Re: One more question about measure
Le 06-juil.-05, à 00:56, Russell Standish a écrit : You are right, my apologies. I read the necessitation rule backwards in your thesis. You do in fact say P = []P. I'll take your word for it that consistency destroys necessitation, but I don't have the intuitive understanding of it yet. Never mind, it is enough for my present purposes. OK. Be careful not to confuse the formula A- B, and the rule A = B. The first is just a formula (equivalent with ~A v B in classical logic). The second is a dynamical rule saying that if the machine proves A it proves B. In general A = B is written A _ B (if this survives its teleportation in the archive!) :-) Bruno PS We loose the necessitation rule for the new box Cp = Bp ~B~p, because although the tautology t is provable, Ct is not. Indeed Ct is Bt ~B~t, but ~B~t = ~Bf, and this is the self-consistency statement no consistent machine can prove. OK? http://iridia.ulb.ac.be/~marchal/
Re: One more question about measure
On Sun, Jun 26, 2005 at 05:30:08PM +0200, Bruno Marchal wrote: This reminds me of something I wanted to ask you Bruno. In your work you axiomatise knowledge and end up with various logical systems that describe variously 1st person knowledge, 1st person communicable knowledge, 3rd person knowledge etc. In some of these, the Deontic axiom comes up, which if translated into Kripke semantics reads all worlds have a successor word (or no worlds are terminal). I recall that for knowledge CP, philosopher asks for both CP - P, and the closure for the necessitation rule. But then this means we can define knowledge of P, CP, by BP P. And then we can interview the machine (through an infinite conversation, ok, but finitely summarized thanks to Solovay's G) about the logic of knowledge CP. This gives a logic of temporal knowledge of a knower verifying the philosophers' most agreed upon definition. How does it give the logic of temporal knowledge? I understand from your points below, that the necessitation rule is necessary for Kripke semantics, and its is clear to me that necessitation follows from Thaetetus 1 3, whereas it doesn't follow from consistency alone (one could consistently prove false things, I guess). I still haven't figured out how to get temporality from a modal logic. Sure I can _interpret_ a logic as having Kripke semantics, and I can interpret the Kripke semantics as a network of observer moments, with the accessibility relation connecting an observer moment to its successor. However, what I don't know is why I should make this interpretation. I take it as the simplest first person notion definable in the language of the machine. [Careful here: CP will appear to be only very indirectly definable by the machine: no machine can give a third person description of its CP logic! The logic of CP is the system known as S4Grz. The subjective temporality aspect come from the fact that on finite transitive frames respecting the Grz formula the Kripke accessibility relation is antisymmetric and reflexive, like in Bergson/Brouwer conception of time. See perhaps: van Stigt, W.?P. (1990). Brouwer's Intuitionism, volume?2 of Studies in the history and philosophy of Mathematics. North Holland, Amsterdam. Boolos, G. (1980b). Provability in Arithmetic and a Schema of Grzegorczyk. Fundamenta Mathematicae, 96:41-45 Goldblatt, R.?I. (1978). Arithmetical Necessity, Provability and Intuitionistic Logic. Theoria, 44:38-46. (also in Goldblatt, R.?I. (1993). Mathematics of Modality. CSLI Lectures Notes, Stanford California). See also http://homepages.inf.ed.ac.uk/v1phanc1/dummet.html Note that BP - P is equivalent to ~P - ~B~ ~P, and if that is true/provable for any P, then it is equivalent to P - ~B~p, so BP - P, as axioms, entails BP - ~B~P (the deontic formula). But, by incompleteness the reverse is false. Now you were just pointing on tis little less simple definition of first person based on the deontic transformation. This one has been studied in my thesis, so I have only my papers in my url for references). Here a new logic is defined by DP = BP ~B~P. It is not used to define a first person knower, but more a first person plural gambler. The logic of DP loses the necessitation rule and loses the Kripke semantics, but get interesting quasi-topological spaces instead. A immediate time notion (re)appear though the combination of the two ideas: define D'P by BP ~B~P P. Do you you grasp the nuance between BP (Theaetetus 0) BP P (Theaetetus 1) BP ~B~P (Theaetetus 2) BP ~B~P P (Theaetetus 3) ? Only Theaetetus 1 gives rise to a temporal subjectivity. (Now if you interview the machine on *comp* itself, by limiting the atomic P to DU accessible truth, the Theaetetus 1, 2 and 3 all leads to different quantum logics. In my thesis of Brussels and Lille I have been wrong, I thought wrongly that the pure (given by Theaetetus 1) first person collapse with comp). -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpDb5ckJdfQu.pgp Description: PGP signature
Re: One more question about measure
Le 05-juil.-05, à 09:39, Russell Standish a écrit : On Sun, Jun 26, 2005 at 05:30:08PM +0200, Bruno Marchal wrote: This reminds me of something I wanted to ask you Bruno. In your work you axiomatise knowledge and end up with various logical systems that describe variously 1st person knowledge, 1st person communicable knowledge, 3rd person knowledge etc. In some of these, the Deontic axiom comes up, which if translated into Kripke semantics reads all worlds have a successor word (or no worlds are terminal). I recall that for knowledge CP, philosopher asks for both CP - P, and the closure for the necessitation rule. But then this means we can define knowledge of P, CP, by BP P. And then we can interview the machine (through an infinite conversation, ok, but finitely summarized thanks to Solovay's G) about the logic of knowledge CP. This gives a logic of temporal knowledge of a knower verifying the philosophers' most agreed upon definition. How does it give the logic of temporal knowledge? I understand from your points below, that the necessitation rule is necessary for Kripke semantics, and its is clear to me that necessitation follows from Thaetetus 1 3, whereas it doesn't follow from consistency alone (one could consistently prove false things, I guess). Right. But then I guess you mean Theaetetus 0 and 1. We loose necessitation once we just add the consistency ~B~P requirement (in Theaetetus 2 and 3). For example from the truth t we can deduce BP, but we cannot deduce Bt ~B~t nor Bt ~B~t t. I recall: BP (Theaetetus 0) BP P (Theaetetus 1) BP ~B~P (Theaetetus 2) BP ~B~P P (Theaetetus 3) ? I still haven't figured out how to get temporality from a modal logic. Sure I can _interpret_ a logic as having Kripke semantics, and I can interpret the Kripke semantics as a network of observer moments, with the accessibility relation connecting an observer moment to its successor. However, what I don't know is why I should make this interpretation. Why not? It is a natural interpretation of S4 type of logic, especially if you accept to interpret the accessibility relation as relation between OMs. It is the case for any interpretation of any theory. Perhaps I miss something here. Of course we could feel even more entitled to take the temporal interpretation once we accept Brouwer temporal analysis of intuitionist logic. Beth and Grzegorczyk have defend similar interpretations. I will come back on the question of interpreting Kripke structure once I will translate a theory by Papaioannou in those terms next week (after a brief explanation of what Kripke structures are for the non-mathematician). Bruno The logic of CP is the system known as S4Grz. The subjective temporality aspect come from the fact that on finite transitive frames respecting the Grz formula the Kripke accessibility relation is antisymmetric and reflexive, like in Bergson/Brouwer conception of time. See perhaps: van Stigt, W.?P. (1990). Brouwer's Intuitionism, volume?2 of Studies in the history and philosophy of Mathematics. North Holland, Amsterdam. Boolos, G. (1980b). Provability in Arithmetic and a Schema of Grzegorczyk. Fundamenta Mathematicae, 96:41-45 Goldblatt, R.?I. (1978). Arithmetical Necessity, Provability and Intuitionistic Logic. Theoria, 44:38-46. (also in Goldblatt, R.?I. (1993). Mathematics of Modality. CSLI Lectures Notes, Stanford California). See also http://homepages.inf.ed.ac.uk/v1phanc1/dummet.html Note that BP - P is equivalent to ~P - ~B~ ~P, and if that is true/provable for any P, then it is equivalent to P - ~B~p, so BP - P, as axioms, entails BP - ~B~P (the deontic formula). But, by incompleteness the reverse is false. Now you were just pointing on tis little less simple definition of first person based on the deontic transformation. This one has been studied in my thesis, so I have only my papers in my url for references). Here a new logic is defined by DP = BP ~B~P. It is not used to define a first person knower, but more a first person plural gambler. The logic of DP loses the necessitation rule and loses the Kripke semantics, but get interesting quasi-topological spaces instead. A immediate time notion (re)appear though the combination of the two ideas: define D'P by BP ~B~P P. Do you you grasp the nuance between BP (Theaetetus 0) BP P (Theaetetus 1) BP ~B~P (Theaetetus 2) BP ~B~P P (Theaetetus 3) ? Only Theaetetus 1 gives rise to a temporal subjectivity. (Now if you interview the machine on *comp* itself, by limiting the atomic P to DU accessible truth, the Theaetetus 1, 2 and 3 all leads to different quantum logics. In my thesis of Brussels and Lille I have been wrong, I thought wrongly that the pure (given by Theaetetus 1) first person collapse with comp). http://iridia.ulb.ac.be/~marchal/
Re: One more question about measure
On Tue, Jul 05, 2005 at 12:09:24PM +0200, Bruno Marchal wrote: How does it give the logic of temporal knowledge? I understand from your points below, that the necessitation rule is necessary for Kripke semantics, and its is clear to me that necessitation follows from Thaetetus 1 3, whereas it doesn't follow from consistency alone (one could consistently prove false things, I guess). Right. But then I guess you mean Theaetetus 0 and 1. We loose necessitation once we just add the consistency ~B~P requirement (in Theaetetus 2 and 3). For example from the truth t we can deduce BP, but we cannot deduce Bt ~B~t nor Bt ~B~t t. I recall: BP (Theaetetus 0) BP P (Theaetetus 1) BP ~B~P (Theaetetus 2) BP ~B~P P (Theaetetus 3) ? If D'P = BP ~B~P P, then D'P = P (ie necessitation). So it seems it is the conjunction of truth of P that gives rise to necessitation, no? I still haven't figured out how to get temporality from a modal logic. Sure I can _interpret_ a logic as having Kripke semantics, and I can interpret the Kripke semantics as a network of observer moments, with the accessibility relation connecting an observer moment to its successor. However, what I don't know is why I should make this interpretation. Why not? It is a natural interpretation of S4 type of logic, especially if you accept to interpret the accessibility relation as relation between OMs. It is the case for any interpretation of any theory. Perhaps I miss something here. Of course we could feel even more entitled to take the temporal interpretation once we accept Brouwer temporal analysis of intuitionist logic. Beth and Grzegorczyk have defend similar interpretations. I will come back on the question of interpreting Kripke structure once I will translate a theory by Papaioannou in those terms next week (after a brief explanation of what Kripke structures are for the non-mathematician). Bruno Fair enough. It is very similar to the situation in my ontology of bitstrings, asking how bitstrings can observe themselves. The way I would probably phrase things is to appeal to something like my TIME axiom as implying a relationship between observer moments. These in turn naturally map into a Kripke structure defining a modal logic for knowlegde contained in each observer moment. Then we can do your Thaetetus move and so on. This is in the reverse order to the way it is presented in your thesis, but it makes more sense to me. Is there some error of logic in thsi process? Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpmAVvAN2DLy.pgp Description: PGP signature
Re: One more question about measure
Le 05-juil.-05, à 12:32, Russell Standish a écrit : On Tue, Jul 05, 2005 at 12:09:24PM +0200, Bruno Marchal wrote: How does it give the logic of temporal knowledge? I understand from your points below, that the necessitation rule is necessary for Kripke semantics, and its is clear to me that necessitation follows from Thaetetus 1 3, whereas it doesn't follow from consistency alone (one could consistently prove false things, I guess). Right. But then I guess you mean Theaetetus 0 and 1. We loose necessitation once we just add the consistency ~B~P requirement (in Theaetetus 2 and 3). For example from the truth t we can deduce BP, but we cannot deduce Bt ~B~t nor Bt ~B~t t. I recall: BP (Theaetetus 0) BP P (Theaetetus 1) BP ~B~P (Theaetetus 2) BP ~B~P P (Theaetetus 3) ? If D'P = BP ~B~P P, then D'P = P (ie necessitation). So it seems it is the conjunction of truth of P that gives rise to necessitation, no? No. Necessitation is the inference rule according to which if the machine proves (soon or later) the proposition p then the machine will prove soon or later D'p. D'p - p is the reflexion axiom for D' (indeed true for the logic obtained by applying Theaetetus 1 and 3 on G). Er ... Russell, if I have been wrong or especially unclear on that point somewhere in SANE or another paper I would be very pleased in case you tell me precisely where. I am quite able to confuse terms myself! I still haven't figured out how to get temporality from a modal logic. Sure I can _interpret_ a logic as having Kripke semantics, and I can interpret the Kripke semantics as a network of observer moments, with the accessibility relation connecting an observer moment to its successor. However, what I don't know is why I should make this interpretation. Why not? It is a natural interpretation of S4 type of logic, especially if you accept to interpret the accessibility relation as relation between OMs. It is the case for any interpretation of any theory. Perhaps I miss something here. Of course we could feel even more entitled to take the temporal interpretation once we accept Brouwer temporal analysis of intuitionist logic. Beth and Grzegorczyk have defend similar interpretations. I will come back on the question of interpreting Kripke structure once I will translate a theory by Papaioannou in those terms next week (after a brief explanation of what Kripke structures are for the non-mathematician). Fair enough. It is very similar to the situation in my ontology of bitstrings, asking how bitstrings can observe themselves. The way I would probably phrase things is to appeal to something like my TIME axiom as implying a relationship between observer moments. These in turn naturally map into a Kripke structure defining a modal logic for knowlegde contained in each observer moment. Then we can do your Thaetetus move and so on. This is in the reverse order to the way it is presented in your thesis, but it makes more sense to me. Is there some error of logic in thsi process? It is ok because the move are not logically related. Note that the first person knowledge axioms S4 are not mine, but are those admitted by almost everyone in the (analytical) philosophical field. But I don't choose them. I am forced to define knowledge by Theaetetus one (it is the simplest way to get the first axiom of S4 which is the reflexion formula and which is obligatory to have a first person), and it is suggested by the fact the (Bp p) *is* equivalent to Bp (as G* told us). It is non trivial because G told us the machine cannot justify that equivalence (although true, this is a consequence of incompleteness). This leads to the soundness of the resulting S4, and that is nice, but not so amazing. But then we get antisymmetry for the Kripke accessibility relation, and this is a truly amazing gift (non trivial to prove). This confirmes the genuine character of the Theaetetus definition in this context because it makes the machine first person notion, not only a knower (in the analytical sense) but a time experiencer sort of knower akin to Brouwer's theory of consciousness. I will say more later. The knower is just a step toward the observer, who gamble on its successor observer-moments. Best regards, Bruno http://iridia.ulb.ac.be/~marchal/
Re: One more question about measure
On Tue, Jul 05, 2005 at 04:03:11PM +0200, Bruno Marchal wrote: If D'P = BP ~B~P P, then D'P = P (ie necessitation). So it seems it is the conjunction of truth of P that gives rise to necessitation, no? No. Necessitation is the inference rule according to which if the machine proves (soon or later) the proposition p then the machine will prove soon or later D'p. D'p - p is the reflexion axiom for D' (indeed true for the logic obtained by applying Theaetetus 1 and 3 on G). Er ... Russell, if I have been wrong or especially unclear on that point somewhere in SANE or another paper I would be very pleased in case you tell me precisely where. I am quite able to confuse terms myself! You are right, my apologies. I read the necessitation rule backwards in your thesis. You do in fact say P = []P. I'll take your word for it that consistency destroys necessitation, but I don't have the intuitive understanding of it yet. Never mind, it is enough for my present purposes. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgp4kHcDcinhc.pgp Description: PGP signature
Re: One more question about measure
On Fri, Jun 24, 2005 at 03:25:29PM +0200, Bruno Marchal wrote: Perhaps. It depends of your definition of OM, and of your everything theory. Let me tell you the Lobian's answer: if I have a successor OM then I have a successor OM which has no successor OM. OK, I am cheating here, but not so much. As I just said to Stathis I must find a way to convince people about the urgency of using the modal logical tools. This reminds me of something I wanted to ask you Bruno. In your work you axiomatise knowledge and end up with various logical systems that describe variously 1st person knowledge, 1st person communicable knowledge, 3rd person knowledge etc. In some of these, the Deontic axiom comes up, which if translated into Kripke semantics reads all worlds have a successor word (or no worlds are terminal). Yet it appears that you would like to identify what I call psychological time as a succession of worlds that follow according to these Kripke semantics. Particularly in comments like the above. Since we have started with an interpretation of knowledge, formalised it to a logical system - how do we make sense of this backwards interpretation, ie treating the Kripke semantics as describing the 1st person appearance of time? You make no reference (or perhaps only hints) in your Lille thesis - I never got around to downloading your Brussels thesis - is there something in that? or is it a more recent development. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics0425 253119 () UNSW SYDNEY 2052 [EMAIL PROTECTED] Australiahttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 pgpA8BeDVEbTX.pgp Description: PGP signature
Re: One more question about measure
Le 26-juin-05, à 03:22, Quentin Anciaux a écrit : Le Samedi 25 Juin 2005 18:51, Bruno Marchal a écrit : Not really because you assume our eyes are bounded. Any finite machine running forever recurs but not infinite or universal one. Bruno Yes I assume my eyes are bounded... because they are, physically speaking they are... Well, it could depend by what you mean by i. And by seeing. Stathis has also assumed in his reasoning that our number of neurons is bounded, but a human can be defined in a more large sense which include the wall on which he draws buffalos. What is important is that we are extendible in principle, at least to make sense of church's thesis and universal machine, and things like all OM. And if I understand you correctly, you are saying that we are universal machine (or we are part of it) so that we can't recurs... I should have said that we don't *necessarily* recur. (And then IF we don't recur, we cannot prove it. We always possibly recur). But as I have showed, what I can see is finite (without taking into accound brain states which is more than 2 states for a neuron, 2 states or electrical states of the brains and not taking in account chemicals properties is not brain states)... what ever event a possible observer which could see all is finite... I take 10x10 resolution, taking an higher resolution will just reveal better and better detail, but we do not see infinite detail... (and I don't conceive my consciousness able to see/understand infinite detail). But if I read that an universal machine runing forever can't repeat, that means that the machine will see better details each time... but what does it means for us ? do you mean that we have to see better and better the world ? has we get asymptotically to an infinite age we should be aware of more details ? Depending on what *you* mean by I you can consider it happens all the time or not. We see more and more details from bacteria to ... Hubble. If you buy an artificial brain you still have the option of path toward amnesia, or attempt to live a long life, seeing more and more detailed but also, and mainly, grasping bigger view on the spectacle. But the price is bigger problems like escaping (or not) black holes, etc. The interest of hypotheses like comp and variants, is not really that it solves such questions, but it can help to formulate them more clearly and it can help to give an idea how complex they are. Bruno http://iridia.ulb.ac.be/~marchal/
Re: One more question about measure
Le 26-juin-05, à 08:47, Russell Standish a écrit : On Fri, Jun 24, 2005 at 03:25:29PM +0200, Bruno Marchal wrote: Perhaps. It depends of your definition of OM, and of your everything theory. Let me tell you the Lobian's answer: if I have a successor OM then I have a successor OM which has no successor OM. OK, I am cheating here, but not so much. As I just said to Stathis I must find a way to convince people about the urgency of using the modal logical tools. This reminds me of something I wanted to ask you Bruno. In your work you axiomatise knowledge and end up with various logical systems that describe variously 1st person knowledge, 1st person communicable knowledge, 3rd person knowledge etc. In some of these, the Deontic axiom comes up, which if translated into Kripke semantics reads all worlds have a successor word (or no worlds are terminal). OK. For the benefit of others I recall that I am interested in what machines can prove and can expect about themselves. The goal is to interview the machine about the measure on the collection of its maximal consistent extensions. The maximal consistent extensions are playing the role of the computational histories, but in a language available to the machine (by Godel technic). I limit myself to sound machine with enough provability power. By a technic akin to godel or meta-programing, we can translate the machine proves some proposition P> *in* the language of the machine. I note that translation Bp. So BP means the machine proves P in the language of the machine. It can be shown that if the machine proves some proposition P, then such a machine has enough introspective power to also prove that she prove P. We have If the machine proves P, then the machine proves BP, which, for a modal logician, is the closer under the necessitation rule. Then, the machine has again enough introspection to know this, in the sense, for any formula P in its language, she can proves the true formula: BP -> BBP She can prove also that she is close for the modus ponens: if she proves A and if she proves A->B, then she proves that B. She know that means that for any A she proves (and its is true) that B(P->Q) -> (BP -> BQ) And it can be proved that she is Loebian, which means she proves BP->P only if she actually proves P. In particular she cannot prove Bf -> f. (where f = 0≠1). Actually she knows she is Loebian: for any P, she proves B(BP -> P) -> BP OK? By a theorem of Solovay, those formula and rules axiomatizes soundly and completely the (propositional) part of the machine's provability logic. Those formula and rules constitutes the logic G). If the formula BP->P is true for any P, this really means the machine is sound. But the machine cannot prove its soundness. She cannot prove BP->P for any P. She cannot prove, for example its own consistency ~Bf (~P is equivalent to P-> f). So although (BP P) is generally equivalent to BP about the machine, she cannot generally prove it, and from its perspective BP is not (necessarily) equivalent to (BP P). So provability, from the machine perspective, behaves like a belief modality, where Bp does not (necessarily) entails P. I recall that for knowledge CP, philosopher asks for both CP -> P, and the closure for the necessitation rule. But then this means we can define knowledge of P, CP, by BP P. And then we can interview the machine (through an infinite conversation, ok, but finitely summarized thanks to Solovay's G) about the logic of knowledge CP. This gives a logic of temporal knowledge of a knower verifying the philosophers' most agreed upon definition. I take it as the simplest first person notion definable in the language of the machine. [Careful here: CP will appear to be only very indirectly definable by the machine: no machine can give a third person description of its CP logic! The logic of CP is the system known as S4Grz. The subjective temporality aspect come from the fact that on finite transitive frames respecting the Grz formula the Kripke accessibility relation is antisymmetric and reflexive, like in Bergson/Brouwer conception of time. See perhaps: van Stigt, W. P. (1990). Brouwer's Intuitionism, volume 2 of Studies in the history and philosophy of Mathematics. North Holland, Amsterdam. Boolos, G. (1980b). Provability in Arithmetic and a Schema of Grzegorczyk. Fundamenta Mathematicae, 96:41-45 Goldblatt, R. I. (1978). Arithmetical Necessity, Provability and Intuitionistic Logic. Theoria, 44:38-46. (also in Goldblatt, R. I. (1993). Mathematics of Modality. CSLI Lectures Notes, Stanford California). See also http://homepages.inf.ed.ac.uk/v1phanc1/dummet.html Note that BP -> P is equivalent to ~P -> ~B~ ~P, and if that is true/provable for any P, then it is equivalent to P -> ~B~p, so BP -> P, as axioms, entails BP -> ~B~P (the deontic formula). But, by incompleteness the reverse is false. Now you were just pointing on tis little less simple definition of first person based on
Re: One more question about measure
Le 26-juin-05, à 08:47, Russell Standish a écrit : On Fri, Jun 24, 2005 at 03:25:29PM +0200, Bruno Marchal wrote: Perhaps. It depends of your definition of OM, and of your everything theory. Let me tell you the Lobian's answer: if I have a successor OM then I have a successor OM which has no successor OM. OK, I am cheating here, but not so much. As I just said to Stathis I must find a way to convince people about the urgency of using the modal logical tools. This reminds me of something I wanted to ask you Bruno. In your work you axiomatise knowledge and end up with various logical systems that describe variously 1st person knowledge, 1st person communicable knowledge, 3rd person knowledge etc. In some of these, the Deontic axiom comes up, which if translated into Kripke semantics reads all worlds have a successor word (or no worlds are terminal). Or, more simply said: with the logic of of BP, G, the logic of third person self-reference, there are cul-de-sac (terminal world) everywhere. All variants of BP (Theatetus 1, 2, 3) are ways of making abstraction of the cul-de-sac worlds, with the goal of getting probabilities. (And those ways are justified by G* which knows more about the machine, if you remember G*). To have probability(P) = one, it is enough to have the truth of P in all accessible OMs. But, alas, in a cul-de-sac world/OM we have that Prob(P) = one trivially (no counterexemples, I assume classical logic in all worlds). To have a probability one we must assure the existence of at least one model, (or one accessible OM, or one consistent extension, etc.). This is provide by the deontic transform where DP is defined by BP ~B~P. Err ... I hope you remember how to see that ~B~p is equivalent with p is consistent for the machine. Modally ~B~P is the dual of BP, it is the diamond which I wrote P. ~B~ gives the simple way to talk on possible world/state/OM... with the machine. The machine stays mute if you ask her if there is one (at least) consistent extension (OM). But she becomes chatty when you ask her what would the worlds look like in case some world exists. Bruno http://iridia.ulb.ac.be/~marchal/
Re: One more question about measure
Quentin Anciaux writes: 1) assume an observer that can see. 2) assume that the observer can see only at a certain resolution/level (it's true that I can't see everything, I do not see quarks for example, nor my cells) Then, I can digitalize every image that I (assuming I'm an observer ;) can see. Now, I'll take an arbitrary image resolution far upper than I details I can actually be aware of. For example : 10x10 pixels, every pixels can have 16.5 millions colors (even if it has been proven that humans can only see less than 20 colors, just for the argument). Then the limit for the eyes to see individual images in a movie is approximately 40hz, so for the argument I will say that I need at least 100 frames by second (higher than what we can perceive). Now how much bits do I need to encode one hour of visual events ? It's simply 10x10x4x100x3600. So the needed number of bits to encode one hour of visual events at a resolution far higher than what we can perceive is finite... It's the same if you replace one hour by the length of a lifetime (+/- 80 years). So even if we are immortal, at a given time in the far away future, the visual events must repeat. You can also work out directly how many possible experiences a human can have. A normal brain has about 10^11 neurons, and each of these neurons can have only one of two states, on or off. This means that the maximum number of possible brain states is 2^10^11, so the number of possible experiences must be less than this. While this is a *huge* number (even if you take into account the fact that the vast majority of possible brain states are nonsense and don't give rise to experiences), it is nevertheless finite, and as you concluded, this means we would start repeating experiences if we lived long enough. However, many people who think about what life would be like if our species survives into the far future - many thousands or millions of years - envisage that our current biological form will be discarded in favour of something more durable and powerful, such as living as sentient software on a computer network. What will happen in this case depends on which cosmological model you follow, but if the network is forever expanding in size in an infinite universe, then there will always be more processing power for new experiences. --Stathis Papaioannou _ REALESTATE: biggest buy/rent/share listings http://ninemsn.realestate.com.au
Re: One more question about measure
Hi Quentin, Hi Bruno, Le Vendredi 24 Juin 2005 15:25, Bruno Marchal a écrit : Because if everything exists... every OM has a successor (and I'd say it must always have more than one), Perhaps. It depends of your definition of OM, and of your everything theory. Let me tell you the Lobian's answer: if I have a successor OM then I have a successor OM which has no successor OM. I don't understand this statement, for me, every OM has a successor, like every integer has. How could it be that an OM can't have a successor ? But I'm firmly convinced that the set of visual OM (I mean by visual, something an observer like a human can see) is finite. OK. But you could take the whole perception field. Our skin is finite too. Etc. Oh Stathis take even the state of each neurons ... Anyway, by the comp hyp I presuppose at once there is such finite description level. I have an example for this : 1) assume an observer that can see. 2) assume that the observer can see only at a certain resolution/level (it's true that I can't see everything, I do not see quarks for example, nor my cells) Then, I can digitalize every image that I (assuming I'm an observer ;) can see. Now, I'll take an arbitrary image resolution far upper than I details I can actually be aware of. For example : 10x10 pixels, every pixels can have 16.5 millions colors (even if it has been proven that humans can only see less than 20 colors, just for the argument). Then the limit for the eyes to see individual images in a movie is approximately 40hz, so for the argument I will say that I need at least 100 frames by second (higher than what we can perceive). Now how much bits do I need to encode one hour of visual events ? It's simply 10x10x4x100x3600. So the needed number of bits to encode one hour of visual events at a resolution far higher than what we can perceive is finite... It's the same if you replace one hour by the length of a lifetime (+/- 80 years). So even if we are immortal, at a given time in the far away future, the visual events must repeat. Not really because you assume our eyes are bounded. Any finite machine running forever recurs but not infinite or universal one. Bruno http://iridia.ulb.ac.be/~marchal/
Re: One more question about measure
Le 22-juin-05, à 19:50, Quentin Anciaux a écrit : I have one more question about measure : I don't understand the concept of 'increasing' and 'decreasing' measure if I assume everything exists. Me neither. Especially when I accept, for the sake of some argument, the ASSA (Absolute Self-Sampling-Assumption) idea. If the measure is relative to your current state/OM, then it makes at least as much sense than Because if everything exists... every OM has a successor (and I'd say it must always have more than one), Perhaps. It depends of your definition of OM, and of your everything theory. Let me tell you the Lobian's answer: if I have a successor OM then I have a successor OM which has no successor OM. OK, I am cheating here, but not so much. As I just said to Stathis I must find a way to convince people about the urgency of using the modal logical tools. and concerning good or bad OM, every OM has good successor and bad successor. What I want to mean is that, I get 100% chance that at least one (I'd say many) of my futur selves will go in hell, and at least one (I'd say also many) will have great experiences. And this, whatever I do... because when I do something, the universe split, and there are branches were I do other thing. I can't constraint the choice. So what is the meaning of increasing and decreasing measure ? What is wrong in every OM has a successor in an everything context ? Here too I could give a precise answer, which is that every OM has a successor, when looking at some absolute third person view, but that that truth is not communicable by the 1-person observer sigh. Have you bought the Smullyan's FU ? (Forever Undecided) Bruno http://iridia.ulb.ac.be/~marchal/
Re: One more question about measure
Hi Bruno, Le Vendredi 24 Juin 2005 15:25, Bruno Marchal a écrit : Because if everything exists... every OM has a successor (and I'd say it must always have more than one), Perhaps. It depends of your definition of OM, and of your everything theory. Let me tell you the Lobian's answer: if I have a successor OM then I have a successor OM which has no successor OM. I don't understand this statement, for me, every OM has a successor, like every integer has. How could it be that an OM can't have a successor ? But I'm firmly convinced that the set of visual OM (I mean by visual, something an observer like a human can see) is finite. I have an example for this : 1) assume an observer that can see. 2) assume that the observer can see only at a certain resolution/level (it's true that I can't see everything, I do not see quarks for example, nor my cells) Then, I can digitalize every image that I (assuming I'm an observer ;) can see. Now, I'll take an arbitrary image resolution far upper than I details I can actually be aware of. For example : 10x10 pixels, every pixels can have 16.5 millions colors (even if it has been proven that humans can only see less than 20 colors, just for the argument). Then the limit for the eyes to see individual images in a movie is approximately 40hz, so for the argument I will say that I need at least 100 frames by second (higher than what we can perceive). Now how much bits do I need to encode one hour of visual events ? It's simply 10x10x4x100x3600. So the needed number of bits to encode one hour of visual events at a resolution far higher than what we can perceive is finite... It's the same if you replace one hour by the length of a lifetime (+/- 80 years). So even if we are immortal, at a given time in the far away future, the visual events must repeat. Quentin Anciaux
Re: One more question about measure
Please replace bits by bytes ;) Quentin Anciaux
Re: One more question about measure
Hi Quentin, Stathis Quentin Anciaux wrote: Hi list, I have one more question about measure : I don't understand the concept of 'increasing' and 'decreasing' measure if I assume everything exists. Because if everything exists... every OM has a successor (and I'd say it must always have more than one), and concerning good or bad OM, every OM has "good" successor and "bad" successor. What I want to mean is that, I get 100% chance that at least one (I'd say many) of my futur selves will go in hell, and at least one (I'd say also many) will have great experiences. And this, whatever I do... because when I do something, the universe split, and there are branches were I do other thing. I can't constraint the choice. So what is the meaning of increasing and decreasing measure ? What is wrong in every OM has a successor in an everything context ? Quentin Hi Quentin In my opinion you are right in suspecting that there is something wrong with increasing or decreasing measure. Since a conscious observer cannot subjectively distinguish between a large (infinite) number of observer moment, he occupies or "surfs" over all of them. Taking a quantum branch does not reduce the number of observer moments because they are still an infinite number of them, and merging branches does not increase the number of observer moment because their sum is also infinite. For this reason I am a firm believer that one can only talk about relative measure (and the RSSA) and not about absolute measure (and the ASSA). Relative measure is the ratio of the number of observer moments before an event and the number after the event. Thus in discussing measure you must define two points: before and after. And you must define an observer and the person or object being observed. If the number of OMs goes to infinity, we can still take a ratio "in the limit". Since the actual number of OMs is infinite, we can normalize measure by defining relative measure for an observer observing himself as equal to 1: that is the number of OMs for an observer divided by the number of OMs for the observer). A given observer can then calculate the relative measure for someone else going between two states as the ratio of the number of OM's between those two states. Thus if an observer carried with him a relative measure measuring instrument (that measures the number of OM's and divides them by themselves) he would find that no matter how risky his behavior is, his own measure remains invariant and fixed at 1. From my own point of view, my relative measure today is not greater or smaller than my relative measure yersterday. The measure of an old and sick man is not greater or smaller than that of a healthy baby that he observes. Some of the other threads in this list (i.e., another puzzle described by Stathis) discuss experiments in which observers are copied and destroyed. Answers to these questions depend on which two points are selected to define relative measure. George Levy Stathis Wrote: Another puzzle: You find yourself in a locked room with no windows, and no memory of how you got there. The room is sparsely furnished: a chair, a desk, pen and paper, and in one corner a light. The light is currently red, but in the time you have been in the room you have observed that it alternates between red and green every 10 minutes. Other than the coloured light, nothing in the room seems to change. Opening one of the desk drawers, you find a piece of paper with incredibly neat handwriting. It turns out to be a letter from God, revealing that you have been placed in the room as part of a philosophical experiment. Every 10 minutes, the system alternates between two states. One state consists of you alone in your room. The other state consists of 10100 exact copies of you, their minds perfectly synchronised with your mind, each copy isolated from all the others in a room just like yours. Whenever the light changes colour, it means that God is either instantaneously creating (10100 - 1) copies, or instantaneously destroying all but one randomly chosen copy. Your task is to guess which colour of the light corresponds with which state and write it down. Then God will send you home. Having absorbed this information, you reason as follows. Suppose that right now you are one of the copies sampled randomly from all the copies that you could possibly be. If you guess that you are one of the 10100 group, you will be right with probability (10100)/(10100+1) (which your calculator tells you equals one). If you guess that you are the sole copy, you will be right with probability 1/(10100+1) (which your calculator tells you equals zero). Therefore, you would be foolish indeed if you don't guess that you in the 10100 group. And since the light right now is red, red must correspond with the 10100 copy state and green with the single copy state. But just as you are about to write down your conclusion, the light changes to green...
