universal computation, it is not its primary modus operandi, hence I would be surprised if the prior distribution of descriptions was given by a universal computer. I don't see the relation with the prior distribution. I guess I miss something here. One can criticse my work on 3 grounds: i) My conclusions do not follow logically from this basis ii) That there are additional hidden assumptions needed, that I've not made clear or precise iii) That one or more of the above assumptions are false [...] I'm am therefore, far more interested in errors of logic or omission. It would seem that you would develop a criticism along the lines of ii) - hidden assumptions, however I've yet to see these spelt out (or perhaps they have, just I haven't understood it because of language barriers). In fact I'm not sure what exactly are your assumptions. Are you talking on the assumptions about consciousness? I have still the feeling that you attach observers to universes. You don't postulate comp, do you? Bruno
Re: The role of logic, planning ...
Just as an example, he says most philosophers would agree that []A-A, where []A is interpreted as knowing A. This is clearly a different meaning of the word to know that we use here in Australia. Provided that A is not a simple artificial construct, meaning: it is a complicity (called generally a complexity), it cannot be known in its entirety. So the [] for knowing is a deficiency rather than an addition. The fact of knowing is added, but the [knowledge of A] is less than A (at least by the Aristotelian part of more.) 'Most' philosophers have yet to learn about complexities. John Mikes [EMAIL PROTECTED] http://pages.prodigy.net/jamikes;
Re: The role of logic, planning ...
Russell Standish wrote: I still believe my general remarks apply to your why Occam's razor. (I reprint it and I will reread it once I have more time). You put to much for me in the hypothesis. Like all physicists you seem not to be aware of the mind body problem. You are right! What is the mind-body problem? I appreciate your frankness! Note that I consider sometimes the UDA as a mean to explain that the mind-body problem is NOT solved automaticaly by COMP (as most materialist believes). The formulation of the mind body problem is dependent of the philosophy you believe in. Well, if you believe in a causal material world, and if you believe in mental sensations and volitions, then the mind-body problem is just the search for an explanation of the link between that causal material world and these mental (subjective, first person) sensations and volitions. A neurophysiologist poetical version is how can grey matter produces feeling of color. Another one is how can just firing of neurons produces feeling of joy or of pain. Etc. An idealist (immaterialist) philosopher must explain the belief in matter. A materialist must explain the belief in beliefs. A cartesian dualist must explain the link between matter and belief (or feeling of belief). Some scientist dismiss it as a non scientific problem. It is easy to show that this dismiss is itself not scientific. Our culture is used to put the mind-body problem in religious matter. You can consider the UDA as a reduction of the mind body problem into the problem of the origin of (the belief in) physical laws. And you can see my UTM interview as a solution of the mind body problem (!). The qualia (internal immediate feelings) appears naturaly thanks to incompleteness, or, more precisely, thanks to the difference between Z1 and Z1* (which itself is inherited from the gap between G and G*). More on this latter. Bruno
Re: The role of logic, planning ...
Ah! You mean the problem of consciousness (or more exactly, the problem of having a theory of conscsiousness). Yes - I'm well aware of this problem, and unlike some, I don't believe it is a non-problem. In Occam's razor, I don't just ignore this problem, I sweep it under the rug. At some stage I have said Theories of Consciousness have bogged down in a quagmire. I think I must have just said it, as I cannot find it written in any publications. Perhaps I want to distance myself somewhat from the mud-slinging going on. The approach in Occam is an axiomatic one (actually that's a little formal for I actually do). Without providing a theory, or even a definition of what consciousness is, in the paper I assume 3 propositions to be true about consciousness: i) It is capable of universal computation (in order to interpret) ii) It experiences time (in order to compute) iii) It projects out actual events from the set of potential events. To support the last point, I rely on the Kolmogorov axioms of probability, which further require (some) results from set theory. Now those are my explicit assumptions. One can criticse my work on 3 grounds: i) My conclusions do not follow logically from this basis ii) That there are additional hidden assumptions needed, that I've not made clear or precise iii) That one or more of the above assumptions are false So far all criticisms of my work of been of category iii) - i.e. James Higgo and Jacques Mallah, and then principally on the time assumption. I've find these criticisms to be ineffective, since it seems to me as a member of homo sapiens, observationally these propositions are true of our species. I'm am therefore, far more interested in errors of logic or omission. It would seem that you would develop a criticism along the lines of ii) - hidden assumptions, however I've yet to see these spelt out (or perhaps they have, just I haven't understood it because of language barriers). Incidently, there is one development of the theory - I believe the arguments in Occam still work if one weakens proposition 1) to equivalence classing (as expounded in On Complexity and Emergence). The resulting prior measure, and complexity measure differ from the universal computing case, but not in any important detail. The reason I say this is that while homo sapiens is capable of universal computation, it is not its primary modus operandi, hence I would be surprised if the prior distribution of descriptions was given by a universal computer. Cheers Marchal wrote: Russell Standish wrote: I still believe my general remarks apply to your why Occam's razor. (I reprint it and I will reread it once I have more time). You put to much for me in the hypothesis. Like all physicists you seem not to be aware of the mind body problem. You are right! What is the mind-body problem? I appreciate your frankness! Note that I consider sometimes the UDA as a mean to explain that the mind-body problem is NOT solved automaticaly by COMP (as most materialist believes). The formulation of the mind body problem is dependent of the philosophy you believe in. Well, if you believe in a causal material world, and if you believe in mental sensations and volitions, then the mind-body problem is just the search for an explanation of the link between that causal material world and these mental (subjective, first person) sensations and volitions. A neurophysiologist poetical version is how can grey matter produces feeling of color. Another one is how can just firing of neurons produces feeling of joy or of pain. Etc. An idealist (immaterialist) philosopher must explain the belief in matter. A materialist must explain the belief in beliefs. A cartesian dualist must explain the link between matter and belief (or feeling of belief). Some scientist dismiss it as a non scientific problem. It is easy to show that this dismiss is itself not scientific. Our culture is used to put the mind-body problem in religious matter. You can consider the UDA as a reduction of the mind body problem into the problem of the origin of (the belief in) physical laws. And you can see my UTM interview as a solution of the mind body problem (!). The qualia (internal immediate feelings) appears naturaly thanks to incompleteness, or, more precisely, thanks to the difference between Z1 and Z1* (which itself is inherited from the gap between G and G*). More on this latter. Bruno 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: The role of logic, planning ...
Russell Standish wrote (to George): I don't think Bruno's conclusion is weird. I come to essentially the same conclusion in Occam, without the need for formalising Knowledge, nor the need to use Modal logic. The fact that you come to the same conclusion does not mean these conclusions are not weird. I hope you realise these conclusions run against the average materialistic aristotelian current scientific paradigm. I would like to think that my exposition is easier to follow than Bruno's, but this could simply be a biased viewpoint on my part. I welcome comment and criticism on that paper. I still believe my general remarks apply to your why Occam's razor. (I reprint it and I will reread it once I have more time). You put to much for me in the hypothesis. Like all physicist you seem not to be aware of the mind body problem. With comp, what the UDA shows (and what the graph movie or Maudlin works proves) is that it is not possible to attach awareness to worlds or histories. The reversal means really that you need first a theory of consciousness, or a psychology for deriving the existence of physical beliefs. I agree that there are similarities in some of our conclusion, but I am not sure we mean the same by psychology. Incidently, I didn't mean to imply that this sort of modeling of Knowlegde was inappropriate, only that there was no discussion as to why one would want to model it in this particular way. The word model is tricky. It means different things for logician and painters (who are using it in the sense of reality) and physicist and toys builder (who are using it as theory or approximation, or reduction). Soemtimes I use it in the physicist sense ... But my approach is more axiomatic. I hope I will be able to give enough illustrations to help understanding ... Its really the same as when Hal Ruhl (and I admit I'm putting words in his mouth here, although its consistent with my understanding of his position) models the universe by cellular automata. Hal Ruhl, like Toffoli, and even like Schmidhuber-2, seems indeed to search for such modelisation. But I do not (and apparently Schmidhuber-1 don't do it either). The UD does NOT depend on the choice of a particular formal systems. The UDA really shows that my awareness will be linked with all implementation of my computationnal extension. By implementation here I just mean the giving of a program and its relative UTM interpretation. And the provability logics (G and G*) is correct and complete for ALL sound classical Universal Machines. In that sense there is no modelisation at all. And comp is not the hypothesis that my brain can be modelised by a Turing Machine, it is the act of faith of telling yes to the (mad) surgeon. I notice Bruno has posted a more detailed discourse on this issue, which I will digest in due course. It is an important one, but it will be fully clear only after I explain Godel and Lob theorem with enough rigor. Perhaps all he was doing was assuming a cultural background of philosophy I have not been exposed to. Just as an example, he says most philosophers would agree that []A-A, where []A is interpreted as knowing A. This is clearly a different meaning of the word to know that we use here in Australia. I know of plenty of people who know that God exists. And I know of a number of other people who know that God doesn't exist. So, by this application of Modal logic, we can conclude that God both exists and doesn't exist at the same time, which seems kind of illogical. To say the least. I must say that I am quite astonished that Australian can know falsities. What is the difference between knowledge and belief for an Australian ? Perhaps the way out of this mess is to say that I'me really talking about belief, ... Yes, I think indeed you were talking of belief. The nice thing with axiomatic approach is that we will define knowledge or knowability by axiom like K, T, 4. Except that formal provability will be defined in arithmetic and then we will look at which formula it obeys. And It does not obeys to knowledge axiom (see below). ...rather than knowledge, however that would imply that knowledge is devoid of meaning, since it is impossible to establish with certainty whether any particular fact is true. But, at least for a non intuitionnist, or a non constructivist, a proposition could be true independently of our belief or knowledge. A platonist (as I am for arithmetics) has no problem with that. Of course the nuance between truth, provable, believable, knowable, .. are subtil. The crazy thing is that Godel (Lob Solovay) will eventually put an immense light on those nuance. In metaphysics the royal argument for explaining that indeed we cannot distinguish knowledge with belief is the dream argument. When we are awake, we cannot know for sure that we are not dreaming. Socrate uses it in his reply to Thaetetus. Descartes, Berkeley and almost all idealist use it also. The beauty of comp is that it gives a
Re: The role of logic, planning ...
Just as an example, he says most philosophers would agree that []A-A, where []A is interpreted as knowing A. This is clearly a different meaning of the word to know that we use here in Australia. I get the impression folks here assume that when one person knows something, that only that person knows that something. For other people to know the same something, they have to discover and assimilate it for themselves. It also seems that folks here assume knowledge is some kind of pattern that exists separate from the truth of surrounding it's existence. From a mystic standpoint, this can't be. To know something is closer to the analogy of a subscriber line. When one *knows* something, anything, they subscribe this pattern. Another issue is how folks seem to thing knowledge is inanimate until someone acts on it, like words on a paper being meaningless until someone read them. From a mystic standpoint, that isn't so. Knowledge and expression is simply manifest from one place to another. The knowledge itself is not constrained to the limits of those that would interpret it. Those entities interpret and then express that understanding wherever they happen to be existing. For someone to try to form the basis for existence based on what one thinks others can know in terms of what I've tried to counter, I feel intuitively that they would fail, or not succeed completely. One analogy to explain this is someone caught in an event horizon of a black-hole. The realm formed by this Event-Horizon can be vast, but is still by definition, limited. I see people trying to define existence by illusionary data like someone trying to understand the universe by what he can see from his vantage point in the Event-Horizon. Drawn out, it would look like someone walking in a circle. Eventually, he'd come back to where he started. He might vary his path slightly to see different things, but he'd simply be make the circle bigger. He can never know what lay outside the circle with his given modus operandi. From my perspective, true knowing, is being what you know. Which implies a great deal on what is truly knowable. If you look at what we're used to here, we have belief and knowing implicitly understood in statistical terms. We know we can walk, we've done it so often, so we don't doubt we can. Belief seems to be predicated on the existence of doubt. True knowing has no constraints of doubt. To know is to be one with that knowledge. This from a mystic standpoint, is true faith. Faith is *not* belief. Faith is knowing. I know of plenty of people who know that God exists. And I know of a number of other people who know that God doesn't exist. So, by this application of Modal logic, we can conclude that God both exists and doesn't exist at the same time, which seems kind of illogical. Perhaps the way out of this mess is to say that I'me really talking about belief, rather than knowledge, however that would imply that knowledge is devoid of meaning, since it is impossible to establish with certainty whether any particular fact is true. Even Mathematical proof is contingent upon belief of the efficacy of the formal proof, Again, I had thought the point of these threads were to try to describe consciousness with the idea in mind of trying to synthesize consciousness in software or some other artificial means. I propose the best way to do this is to know what one is after specifically, then solve the problem of achieving it. If one attempts to use a limited thinking style to implement something interpreted with that same thinking style, the end result would seem to necessarily be limited to perceptional constraints of that thinking style. I get the intuitive sense, that linear or sequential thinking will not result in the kind of achievement we're talking about. Robert W. __ Do You Yahoo!? Yahoo! Auctions - buy the things you want at great prices http://auctions.yahoo.com/
Re: The role of logic, planning ...
Hi Russell, I spent a while poring over Bruno's thesis, and borrowed Boolos from a local university library to udnerstand more what it was about. I didn't go into too great a length into the results and structure of Modal logic, although I gained an appreciation, and an understanding of the symbology. However, my main problem with Bruno's work lay not in the technical details of Model logic, rather with the phrases of the ilk We modelise knowledge by Bew(|p|). I can appreciate its only a model, but why should I believe that model of knowing has any connection with reality? I'm afraid none of the Booloses, nor Bruno's posting helped me with this. I am glad you borrowed Booloses from a library and that you spent a while poring over my thesis. I want just made precise that I have never try to modelise knowledge by Bew(|p|). This is, actually, a rather sensible point. Most philosopher agree that S4 is a good *axiomatic* of knowledge. Precisely S4 is KT4 + MP,NEC or, explicitely (added to the Hilbert Ackerman axioms) : AXIOMS [](A - B) - ([]A -[]B) K []A - AT []A - [][]B4 RULES A/[]A(A (A-B)) / BNEC MP. That is, most philosopher (since Plato, but I remember having seen a Buddhist similar writing) agree that: -if A-B is knowable and if A is knowable, then B is knowable. (K) - if A is knowable then A is true. (T) - if A is knowable than that very fact (that A is knowable) is knowable (4) Would you agree with that? 4 makes that knowledge somehow introspective. Now we will see that if []A represent the formal provability of A, or (provability by a sound machine), i.e. Bew(|A|), although 4 and K are verified, we don't have T, that is, we don't have []A - A provable for all sentence A. Bew(|A|) - A is not always provable. This entails that formal provability cannot and should not be used for the formalisation of knowledge. You can guess the reason. Consider []FALSE - FALSE, this is equivalent to -[]FALSE which is the statement of (self-) consistency (by the machine or the formal system), that is TRUE, which by Godel's second theorem is NOT provable (by the sound machine). But then, how to formalize knowledge ? When Socrate asked Thaetetus what is knowledge of p, Thaetetus replied justification of p. But then Socrate argues that a justification of p can be wrong. Thaetetus proposed then to define knowledge by justification of p *and* truth of p, by definition ! We will see that it is impossible to define truth of p in the language of the machine (Tarski theorem), but still we can define knowledge of p (for the machine) by Bew(|p|) *and* p If we define KNOW(A) by []A A, then the modal KNOW obeys S4, that is KNOW(A - B) -(KNOW A - KNOW B), (KNOW A) - A, etc. (see above). To sum up: I never modelized knowledge of p by Bew(|p|), but I will indeed define knowledge of p (in the language of the machine) by Bew(|p|) p. How come? Is not Bew(|p|), for the sound machine, trivially equivalent with Bew(|p|) p ? Yes. But the point is that the sound machine neither can bew it, nor know it! We will see how precisely the epistemological nuance between Bew(|p|) p and Bew(|p|) are made necessary by the incompleteness phenomenon. All this will be made transparent with the modal logic G and G* and their arithmetical interpretations. The atomic sentences are interpreted by arithmetical sentences. I can appreciate its only a model, but why should I believe that model of knowing has any connection with reality? I'm afraid none of the Booloses, nor Bruno's posting helped me with this. The connection with the reality, as you see, is done in the most platonist superb manner, I just add it by definition. Nuancing Bew(|p|) by Bew(|p|) truth(of p). Well, later I will propose another nuancing of Bew(|p|), more appropriate for measuring probability one on possible consistent extension). Bew(|p|) is nuanced by Bew(|p|) consistency of p. (a necessary step by UDA actually). The embedding in UD* will be translated in the language of the machine by restricting the arithmetical interpretation of p. And to get George's prize I will still need to extract LASE (the little abstract schroedinger equation) from that embedding. And of course i will need to make clear the relationship between LASE and the quantum histories. Bruno
Re: The role of logic, planning ...
Hi Russell, I spent a while poring over Bruno's thesis, and borrowed Boolos from a local university library to udnerstand more what it was about. I didn't go into too great a length into the results and structure of Modal logic, although I gained an appreciation, and an understanding of the symbology. However, my main problem with Bruno's work lay not in the technical details of Model logic, rather with the phrases of the ilk We modelise knowledge by Bew(|p|). I can appreciate its only a model, but why should I believe that model of knowing has any connection with reality? I'm afraid none of the Booloses, nor Bruno's posting helped me with this. I am glad you borrowed Booloses from a library and that you spent a while poring over my thesis. I want just made precise that I have never try to modelise knowledge by Bew(|p|). This is, actually, a rather sensible point. Most philosopher agree that S4 is a good *axiomatic* of knowledge. Precisely S4 is KT4 + MP,NEC or, explicitely (added to the Hilbert Ackerman axioms) : AXIOMS [](A - B) - ([]A -[]B) K []A - AT []A - [][]B4 RULES A/[]A(A (A-B)) / BNEC MP. That is, most philosopher (since Plato, but I remember having seen a Buddhist similar writing) agree that: -if A-B is knowable and if A is knowable, then B is knowable. (K) - if A is knowable then A is true. (T) - if A is knowable than that very fact (that A is knowable) is knowable (4) Would you agree with that? 4 makes that knowledge somehow introspective. Now we will see that if []A represent the formal provability of A, or (provability by a sound machine), i.e. Bew(|A|), although 4 and K are verified, we don't have T, that is, we don't have []A - A provable for all sentence A. Bew(|A|) - A is not always provable. This entails that formal provability cannot and should not be used for the formalisation of knowledge. You can guess the reason. Consider []FALSE - FALSE, this is equivalent to -[]FALSE which is the statement of (self-) consistency (by the machine or the formal system), that is TRUE, which by Godel's second theorem is NOT provable (by the sound machine). But then, how to formalize knowledge ? When Socrate asked Thaetetus what is knowledge of p, Thaetetus replied justification of p. But then Socrate argues that a justification of p can be wrong. Thaetetus proposed then to define knowledge by justification of p *and* truth of p, by definition ! We will see that it is impossible to define truth of p in the language of the machine (Tarski theorem), but still we can define knowledge of p (for the machine) by Bew(|p|) *and* p If we define KNOW(A) by []A A, then the modal KNOW obeys S4, that is KNOW(A - B) -(KNOW A - KNOW B), (KNOW A) - A, etc. (see above). To sum up: I never modelized knowledge of p by Bew(|p|), but I will indeed define knowledge of p (in the language of the machine) by Bew(|p|) p. How come? Is not Bew(|p|), for the sound machine, trivially equivalent with Bew(|p|) p ? Yes. But the point is that the sound machine neither can bew it, nor know it! We will see how precisely the epistemological nuance between Bew(|p|) p and Bew(|p|) are made necessary by the incompleteness phenomenon. All this will be made transparent with the modal logic G and G* and their arithmetical interpretations. The atomic sentences are interpreted by arithmetical sentences. I can appreciate its only a model, but why should I believe that model of knowing has any connection with reality? I'm afraid none of the Booloses, nor Bruno's posting helped me with this. The connection with the reality, as you see, is done in the most platonist superb manner, I just add it by definition. Nuancing Bew(|p|) by Bew(|p|) truth(of p). Well, later I will propose another nuancing of Bew(|p|), more appropriate for measuring probability one on possible consistent extension). Bew(|p|) is nuanced by Bew(|p|) consistency of p. (a necessary step by UDA actually). The embedding in UD* will be translated in the language of the machine by restricting the arithmetical interpretation of p. And to get George's prize I will still need to extract LASE (the little abstract schroedinger equation) from that embedding. And of course i will need to make clear the relationship between LASE and the quantum histories. Bruno
Re: The role of logic, planning ...
George Levy wrote: Marchal wrote: And we have as results (including the exercices!): Any frame (W,R) respects K A frame (W,R) respects T iff R is reflexive A frame (W,R) respects 4 iff R is transitive A frame (W,R) respects 5 iff R is euclidian (where R is Euclidian means that if xRy and xRz then yRz, for x, y z in W). A frame (W,R) respects D iff (W,R) is ideal A frame (W,R) respects C iff (W,R) is realist. We will talk on the semantics of L and Grz later. I do not think you defined euclidian There is obviously a connection to geometry but I dn't see it. I just have defined it above. R is Euclidian means that if xRy and xRz then yRz. A more concrete euclidian relation: W = the plane, i.e. the worlds are the point of the plane. xRy = there is a straight line from x to y. It is clearly euclidian because if there is a straight line from x to y, and straight line from x to z, there is a straight line from y to z. You can forget it because 5 is the only formula we will never meet. I guess we have to visit the whole Louvre to get to the Mona Lisa :-). Any short cut? Thanks for Mona Lisa !. A short cut? Gosh! My machine interview *is* a terrible short cut :-) Well I will try to follow a spirale, not giving you all technical details (at once). Don't forget we are going from the psychology of the machines, *by* the machines (and by their angels!) to their most probable physical beliefs. So there is some need to be cautious with the vocabulary, to say the least. Bruno
Re: The role of logic, planning ...
George Levy wrote: Marchal wrote: And we have as results (including the exercices!): Any frame (W,R) respects K A frame (W,R) respects T iff R is reflexive A frame (W,R) respects 4 iff R is transitive A frame (W,R) respects 5 iff R is euclidian (where R is Euclidian means that if xRy and xRz then yRz, for x, y z in W). A frame (W,R) respects D iff (W,R) is ideal A frame (W,R) respects C iff (W,R) is realist. We will talk on the semantics of L and Grz later. I do not think you defined euclidian There is obviously a connection to geometry but I dn't see it. I just have defined it above. R is Euclidian means that if xRy and xRz then yRz. A more concrete euclidian relation: W = the plane, i.e. the worlds are the point of the plane. xRy = there is a straight line from x to y. It is clearly euclidian because if there is a straight line from x to y, and straight line from x to z, there is a straight line from y to z. You can forget it because 5 is the only formula we will never meet. I guess we have to visit the whole Louvre to get to the Mona Lisa :-). Any short cut? Thanks for Mona Lisa !. A short cut? Gosh! My machine interview *is* a terrible short cut :-) Well I will try to follow a spirale, not giving you all technical details (at once). Don't forget we are going from the psychology of the machines, *by* the machines (and by their angels!) to their most probable physical beliefs. So there is some need to be cautious with the vocabulary, to say the least. Bruno
Re: The role of logic, planning ...
Dear Russell: At 5/2/01, you wrote: Incidently, I didn't mean to imply that this sort of modeling of Knowlegde was inappropriate, only that there was no discussion as to why one would want to model it in this particular way. Its really the same as when Hal Ruhl (and I admit I'm putting words in his mouth here, although its consistent with my understanding of his position) models the universe by cellular automata. Yes I can agree its a consistent model, and possibly one that's testable. However, I fail to see why one would want to do that. Good physical models ought to have some understandable basis (explanatory power perhaps). Indeed one could at first cut consider my model to be a collection of differently configured cellular automata hopping from acceptable sub pattern to acceptable sub pattern on a huge preexisting meta pattern. However, I try to show that none of these can be deterministic cascades i.e. single valued non halting machines with static definitions. As to the usefulness of deterministic cellular automata to model physics I cite Tommaso Toffoli's paper: Occam, Turing, von Neumann, Jaynes: How much can you get for how little (A conceptual introduction to cellular automata) at: http://www.interjournal.org/cgi-bin/manuscript_abstract.cgi?345678901 I do try to build a derivation of our universe's physics on my somewhat different base. Perhaps the way out of this mess is to say that I'me really talking about belief, rather than knowledge, however that would imply that knowledge is devoid of meaning, since it is impossible to establish with certainty whether any particular fact is true. Even Mathematical proof is contingent upon belief of the efficacy of the formal proof, something that has been called into doubt, particularly for more complex proofs like Fermat's last theorem, or the 4 colour theorem. I don't mean to be picky, but its just these sorts of considerations and misunderstandings that throw me off the track every time. Cheers I think there are rather few universal [true?] facts in my model but that awaits my making it comprehensible. Most truth and meaning will be the self referential province of individual cellular actors [actors indicating non deterministic constructs]. Hal
Re: The role of logic, planning ...
Hi Russell, I spent a while poring over Bruno's thesis, and borrowed Boolos from a local university library to udnerstand more what it was about. I didn't go into too great a length into the results and structure of Modal logic, although I gained an appreciation, and an understanding of the symbology. However, my main problem with Bruno's work lay not in the technical details of Model logic, rather with the phrases of the ilk We modelise knowledge by Bew(|p|). I can appreciate its only a model, but why should I believe that model of knowing has any connection with reality? I'm afraid none of the Booloses, nor Bruno's posting helped me with this. I am glad you borrowed Booloses from a library and that you spent a while poring over my thesis. I want just made precise that I have never try to modelise knowledge by Bew(|p|). This is, actually, a rather sensible point. Most philosopher agree that S4 is a good *axiomatic* of knowledge. Precisely S4 is KT4 + MP,NEC or, explicitely (added to the Hilbert Ackerman axioms) : AXIOMS [](A - B) - ([]A -[]B) K []A - AT []A - [][]B4 RULES A/[]A(A (A-B)) / BNEC MP. That is, most philosopher (since Plato, but I remember having seen a Buddhist similar writing) agree that: -if A-B is knowable and if A is knowable, then B is knowable. (K) - if A is knowable then A is true. (T) - if A is knowable than that very fact (that A is knowable) is knowable (4) Would you agree with that? 4 makes that knowledge somehow introspective. Now we will see that if []A represent the formal provability of A, or (provability by a sound machine), i.e. Bew(|A|), although 4 and K are verified, we don't have T, that is, we don't have []A - A provable for all sentence A. Bew(|A|) - A is not always provable. This entails that formal provability cannot and should not be used for the formalisation of knowledge. You can guess the reason. Consider []FALSE - FALSE, this is equivalent to -[]FALSE which is the statement of (self-) consistency (by the machine or the formal system), that is TRUE, which by Godel's second theorem is NOT provable (by the sound machine). But then, how to formalize knowledge ? When Socrate asked Thaetetus what is knowledge of p, Thaetetus replied justification of p. But then Socrate argues that a justification of p can be wrong. Thaetetus proposed then to define knowledge by justification of p *and* truth of p, by definition ! We will see that it is impossible to define truth of p in the language of the machine (Tarski theorem), but still we can define knowledge of p (for the machine) by Bew(|p|) *and* p If we define KNOW(A) by []A A, then the modal KNOW obeys S4, that is KNOW(A - B) -(KNOW A - KNOW B), (KNOW A) - A, etc. (see above). To sum up: I never modelized knowledge of p by Bew(|p|), but I will indeed define knowledge of p (in the language of the machine) by Bew(|p|) p. How come? Is not Bew(|p|), for the sound machine, trivially equivalent with Bew(|p|) p ? Yes. But the point is that the sound machine neither can bew it, nor know it! We will see how precisely the epistemological nuance between Bew(|p|) p and Bew(|p|) are made necessary by the incompleteness phenomenon. All this will be made transparent with the modal logic G and G* and their arithmetical interpretations. The atomic sentences are interpreted by arithmetical sentences. I can appreciate its only a model, but why should I believe that model of knowing has any connection with reality? I'm afraid none of the Booloses, nor Bruno's posting helped me with this. The connection with the reality, as you see, is done in the most platonist superb manner, I just add it by definition. Nuancing Bew(|p|) by Bew(|p|) truth(of p). Well, later I will propose another nuancing of Bew(|p|), more appropriate for measuring probability one on possible consistent extension). Bew(|p|) is nuanced by Bew(|p|) consistency of p. (a necessary step by UDA actually). The embedding in UD* will be translated in the language of the machine by restricting the arithmetical interpretation of p. And to get George's prize I will still need to extract LASE (the little abstract schroedinger equation) from that embedding. And of course i will need to make clear the relationship between LASE and the quantum histories. Bruno
Re: The role of logic, planning ...
Hi Marchal, This is a reply to your last two posts. I hope other everythingers beside myself are attempting to follow this adventure in logic. It appears to be really worth the effort. Please feel free to contribute to this exchange. Marchal wrote: And we have as results (including the exercices!): Any frame (W,R) respects K A frame (W,R) respects T iff R is reflexive A frame (W,R) respects 4 iff R is transitive A frame (W,R) respects 5 iff R is euclidian (where R is Euclidian means that if xRy and xRz then yRz, for x, y z in W). A frame (W,R) respects D iff (W,R) is ideal A frame (W,R) respects C iff (W,R) is realist. We will talk on the semantics of L and Grz later. I do not think you defined euclidian There is obviously a connection to geometry but I dn't see it. Actually we will need also -Predicate logic, and arithmetics -weak logics (intuitionist logic, quantum logic) -Algebraic semantics of weak logics -Kripke semantics of weak logics I guess we have to visit the whole Louvre to get to the Mona Lisa :-). Any short cut? Then the interview itself will begin. We can follow the historical progress of that interview: -Goedel's theorem; -Loeb's theorem; (just this one makes the travel worth!) -Solovay's theorem; -Muravitski Kusnetsov, Boolos, Goldblatt theorems; -Other theorems by Goldblatt -Still Other theorems by Goldblatt. -Visser's theorem; It is the theorem by Solovay which will make clear the relation between provability logic and some modal logics. Boolos, Goldblatt, Visser has found result which will make part of our the translation of the UDA argument almost transparent. Thank you for outlining a itinirary for our journey into logic I thought our destination was much closer.. Does it have to be that complicated? Thanks for the effort. George
