pjn@brillig (P. J. Narayanan) (05/22/87)
I have this opinion about Heisenberg's uncertainty principle. The principle says that (in one of its forms), DELTA-p * DELTA-x >= h-bar / 2. I have complaints about the popular interpretation of this that you cannot know for certain the position and momentum of any body. This, while true, doesnot, in my opinion, imbibe all the deep meanings of the principle. Since the Right Hand Side is a number greater than 0, this equation suggests that you cannot measure the position (or momentum) of any body without any uncertainty in it. That is, DELTA-p or DELTA-x can never be *EQUAL* to 0. This suggests that you cannot know anything, repeat *ANYTHING*, for certain in this world. It seems to me that this inference is too strong a one philosophically, but really unavoidable from the equation. It also makes one think sbout the various notions men have about knowing things for certain and the practice of taking oath and testifying in a court of law etc. I would like to see some opinions on the net, highlighting the issues involved. How can we sort of "tune" the above inference to everyday life; avoiding very strong inferences?? P J Narayanan ------------- ______________________________________________________________________________ God told me yesterday that he really doesnot play dice! ______________________________________________________________________________
ogil@sphinx.uchicago.edu (Lord Julius) (05/22/87)
In article <6762@mimsy.UUCP> pjn@brillig.UUCP (P. J. Narayanan) writes: > >I have this opinion about Heisenberg's uncertainty principle. > [Much deleted] > >This suggests that you cannot know anything, repeat *ANYTHING*, for certain >in this world. It seems to me that this inference is too strong a one >philosophically, but really unavoidable from the equation. It also makes one >think sbout the various notions men have about knowing things for certain >and the practice of taking oath and testifying in a court of law etc. > [Deleted] > >P J Narayanan >------------- While it is true that Heisenberg's principle says we cannot know exactly the position or the momentum of any particle, applying it to the macro- scopic world in the form you propose makes little sense. It all depends on what you mean by "certainty." There are things which I consider certain. To take a silly example, let's look at gravity. I am certain that if I drop a stone it will fall to the ground. Of course, that's trivial. But I also know that, neglecting air resistance, it will fall with an acceleration of 9.8 meters per second squared. This is not _exact_, but it is certain. Within a very small (microscopic) degree of error, I can tell you where the center of mass of that rock is going to be at any given point in time. I can never tell exactly where it is or how fast it's moving, but that's OK. I know enough for my peace of mind. In a recent _Fantasy and Science Fiction_ column, Isaac Asimov wrote that he is glad to be living in an age where the basic laws of nature are known. I don't completely agree with that, but his justification is decent. He said that we know the rules of the universe very well on a macroscopic scale, and that is the scale we have to deal with. We even know what's going on on the atomic level, and we have a good idea about the nuclear level. The business of science is to refine our knowledge. Of course, major new ideas will result from this refinement, but they're not likely to change the macroscopic application of the laws we know now. With regard to moral certainty, we are entering an entirely different ballpark. Unless you believe humans are automatons whose every motion is decided by the particles in our body, you cannot apply Heisenberg's principle. I'm not qualified to discuss moral certainty with any regard to its history, but I do believe it cannot be equated with physical certainty. Certainty in courtroom testimony, regardless of its philosophical problems, seems to be unattainable in most cases, thanks to the fallibility of human memory. Studies show that less than one half (I can't recall exact percentages) of eyewitnesses disagree on the details of observed mock crimes. Perhaps we need a Heisenberg principle for memory: (uncertainty in detail) * (uncertainty in time & location) > n, where n is any natural number you care to choose (:-). I am interested to see what some philosophers have to say out there. Any takers? (he says, climbing into his asbestos suit...) -- Brian W. Ogilvie | "Tenants of the house, ...{uwvax,hao}!oddjob!sphinx!ogil | Thoughts of a dry brain in a dry season."
erhoogerbeet@watmath.UUCP (05/23/87)
In article <6762@mimsy.UUCP> pjn@brillig.UUCP (P. J. Narayanan) writes: >I have this opinion about Heisenberg's uncertainty principle. > >The principle says that (in one of its forms), > > DELTA-p * DELTA-x >= h-bar / 2. >I have complaints about the popular interpretation of this that you cannot >know for certain the position and momentum of any body. This, while true, >doesnot, in my opinion, imbibe all the deep meanings of the principle. Since >the Right Hand Side is a number greater than 0, this equation suggests that >you cannot measure the position (or momentum) of any body without any >uncertainty in it. That is, DELTA-p or DELTA-x can never be *EQUAL* to 0. > >This suggests that you cannot know anything, repeat *ANYTHING*, for certain >in this world. It seems to me that this inference is too strong a one >philosophically, but really unavoidable from the equation. It also makes one >think sbout the various notions men have about knowing things for certain >and the practice of taking oath and testifying in a court of law etc. [some deleted] Though I am certainly no philosopher, I seem to remember this (heard from someone about Descartes' ramblings). Let us assume that there is nothing that can be known for certain. Then, a contradiction is reached because we have taken as certain the hypothesis that nothing can be known for certain. Therefore, it must be so that you can know something for certain. The actual argument continues and finally ends up in the famous "cogito ergo sum." - I think therefore I am. I don't know what this means in relation to Heisenberg's principle, but I think we can know *SOMETHING*. Also, I think Heisenberg's principle relates to subatomic particles only. Of course, you seem to be implying that since matter (ie you and me) is composed of subatomic particles, there is a small probability that matter will not neccessarily be where it is supposed to be and therefore not react as it should. As for testimony: human memory is imperfect enough that even the most objective of people have biological, cultural and psychological biases. I wonder what a real (as opposed to supposed, arm-chair or imaginary) philosopher would think. Just thought I'd get my piece in. ------ --------- ------------------------------------------ erhoogerbeet@watmath.uucp "`The Guide says there is an art to flying,' ehoogerbeets@wateuler.uucp said Ford,`or at least a knack. The knack lies Edwin (Deepthot) in learning how to throw yourself at the ground Hoogerbeets and miss.' He smiled weakly."
ma188saa@sdcc3.ucsd.EDU (Steve Bloch) (05/25/87)
In article <6762@mimsy.UUCP> pjn@brillig.UUCP (P. J. Narayanan) writes: > [Heisenberg's uncertainty principle, sorta] > >This suggests that you cannot know anything, repeat *ANYTHING*, for certain >in this world. It seems to me that this inference is too strong a one >philosophically, but really unavoidable from the equation. It also makes one >think sbout the various notions men have about knowing things for certain >and the practice of taking oath and testifying in a court of law etc. It suggests to me that you cannot know anything BY OBSERVATION for certain. Things that you make up and derive (i.e. most of mathematics) can be known with the same degree of confidence that you have in your own logical consistency. Heisenberg alone doesn't destroy human knowledge; it's the one-two punch of Heisenberg and Goedel. I just thought of something: a consistent logician cannot believe in its own consistency IF IT HAS READ GOEDEL. But a person with a mental block against Goedel's proof is perfectly capable of consistently believing in its own consistency. Anyone for banning all publication of Goedel's proof, or anything inspired by it, so that humans can be consistent again? Sorry; I'm falling into Russell-and-Whitehead, who avoided self-referential statements by making them inadmissible in the same way that I'm making Goedel inadmissible. I suspect any procedure capable of RECOGNIZING Goedelese reasoning would necessarily believe it, and therefore not consistently believe in its own consistency.
ma188saa@sdcc3.ucsd.EDU (Steve Bloch) (05/25/87)
In article <13261@watmath.UUCP> erhoogerbeet@watmath.UUCP (Edwin (Deepthot)) writes: >Though I am certainly no philosopher, I seem to remember this (heard from >someone about Descartes' ramblings). > >Let us assume that there is nothing that can be known for certain. Then, >a contradiction is reached because we have taken as certain the hypothesis >that nothing can be known for certain. Therefore, it must be so that you can >know something for certain. > Did Descartes really say that? I'm disappointed. Let X, in the following sentence, be "nothing can be known for certain." The truth of X does not imply that anybody KNOWS X. (In the same way, there could well be a God or three without anyone knowing for certain that such existed.) Descartes should have been familiar enough with the difference between truth and knowledge to realize that. Playing Devil's advocate, I guess I should point out that a statement which inherently cannot be proven is necessarily irrelevant. If X cannot be proven, then ~X cannot be disproven, so ~X is consistent with any possible universe and so tells us nothing about which possible universe we're in. (I would use this on Creationists if I could get one to hold still long enough.)
obnoxio@BRAHMS.BERKELEY.EDU (Obnoxious Math Grad Student) (05/25/87)
In article <3977@sdcc3.ucsd.EDU>, ma188saa@sdcc3 (Steve Bloch) writes: >In article <6762@mimsy.UUCP> pjn@brillig.UUCP (P. J. Narayanan) writes: >> [Heisenberg's uncertainty principle, sorta] Most comments about P J Narayanan's reasoning have restricted themselves to the stance that momentum and position are not all there is in the world. I tend to agree, and found no way of reading his statement but as extreme positivism. (Or possibly an attempt to stir up trouble.) But there's a more fundamental objection, even within such a strong and uncompromising philosophical outlook: Namely, one has no "right" to attach "certain" position/momentum to a particle in the first place. We do so only out of habit from the every- day macroscopic world. We all recognize instantly that it's nonsense to talk about, say, the marital status of electrons. What has to be (un)learned is that it is also nonsensical to talk about *the* momentum, etc, of electrons, ex- cept according to the rules of QM. They no longer become simple num- bers, attached to particles like so many pin-the-tail-on-the-donkey games, but are--and this is speaking loosely--probabilistic mixes of values. These mixtures are all that can be said, and there are no fundamental limitations on knowing just what these mixtures are. (The above is, of course, how things go within standard interpretations of QM. If someone doesn't want to believe in QM or Copenhagen, etc, fine. That's a different question; I'm just summarizing the party line.) >I just thought of something: a consistent logician cannot believe >in its own consistency IF IT HAS READ GOEDEL. [...] > I suspect any procedure >capable of RECOGNIZING Goedelese reasoning would necessarily believe >it, and therefore not consistently believe in its own consistency. Maybe, maybe not. Goedel's proof is highly constructive, but that never stopped anyone. A logician I know once received a letter from an ultra-ultra fanatical constructivist, who rejected Goedel's theor- em for not being constructive enough. Moreover, having detected this fatal flaw, the letter writer claimed that he had gone on and had a (very constructive) proof of the consistency of Peano Arithmetic. The mind boggles. ucbvax!brahms!weemba Matthew P Wiener/Brahms Gang/Berkeley CA 94720 The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the Devil to darken the spirit and to confine man in the bonds of Hell ... -Saint Augustine
g-rh@cca.CCA.COM (Richard Harter) (05/25/87)
In article <3977@sdcc3.ucsd.EDU> ma188saa@sdcc3.ucsd.edu.UUCP (Steve Bloch) writes: >I just thought of something: a consistent logician cannot believe >in its own consistency IF IT HAS READ GOEDEL. But a person with a >mental block against Goedel's proof is perfectly capable of >consistently believing in its own consistency. Anyone for banning >all publication of Goedel's proof, or anything inspired by it, so >that humans can be consistent again? >Sorry; I'm falling into Russell-and-Whitehead, who avoided >self-referential statements by making them inadmissible in the same >way that I'm making Goedel inadmissible. I suspect any procedure >capable of RECOGNIZING Goedelese reasoning would necessarily believe >it, and therefore not consistently believe in its own consistency. Er, well, no. It's not that simple. You have to explain what you mean by "consistently believe". (Hint -- don't try :-)). If the logician is consistent, then she may consistently believe in her consistency. This remains true, even though she knows that she cannot demonstrate her consistency. This is much the situation modern logic is actually in. We believe that arithmetic is consistent. Yet we know that we cannot prove the consistency of arithmetic without using methods less elementary than arithmetic. Whence the belief in the consistency of arithmetic is a matter of faith. Incidentally, there is a school that believes that arithmetic is, in fact, inconsistent, but that the smallest demonstration of this inconsistency would take more time than the entire lifetime of the universe. Members of this school consistently believe in the inconsistency of logic. I am not quite up to determining whether they consistently believe in their own inconsistency. -- Richard Harter, SMDS Inc. [Disclaimers not permitted by company policy.]
biep@cs.vu.nl (J. A. "Biep" Durieux) (05/26/87)
In article <3977@sdcc3.ucsd.EDU>, ma188saa@sdcc3.ucsd.edu.UUCP (Steve Bloch) writes: >Sorry; I'm falling into Russell-and-Whitehead, who avoided >self-referential statements by making them inadmissible in the same >way that I'm making Goedel inadmissible. How can that be done? "Any self-referring statement is inadmissible!" Now, is that statement self-referring? Don't you need an appropriate variant of Loeb's theorem ("any sentence which would be self-referring if it were self-referring is self-referring") or its negation to decide on that? And then: is this Loeb-variant self-referring? According to LV (the Loeb-variant) itself, yes, but then ... [Thanks, Ob(noxio), for introducing me to another subtlety of self-reference - is uncle Doug (Hofstadter) on the net?] So the main question remains: how can a non-decidable class of objects be labeled inadmissible? -- Biep. (biep@cs.vu.nl via mcvax) "Law" is the name given to a collection of rules describing how to act with people that do not follow the law.
jiml@alberta.UUCP (05/26/87)
In article <1782@sphinx.uchicago.edu> ogil@sphinx.UUCP (Lord Julius) writes: >In article <6762@mimsy.UUCP> pjn@brillig.UUCP (P. J. Narayanan) writes: >> >>I have this opinion about Heisenberg's uncertainty principle. >[Much deleted] >>This suggests that you cannot know anything, repeat *ANYTHING*, for certain >>in this world. It seems to me that this inference is too strong a one >>philosophically, but really unavoidable from the equation. > >While it is true that Heisenberg's principle says we cannot know exactly >the position or the momentum of any particle, applying it to the macro- >scopic world in the form you propose makes little sense. It all depends >on what you mean by "certainty." > >There are things which I consider certain. To take a silly example, >let's look at gravity. I am certain that if I drop a stone it will fall >to the ground. Of course, that's trivial. But I also know that, neglecting >air resistance, it will fall with an acceleration of 9.8 meters per second >squared. This is not _exact_, but it is certain. Within a very small >(microscopic) degree of error, I can tell you where the center of mass of >that rock is going to be at any given point in time. I can never tell >exactly where it is or how fast it's moving, but that's OK. I know >enough for my peace of mind. >[Much deleted] >I am interested to see what some philosophers have to say out there. >Any takers? (he says, climbing into his asbestos suit...) I'll bite, but first a bit of a disclaimer: I'm a skeptic. I'm not sure how far to push the idea that "no knowledge is possible" (DesCartes' cogito seems pretty reasonable, moreover for practical purposes I'm a realist), but I'll take a stab at the above notion of certainty for starters... Epistemologists often define knowledge as "justified true belief". One can claim to know a proposition if one believes in it, is justified in doing so, and if it also happens to be true. Notwithstanding Gettier's famous counter-example (which I can discuss upon request), I accept this definition. Note that it differs appreciably from the commonplace usage; people often claim to have knowledge if they have a particularly powerful belief, or have little cause to doubt the truth of a claim. The problem is that no matter how strong one's conviction might be, and no matter how justified one might be in holding some belief, there is still a chance of error. As long as that possibility exists, I frown upon any attribution of 'certainty' as a relation between believer and proposition. Richard Dawkins, in his new book, _The Blind Watchmaker_ has a rather vivid example concerning human certainty and statue-waving. He recognizes that the macroscopic phenomena we observe daily are subject to probabalistic events at the sub-atomic level. All of our laws of classical physics are statistical generalizations of quantum events. As statistical epiphenomena, macroscopic occurrances are relatively stable and predictable. But it is conceivable that gross disturbances might arise if sub-atomic events are "just right." None of us have ever seen a statue wave at us. I say this with all the 'certainty' I can muster, and I'm extremely justified in making the claim, given that the probability that the molecules that make up the arm of the statue might co-ordinate themselves in just the right fashion so as to cause the arm to wave is infinitesimal. Yet given enough time, and enough statue-watchers, such an event will occur, whether the observers are certain it will happen or not. Of what value is 99.9999999999999999999999999% certainty/knowledge? It is of immesurable worth, as far as leading our lives go. The decisions we make from minute to minute rely upon information that is far less 'certain' than that. But we shouldn't be using 'certain' in a comparative sense as I did above, but I recognize that that is how it is commonly conceived. For a technical discussion, we need absolute senses of 'certainty' and 'knowledge'. Something is known if and only if it is true. The trouble is, the truth is INACCESSIBLE. Before anyone jumps on me with the rebuttal, "well if you can't know anything then how can you lead your life/do anything at all?", let me emphasize that I have focused on the absolute notion of knowledge. There are lots of claims and theories that I accept and believe, and I'll argue 'til I'm blue in the face why a particular position is a reasonable one to hold, or why it would be utterly foolish to hold a contrary view, but I shy away from any talk about 'truth' in such discussions. So many people have claimed to have the truth. They can't all be right. Oh, one last word: What do I think about the truth or the certainty of statements in logic? Do I hold that ~(p&~p)? Well, I think to deny the law of non-contradiction is an act of complete lunacy, but I have yet to commit myself to the belief that such a statement is true. Jim Laycock ihnp4!alberta!cavell!jiml OR alberta!uqv-mts!Jim_Laycock Philosophy grad, University of Alberta (soon to be at Western Ontario) Interests: Philosophies of Logic, Language, Mind, AI, and everything else. Summer Job: Writing a parser to convert English sentences to sentences in predicate calculus (a la Montague, Gazdar). Supervisor: Jeff Pelletier
turpin@ut-sally.UUCP (05/26/87)
In article <3977@sdcc3.ucsd.EDU>, ma188saa@sdcc3.ucsd.EDU (Steve Bloch) writes: > I just thought of something: a consistent logician cannot believe > in its own consistency IF IT HAS READ GOEDEL. Think again. (Perhaps after studying a logic text.) Godel's *incompleteness* theorems simply say that your hypothetical logician, if consistent, might come across unprovable statements. In the first order predicate calculus, for example, there are fully quantified statements that can neither be proven or disproven. Such a statement, or its denial, can be as an axiom to the system, which remains consistent thereby. Godel is second only to Einstein in having pseudophilosophical claptrap "based" on his work. Poor Godel. Russell
wex@milano.UUCP (05/27/87)
In article <13261@watmath.UUCP> erhoogerbeet@watmath.UUCP (Edwin (Deepthot)) writes: > Let us assume that there is nothing that can be known for certain. Then, > a contradiction is reached because we have taken as certain the hypothesis > that nothing can be known for certain. Therefore, it must be so that you can > know something for certain. Readers of talk.philosophy.misc have argued this on out months ago. The fallacy lies not in knowledge (as Steve assumes below) but in a linguistic trick. Try the following formulation: "I am reasonably sure (P > .999999) that nothing can be known for certain." This claim does not itself contain a claim of certainty, merely an assertion of probability. Thus, no contradiction. In article <3978@sdcc3.ucsd.EDU>, ma188saa@sdcc3.ucsd.EDU (Steve Bloch) writes: > If X cannot be proven, then ~X cannot be disproven... I can see I'm going to have to dig out my archives on intuitionist logic. Steve's statement is problematic only in two valued logic. There are logics in which X can be any of {True, False, Unproven}. Thus, I can show that X is not false and not have shown that it is true. More formally, in a three-valued logic, ~(~P) ==> P is no longer the case. -- Alan Wexelblat ARPA: WEX@MCC.COM UUCP: {seismo, harvard, gatech, pyramid, &c.}!sally!im4u!milano!wex Shit happens.
cracraft@ccicpg.UUCP (05/27/87)
In article <3977@sdcc3.ucsd.EDU>, ma188saa@sdcc3.ucsd.EDU (Steve Bloch) writes: > Heisenberg alone doesn't destroy human knowledge; it's the > one-two punch of Heisenberg and Goedel. > Actually, it was a one-two-three punch (Kant-Heisenberg-Goedel). Also, we seem to be dealing with just Westerners here. To get a complete picture, toss in Easterners like Buddha and the Madhyamika dialectic. Stuart
jbuck@epimass.UUCP (05/27/87)
In article <8135@ut-sally.UUCP> turpin@ut-sally.UUCP (Russell Turpin) writes: >In article <3977@sdcc3.ucsd.EDU>, ma188saa@sdcc3.ucsd.EDU (Steve Bloch) writes: >Godel's *incompleteness* theorems simply say that your >hypothetical logician, if consistent, might come across >unprovable statements. In the first order predicate calculus, for >example, there are fully quantified statements that can neither >be proven or disproven. Such a statement, or its denial, can be >as an axiom to the system, which remains consistent thereby. >Godel is second only to Einstein in having pseudophilosophical >claptrap "based" on his work. Poor Godel. How ironic that your name is Russell! Too bad it's not your last name :-). Actually Einstein isn't even in the running; the other big scientific discovery with philosophical implications is quantum theory; by contrast relativity is only a slight adjustment of Newton. Godel's first incompleteness theorem states that you WILL, not MIGHT, come up with unproveable statements if your axiom system is powerful enough. But his second theorem is more interesting, in that it asserts that a sufficiently powerful system (one that includes Peano arithmetic, for example) can never prove its own consistency. This has far more philosophical consequences than the first. Some people still seem to want to hold on to 19th century clockwork-universe objective (or should I say Objectivist? Better not :-)) viewpoints despite the fact that Heisenburg and Godel blasted these notions quite effectively in the 20s and 30s. You can't know that you're right. You can't observe something without changing it. There is no such thing as an objective observer. C'est la vie. -- - Joe Buck jbuck@epimass.EPI.COM (in the brave new world of domains!) {seismo,ucbvax,sun,decwrl,<smart-site>}!epimass.epi.com!jbuck
flink@mimsy.UUCP (05/27/87)
In article <1146@cavell.UUCP> jiml@cavell.UUCP (Jim Laycock) writes: >There are lots of claims and theories that I accept and believe, and >I'll argue 'til I'm blue in the face why a particular position is a >reasonable one to hold, or why it would be utterly foolish to hold a >contrary view, but I shy away from any talk about 'truth' in such >discussions. So many people have claimed to have the truth. They can't >all be right. I may be missing some technical sense of `truth' which you are using, but isn't believing a proposition pretty much the same as believing that the proposition is true? I'm not necessarily endorsing a redundancy theory of truth, just the good old-fashioned condition that "p is true" iff p. So, if you really believe p, I don't see any reason to shy away from saying that p is true. Perhaps we don't disagree. I think that it is rational sometimes to believe propositions, _with_(psychological)_certainty_, even though they are possibly false. In other words, I'm a fallibilist. And one reason I'm a fallibilist is that I think that in accepting empirical propositions one always, from an epistemic viewpoint, risks falsehood; yet one cannot go about life without believing any empirical propositions. Is this newsgroup an appropriate place to discuss fallibilism? Any takers? -- P.S. I've conspired with news sites everywhere to suppress opposition from anti-certainty philosophers: This program posts news to many thousands of machines throughout... Are you absolutely sure that you want to do this? [ny] :-) Paul Torek flink@mimsy.umd.edu
jiml@alberta.UUCP (05/28/87)
]In article <1146@cavell.UUCP> jiml@cavell.UUCP (Jim Laycock) writes: ]There are lots of claims and theories that I accept and believe, and ]I'll argue 'til I'm blue in the face why a particular position is a ]reasonable one to hold, or why it would be utterly foolish to hold a ]contrary view, but I shy away from any talk about 'truth' in such ]discussions. ] In article <6794@mimsy.UUCP> flink@mimsy.UUCP (Paul V Torek) writes: >I may be missing some technical sense of `truth' which you are using, but >isn't believing a proposition pretty much the same as believing that the >proposition is true? I'm not necessarily endorsing a redundancy theory of >truth, just the good old-fashioned condition that > "p is true" iff p. >So, if you really believe p, I don't see any reason to shy away from saying >that p is true. > >Perhaps we don't disagree. I think that it is rational sometimes to believe >propositions, _with_(psychological)_certainty_, even though they are possibly >false. In other words, I'm a fallibilist. And one reason I'm a fallibilist >is that I think that in accepting empirical propositions one always, from an >epistemic viewpoint, risks falsehood; yet one cannot go about life without >believing any empirical propositions. When I toss a fair coin, I am epistemically indifferent about the outcome of the toss. Do I believe in one outcome or the other? No, I commit myself to neither alternative. So, Bel(Jim,Prob(heads)=.5) and Bel(Jim,Prob(tails)=.5) I believed that the Oilers would win Game 5 of the Cup Finals. I could construct an argument to lend credence to my belief. I was by no means certain that they would win (I say in hindsight), but I thought they stood a pretty good chance. Let's assign a measure of probability, say .75. So, Bel(Jim,Prob(win)=.75) and Bel(Jim,Prob(lose)=.25). I can't say I entertained the latter belief a great deal (until the third period, but by then I had changed my assessment of probability), and one might wish to argue that I had no such belief at all. But I definitely had the former belief in mind. So, while it is the case that I believed that the Oilers would win the game, I would not admit to Bel(Jim,Prob(win)=1.0), and hence I'd never assert that it's true that the Oilers were going to win Game 5. With regard to more reliable empirical observations/predictions, I'd assign much higher probabilities, and would likely reach a point whereat the negation of my belief would never be entertained at all. But, being a skeptic, upon being asked whether a particular proposition was true, I'd be reluctant to agree even to the most blatantly obvious claims.
news@rlvd.UUCP (News) (06/01/87)
In article <8135@ut-sally.UUCP> turpin@ut-sally.UUCP (Russell Turpin) writes: >In article <3977@sdcc3.ucsd.EDU>, ma188saa@sdcc3.ucsd.EDU (Steve Bloch) writes: > >> I just thought of something: a consistent logician cannot believe >> in its own consistency IF IT HAS READ GOEDEL. > >Godel's *incompleteness* theorems simply say that your >hypothetical logician, if consistent, might come across >unprovable statements. In the first order predicate calculus, for >example, there are fully quantified statements that can neither >be proven or disproven. Such a statement, or its denial, can be >as an axiom to the system, which remains consistent thereby. Paraphrasing Godel on incompleteness: "A formal system is either incomplete or inconsistent". Taking the example of the predicate calculus, there are statements that can neither be proven or disproven, but you cannot tell me how many of them there are, and you cannot prove to me that there are no more such statements. Thus the system MAY be consistent, but you cannot know that it is for sure. You may end up with an infinite number of axioms..... Someone should have shot Godel at birth. Ian "Motorcycle Maaaaan" Gunn UK JANET : ian@uk.ac.rl.vd Rutherford Appleton Laboratory UUCP : ..!mcvax!ukc!rlvd!ian Chilton, Didcot, Oxon OX11 0QX ARPA : @ucl.cs.arpa:ian@vd.rl.ac.uk England. 'phone : (0235) 21900 ext: 5707