tag:blogger.com,1999:blog-10584495.post5159597103542679960..comments2024-03-28T12:34:14.649-07:00Comments on dangerous idea: The Case Against FrequentismVictor Repperthttp://www.blogger.com/profile/10962948073162156902noreply@blogger.comBlogger43125tag:blogger.com,1999:blog-10584495.post-89428884549170929912010-12-04T01:07:52.061-07:002010-12-04T01:07:52.061-07:00I wonder why Christians think of the resurrection ...I wonder why Christians think of the resurrection as such an astonishing event?<br /><br />If there is a God who had the astounding power to create everything in existence-billions of planets,galaxies-and the astonishing complexity of life, then wouldn't bringing a three day old corpse back to life be the equivalent of a two-bit magician pulling a rabbit out of a hat? <br /><br />I once saw illusionist David Blaine bring a dead fly back to life. The resurrection of Jesus by the Christian god would be less impressive than that.kilo papahttps://www.blogger.com/profile/15112057471953902453noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-30762002129269730972010-12-02T23:55:54.062-07:002010-12-02T23:55:54.062-07:00"Yet even in the absence of any explanation, ..."Yet even in the absence of any explanation, or any set of priors that would significantly raise the probability of the event, we would still infer that the event probably happened."<br /><br />Graham - I don't think Hume or I would disagree with you. The issue is not the absolute value of prior beliefs but the relative value.Mark Frankhttps://www.blogger.com/profile/07117994136165938870noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-32647656375057453072010-12-02T12:42:22.596-07:002010-12-02T12:42:22.596-07:00BTW: It's not that there aren't answers av...BTW: It's not that there aren't answers available to the Christian evidentialist<br /><br />Eg: Swinburne "Was Jesus God?"pp100-102 & 125-127Mr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-75178547748851130192010-12-02T12:31:20.580-07:002010-12-02T12:31:20.580-07:00Re: "Odin's evangelist"
- yes, I im...Re: "Odin's evangelist"<br /> - yes, I imagine that would provide good evidence that a <i>supernatural event</i> had occurred.<br />However here's something for Christians to ponder, especially those who are inclined to evidentialism. <i>A Biblically informed epistemology would insist that we <b>reject</b> the truth of the Odinists claims.</i> Even though we would have better evidence for Odinism than we have for the Resurrection.<br /> <br />Deut13<br /> 1 If a prophet, or one who foretells by dreams, appears among you and announces to you a sign or wonder, 2 and if the sign or wonder spoken of takes place, and the prophet says, “Let us follow other gods” (gods you have not known) “and let us worship them,” 3 you must not listen to the words of that prophet or dreamer. The LORD your God is testing you to find out whether you love him with all your heart and with all your soul. 4 It is the LORD your God you must follow, and him you must revere. Keep his commands and obey him; serve him and hold fast to him.<br /> <br />Matthew 7<br /> <br />15 “Watch out for false prophets. They come to you in sheep’s clothing, but inwardly they are ferocious wolves. 16 By their fruit you will recognize them. .... 21 “Not everyone who says to me, ‘Lord, Lord,’ will enter the kingdom of heaven, but only the one who does the will of my Father who is in heaven. 22 Many will say to me on that day, ‘Lord, Lord, did we not prophesy in your name and in your name drive out demons and in your name perform many miracles?’ 23 Then I will tell them plainly, ‘I never knew you. Away from me, you evildoers!<br /> <br />The point being that attaching a miracle to a message does not establish the truth of that message.<br /> <br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-50845454987004503352010-12-02T12:30:06.612-07:002010-12-02T12:30:06.612-07:00Mark
I designed the thought experiment so that
(...Mark<br /> <br />I designed the thought experiment so that<br />(a) the form in which the testimony is given by each witness reduces the probability of lying <br />(b) the odds of all these witnesses lying about the same event would be vanishingly low<br />(c) there was no available naturalistic explanation for the purported event (i. journalists, ii. James Randi and Michael Shermer are present to rule out trickery & geologists and other scientists are witnesses iii. geologists cannot replicate the results even though they can replicate the conditions.) If you like, change the purported event to make naturalistic explanations more difficult.<br />(d) the physical traces rule out hallucination etc <br />(e) the point of the thought experiment is that we cannot give an explanation for the event, theological or otherwise. Yet even in the absence of any explanation, or any set of priors that would significantly raise the probability of the event, we would still infer that the event probably happened. <br /><br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-15960835604939247192010-12-02T00:49:22.557-07:002010-12-02T00:49:22.557-07:00"My point is that testimony can at least in t..."My point is that testimony can at least in theory provide evidence of a 'miracle/paranormal event'. Even when the prior probability is vanishingly low on everyone's reasoning"<br /><br />I agree. However, and I think this is Hume's point, the prior probability has to be greater than the prior probability of the alternative that the witnesses were either lying, somehow deluded or misinterpreted. <br /><br />Also you need to be careful about the definition of the priors. We might conclude that the stone did indeed roll up hill. The interesting question is why. <br />One prior might be that he was right, another might be that the centre of the stone is iron and there was a large magnet at the top of the hill the witnesses were not aware of. A third is to recognise we don't yet know why.<br /><br />Let us add one more thing to your example. Suppose immediately before the event an evangelist held an intense session among the assembled audience with much prayer and music and shouting and announced he would cause the stone to roll up hill. Now the priors really kick in - the religious types in the audience would find it far more plausible that supernatural forces caused the stone to roll. The Dawkins, Goldacre crowd would now find it far more likely there was some kind of mass delusion/trick to which they had been subject a la Darren Brown. Your own prior might change if you were told that the evangelist was not Christian but some never heard of sect that worshipped the old Norse Gods.Mark Frankhttps://www.blogger.com/profile/07117994136165938870noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-54834774572591166712010-12-01T15:29:08.290-07:002010-12-01T15:29:08.290-07:00Let's not get too caught up with the issues of...Let's not get too caught up with the issues of priors. Here's a little thought experiment.<br /><br /> Suppose a group consisting of several investigative journalists, police officers, Martin Amis, Reza Aslan, EJ Hobsbawm, Simon Conway Morris, Richard Fortey, Ben Goldacre, Elton John, James Randi, Tim Haggard, Michael Shermer and Stephen Weinberg are all at one event, and all report that they observed a rock roll a short distance up a hill at that event. (Maybe they're all at an outdoor arts festival or something..use your imagination.)<br /><br />Suppose the sceptics are reluctant to report this, as they were unable to explain the event (they could detect no trickery) and we have to do a little digging to discover their testimony. Say it pops up in personal journals and diaries, and a few of the record comments to journalists. But <i>every sceptic on the list unequivocally testifies to the event in some way.</i><br />The few religious witnesses present could conceive of no religious significance. The stone just unexpectedly rolled up a hill for a few metres, then came to a stop. So they mention it in a few interviews and such, but they don't make a fuss in public. They, too, lack an explanation - and they're a little worried that this is a hoax at their expense. <br />Only the journalists and the police seem to get excited over the event. Say they had some photographic evidence, and could point to tracks left in the ground. Yet every effort to recreate the conditions of the event fails to reproduce the event. That is, geologists and physicists can't explain why this rock would behave this way, and they can't make any other rock behave this way. <br /><br />Now, if he had that sort of testimony, with some corroboration, wouldn't John agree that "a rock had rolled up a hill at a public event"? Even though the prior probability is extraordinarily low on his assumptions - <b>and on Vics</b>.<br />The testimony is so improbable if the event did not happen, that it overwhelms the vanishingly low prior probability of the event. <br /><br />Now the evidence for the Resurrection is not that good. But that's not my point. My point is that testimony can <i> at least in theory</i> provide evidence of a 'miracle/paranormal event'. Even when the prior probability is vanishingly low <i> on everyone's reasoning</i><br /><br /><br />Or is John maintaining that he would deny that the event occurred in the face of the testimony? <br /><br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-9364639168666838912010-12-01T15:28:09.576-07:002010-12-01T15:28:09.576-07:00The point being, of course, that banging on about ...The point being, of course, that banging on about "analogy" is pointless. Yes, there must be some 'point of contact' with history - 'a negotiation between familiarity and strangeness' - but historians seek to re-create unrepeatable experiences as a matter of course. <br /><br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-66726978149363845912010-12-01T14:42:41.145-07:002010-12-01T14:42:41.145-07:00I should also point out that historians have to ca...I should also point out that historians have to capture unique and strange experiences all the time. <br /><br />Tosh makes it clear that the historians first duty is to respect the <i>otherness</i> of the past. Many cultures have experienced the world in ways that we struggle to imagine (merely consider the way that we experience <i>time</i> in seconds, minutes and hours. Now compare this to societies that did not have mechanical clocks.)<br /><br />The historian, according to Tosh, must use her imagination and experience to understand and re-create these experiences for the modern reader. <br />"Their purpose is not only to uncover the strangeness of the past but to explain it."<br /><br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-4825775408636352812010-12-01T08:06:02.142-07:002010-12-01T08:06:02.142-07:00Someone on this thread said -- Philosophy is dead....Someone on this thread said -- Philosophy is dead.<br /><br />Oh that is good for a chuckle.GREVhttps://www.blogger.com/profile/10415494137313565242noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-71099308750009520002010-12-01T04:36:02.304-07:002010-12-01T04:36:02.304-07:00You say that frequency theory has falled on hard t...You say that frequency theory has falled on hard times - but there is no theory of probability that is doing better. The philosophy of probability is unsettled.<br /><br />However, whatever your philosophy, the probability of an event depends on what you know about it. It is not a fixed value of the event.<br /><br />Take a dice throw. I throw a dice and it lands six uppermost. What is the probability of that event? <br /><br />Mostly all we know about a dice is most dice are manufactured so that they are roughly symmetrical in shape and weight and that such dice when tossed repeatedly come down with each side in approximately equal frequency. However, we might know more about this dice. We might know that is assymmetrical. We might know that the person tossing it likes to have the one uppermost before tossing and that this leads to sixes being less frequent in the long run. In the extreme we might already know the result of the toss!<br /><br />I believe this is the answer to what class of events a miracle belongs. The class is defined by the amount of relevant knowledge we have about the event (what is relevant is based on our experience of what is likely to influence the outcome). <br />So applying a Bayesian approach to a miracle e.g. water into wine.<br /><br />We have an observation:<br /><br />A small number of people report observing Jesus turning water into wine.<br /><br />We have two competing hypotheses:<br /><br />H1. The water did turn into wine.<br />H2. The reports were erroneous - either the observers were mistaken or they misled us.<br /><br />The prior probability for H1 is based on what we know. For most people this is very little except there was some water and some people wanted it to turn to wine. The observed frequency of water turning into wine under those circumstances other than the case under consideration is zero. <br />The prior probability for H2 is a lot higher. Our knowledge includes that people often make false reports intentionally and unintentionally.<br /><br />Of course you may believe that Jesus was capable of turning water into wine and wanted to do so - so you believe that you have relevant "knowledge" which raises the prior probability of H1 dramatically.Mark Frankhttps://www.blogger.com/profile/07117994136165938870noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-10700932118220889932010-11-30T14:16:41.989-07:002010-11-30T14:16:41.989-07:00"Who is recommending that we put ourselves in..."Who is recommending that we put ourselves in our opponents shoes before we fix a prior probability?"<br /><br />Isn't that implied by John's OTF?David Parkerhttps://www.blogger.com/profile/13714637134009580948noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-76167993132601378452010-11-30T11:15:13.792-07:002010-11-30T11:15:13.792-07:00"I don't think it's even possible to ..."I don't think it's even possible to imagine how the evidence might look to someone with a different belief system"<br /> <br />That's a very odd claim. <br /> <br />History and the social sciences crucially depend on our ability to put ourselves in another person's shoes. To imagine how events would look to individuals with alien belief systems is essential to history.<br />And psychologists commonly infer how events appear to individuals with cognitive structures different to our own - young infants and even primates. Are you arguing that cognitive science, for example, is impossible?<br /> <br />And "you only think that because..." is levelled at believers and non-believers on this blog ad nauseam. <br /> <br />In any case, what on earth does this have to do with fixing priors? Who is recommending that we put ourselves in our opponents shoes before we fix a prior probability?<br /> <br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-11735574965646050552010-11-30T11:14:37.085-07:002010-11-30T11:14:37.085-07:00"so there is independent corroboration of the..."so there is independent corroboration of the premise that someone won the lottery. Not so in the case of miracles."<br />1)<br />Victor says - I have evidence that corroborates that a miracle took place<br /> The sceptic says - no you don't the prior probability of a miracle is too low<br />Victor says - But in many cases, events with an astonishingly low prior probability do occur<br /> The sceptic says - Yes, but we have evidence that corroborates that these events took place!<br />2)<br />The reference to the photograph is not helpful, given that written testimony is the primary source of our knowledge of the past. Suppose we only had written testimony that an individual won a lottery, and some evidence that their wealth increased. <br />Would we ceteris paribus accept the recipient's testimony that they were, indeed, incredibly fortunate?<br /> <br />(Or suppose it was an event with an even lower probability - a eccentric aristocrat decided to donate huge funds to a peasant chosen at random - would we accept testimony that such an event had occurred?)<br /> <br />3)<br />I'm not sure that objecting to subjective probabilities simply because they are subjective is a great strategy for the Frequentist.<br /> <br />4) In any case we do,commonly, have degrees of belief in various propositions. In the absence of an argument that this is irrational, Bayesians who reason from degrees of belief seem to be on safe ground.<br /> <br />5)<br />Another popular objection is "This isn't how I use Bayes Theorem in my day job." I'm not sure if that's what's behind Ed's arguments. But ignorance of Bayesian epistemology isn't an argument against any version of it. Neither is an air of condescension.<br /> <br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-3128095068404206432010-11-29T18:26:22.915-07:002010-11-29T18:26:22.915-07:00Victor also said,
"...Experimentally imagini...Victor also said,<br /><br />"...Experimentally imagining how the evidence might look to someone with a different belief system, which is what's involved in the "outsider test" is something that is worth doing, but I don't think it represents the ultimate test of the adequacy of one's religious (or non-religious) beliefs."<br /><br />I'll go you one further: I don't think it's even <i>possible</i> to imagine how the evidence might look to someone with a different belief system, and is therefore <i>im</i>possible to assign any meaningful Bayesian values to the initial conditions, thus making probability analysis a futile exercise.GearHedEdhttps://www.blogger.com/profile/09288513835630145996noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-58878088854642417172010-11-29T18:22:09.620-07:002010-11-29T18:22:09.620-07:00Victor also said,
"...The question is whethe...Victor also said,<br /><br />"...The question is whether it is really possible to find a "neutral" or "unbiased" position from which to evaluate just how strong the antecedent improbability of a miracle really is. I don't see how you can evaluate the likelihood of miracles independently of a) the antecedent probability of theism and b) the overall likelihood that God, if He exists, would do something along these lines as part of his way of communicating with human beings."<br /><br />This problem you have pointed out is at the root of why Bayesian logic can't prove miraculous claims: You have no basis for assigning any initial probability vlaues.GearHedEdhttps://www.blogger.com/profile/09288513835630145996noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-35537634270564093672010-11-29T18:18:17.779-07:002010-11-29T18:18:17.779-07:00Victor said,
"It's pretty much an open s...Victor said,<br /><br />"It's pretty much an open secret in the study of Hume's essay that a straightforward application of the mathematical probability theory he lays out would lead to, for example, skepticism about newspaper reports that person X won the lottery."<br /><br />Except that we can see the grinning redneck holding the giant fake winnings check in all the publicity photos, so there is independent corroboration of the premise that someone won the lottery. Not so in the case of miracles.GearHedEdhttps://www.blogger.com/profile/09288513835630145996noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-83879633066215695702010-11-29T12:45:00.175-07:002010-11-29T12:45:00.175-07:00Re: Extraordinary events in history.
In "A ...Re: Extraordinary events in history.<br /> <br />In "A Problem from Hell" Samantha Power tries to explain why the allies did not react to the German attempt to exterminate the Jewish people in World War II. Naturally, she points out that the allies suppressed evidence and intelligence of the Holocaust as they did not want to be dragged into any campaign that would interfere with their central objective - the military defeat of Germany. (So no bombers could be spared for Auschwitz etc.)<br /> <br />But Power notes, in an aside, that this does not quite explain all the facts. Enough evidence existed in the public realm to convince the average citizen in the West that the Holocaust was underway. And anti-semitism does not explain why the average Westerner was so disinterested in the plight of the Jews in Europe. Isaiah Berlin could only see a massive anti-Semitic pogrom. This was the analysis of many Zionists also.<br /> <br />So what explains their disbelief? <i>""A plot for outright annihilation had never been seen before and therefore could not be imagined.</i> The tales of German cremation factories and gas chambers sounded far-fetched. The deportations could be explained: Hitler needed Jewish slave labour for the war effort...In the 1920's and 1930s, the press had debunked many of the allies wartime reports of German savagery, yielding "a hangover of skepticism" "<br /> <br />Power also tells the story of Raphael Lemkin, who fled Nazi Germany, and Soviet Poland, as he had read the evidence correctly. Hitler would invade the East, and annihilate the Jewish people. Jewish communities, whether intellectual or traditional, refused to believe him. Pogroms could be expected, and great suffering. But it simply made no sense for a modern nation to turn the engines of total war to an unreasonable goal - the elimination of an ethnic group that posed no threat. The Nazis had something to lose in making such an effort, and they would gain nothing in return.<br /> <br />Samantha Power "A Problem from Hell: America and the Age of Genocide" pp17-61, esp 34-36<br /> <br />Tragically, <i>the Holocaust</i> was a unique historical event. <i>Genocides</i> were common - Lemkin knew this. He studied the Turkish genocide of the Armenians, the Mongol incursions into the West etc. But never before had the machinery of an industrialised state been turned to such an irrational goal. Lemkin knew what was coming. Those who kept their thoughts within the realm of common experience did not.<br />For this reason (and so many, many others) I doubt that simple rules about 'analogy' capture the complexity of history. In fact "analogy" does not even feature in the index of "The Pursuit of History". It's easy to see why.<br />GrahamMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-12049271023362632762010-11-29T12:44:27.771-07:002010-11-29T12:44:27.771-07:00Re: extraordinary events and eyewitness testimony ...Re: extraordinary events and eyewitness testimony - according to BBC's "Horizon" scientists dismissed the possibility of Freak Waves, despite excellent eyewitness evidence of their occurrence. "Freak waves are the stuff of legend. They aren't just rare, according to traditional views of the sea, they shouldn't exist at all." Then on New Year's Day, 1995 a wave of 26m was measured hitting the Draupner oil rig in the North Sea off Norway. <br /> <br />Subsequent research vindicated the eyewitnesses. Freak waves are alarmingly common. As yet, there is no consensus on what causes these waves.<br /> <br />http://www.bbc.co.uk/science/horizon/2002/freakwave.shtmlMr Vealehttps://www.blogger.com/profile/12931446615905211560noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-34167666794659349862010-11-29T11:08:15.049-07:002010-11-29T11:08:15.049-07:00Tristan: The Obama example was a counterexample to...Tristan: The Obama example was a counterexample to John, who goes from the lack of corroboration for miracles in his own experience to a zero probability. It's pretty much on open secret in the study of Hume's essay that a straightforward application of the mathematical probability theory he lays out would lead to, for example, skepticism about newspaper reports that person X won the lottery. <br /><br />The title links to an full-length essay that I wrote on Bayesianism and miracles, which was published on Infidels. I also published a paper for the peer-reviewed International Journal for Philosophy of Religion in February 1989. So I think I have done some brushing up on Hume. <br /><br />In fact, I explicitly argue that it's quite to be expected that the evidence for miracles like the Resurrection will be insufficient to persuade atheists. <br /><br />The question is whether it is really possible to find a "neutral" or "unbiased" position from which to evaluate just how strong the antecedent improbability of a miracle really is. I don't see how you can evaluate the likelihood of miracles independently of a) the antecedent probability of theism and b) the overall likelihood that God, if He exists, would do something along these lines as part of his way of communicating with human beings. <br /><br />The model of reasoning that I am inclined to accept is that everyone starts where they start, and then conditionalizes their belief systems on the evidence. And then, maybe, however many generations it takes, we can reach a consensus. With issues like God or Christianity I don't see the consensus coming anytime soon. Experimentally imagining how the evidence might look to someone with a different belief system, which is what's involved in the "outsider test" is something that is worth doing, but I don't think it represents the ultimate test of the adequacy of one's religious (or non-religious) beliefs.Victor Repperthttps://www.blogger.com/profile/10962948073162156902noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-22437625261345129122010-11-29T09:45:49.578-07:002010-11-29T09:45:49.578-07:00Victor-
Your Obama example misses it's mark. ...Victor-<br /><br />Your Obama example misses it's mark. There is nothing in nature which says a black man getting elected President is impossible let alone improbable.<br /><br />It's by no means miraculous that in a free Democracy a black man who runs for President has the possibility of being elected into the Presidency.<br /><br />Whereas miracles, the sort of magic and mystical, as talked about by believers are not verifiable let alone possible (according to the laws of nature). Thus most miracles must be considered highly unbelievable.<br /><br />As a philosopher you may want to brush up on your David Hume, but also perhaps Kant. Because I think you are confusing probability with plausibility, that is to say given the circumstances it was highly probable Obama would have a chance at getting elected, however, it was highly implausible that Mickey Mouse would take corporeal form and would be elected President. <br /><br />Regardless, it comes down to a matter of proof and validation--otherwise all you have is conjecture minus a basis for support (in reality). If you say Jesus resurrected that's the same as me saying Mickey Mouse was elected President. You need some mighty hefty evidence to prove such a claim, because typically men don't resurrect and fictional mice don't become president. Whose to say it didn't happen? All we can say it is doesn't usually happen, we've never witnessed it ourselves, and so it's probably a better bet to assume it didn't happen. <br /><br />Obama being elected President of the United State, however, has a strong basis in reality--1) it was a totally probable event, 2) it was totally possible, and 3) it happened. Moreover, all this can be verified and collaborated.<br /><br />Now if miracles are to be believed, wouldn't you think that they must meet the same prerequisites? If not, then why the double standard?Tristan Vickhttps://www.blogger.com/profile/05348780254008374268noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-83628140117457629012010-11-29T08:12:49.240-07:002010-11-29T08:12:49.240-07:00To all your "philosophers": STOP USING P...To all your "philosophers": STOP USING PROBABILITIES WHEN YOU KNOW NOT EVEN THE FIRST THING ABOUT THE SUBJECT.<br /><br />If you want to know about Bayes' Theorem, TAKE A MATH CLASS, not a philosophy class.<br /><br />Philosophy is dead.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10584495.post-62233556071432194482010-11-28T21:24:42.084-07:002010-11-28T21:24:42.084-07:00But this is off topic.But this is off topic.GearHedEdhttps://www.blogger.com/profile/09288513835630145996noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-41245683691151900602010-11-28T21:24:15.444-07:002010-11-28T21:24:15.444-07:00This works, too:
A = John MAY HAVE finished the r...This works, too:<br /><br />A = John MAY HAVE finished the race in first place<br />B = John finished the race<br /><br />If A=>B, if P(B)=1 (full success, i.e., finished first), P(A)=1 (DID finish the race in first place).<br /><br />But again, John can only have finished the race in one spot, making ">" extraneous.GearHedEdhttps://www.blogger.com/profile/09288513835630145996noreply@blogger.comtag:blogger.com,1999:blog-10584495.post-40315085992709041882010-11-28T21:18:35.844-07:002010-11-28T21:18:35.844-07:00Then again, if P(B) = 1 (complete success), and P(...Then again, if P(B) = 1 (complete success), and P(A)>=P(B), then P(A) must = 1 (first place); however, this may not be a good example since Johnny can only finish the race in one position (so the ">" symbol would be unnecessary in your example).GearHedEdhttps://www.blogger.com/profile/09288513835630145996noreply@blogger.com