In a previous post<\/a>, I argued that falsifiability is not the be-all and end-all<\/em> of Science.\u00a0\u00a0 There are valid scientific beliefs that are not falsifiable.<\/p>\n However, there is something to the idea that beliefs should be falsifiable.\u00a0 One way to make this precise is to use Bayes’ Theorem<\/a>.\u00a0 This is a rule which says how to update your probabilities when you get some new evidence E.\u00a0 It says that your belief in some idea X should be proportional to your prior probability <\/em>(how strongly you believed in before the evidence), times the likelihood <\/em>of having measured the new evidence given X.\u00a0\u00a0 (You also have to divide by the probability of having measured the new evidence, but this is the same no matter what X is, so it doesn’t affect the ratio of odds between two competing hypotheses X and Y.\u00a0 It’s just needed to get the probabilities to add up to 1).\u00a0 As an equation: $$!P(X|E) = P(X) P(E|X) \/ P(E).$$We won’t actually plug any numbers into this equation in this post.\u00a0 Instead, I’ll just point out a general property which this equation has.<\/em>\u00a0 Suppose you are about to perform an experiment.\u00a0 On average, you expect that your probability is going to be the same after the experiment as before.<\/p>\n For example, suppose you believe there is a 1\/50 chance that there exists a hypothetical Bozo particle (I just made that up right now).\u00a0 And suppose you perform an experiment which has a 50% chance of detecting the Bozo if it exists.\u00a0 Just for simplicity in this example let’s suppose there are no false positives: if you happen to see the Bozo, it leaves a trail in your particle detector which can’t be faked.<\/p>\n There are two possible outcomes: you see the Bozo or you don’t.\u00a0 In order to see the Bozo, it needs to (a) exist and (b) deign to appear, so you have a 1% chance of seeing it.\u00a0 In that case, the probability that the Bozo increases to 1.<\/p>\n On the other hand, you have a .99 chance of not <\/em>seeing the Bozo.\u00a0 In that case, your probabilty ratio goes from 49:1 to 98:1 since the Bozo exists possibilities just got halved.\u00a0 This corresponds to a 1\/99 probability that the Bozo exists.<\/p>\n On average<\/em>, your final probability is $$(.01 \\times 1) + (.99 \\times 1\/99) = .02$$.\u00a0 Miraculously, this is exactly the same as the intitial probability 1\/50 of the Bozo existing! Or maybe it isn’t so much of a miracle after all.\u00a0 On reflection, it’s pretty obvious that this had to happen.\u00a0 If you could somehow know in advance that performing an experiment would tend to increase (or decrease) your belief in the Bozo, that would mean you that just knowing that the experiment has been done (without looking at the result) should increase or decrease your probability.\u00a0 That would be weird.\u00a0 So really, it had to be the same.<\/p>\n We call this property of probabilities Reflection,<\/em> because it says that if you imagine yourself reflecting on a future experiment and thinking about the possible outcomes, your probabilities shouldn’t change as a result.<\/p>\n Now Reflection has an interesting consquence.\u00a0 Since on average your probabilities remain the same, if an experiment has some chance of increasing <\/em>your confidence in some hypothesis X, it must necessarily also have some chance of decreasing <\/em>your confidence in X.\u00a0 And vice versa.\u00a0 They have to be in perfect balance.<\/p>\n This means, you can show that it is impossible for an observation to confirm a hypothesis, unless it also had some chance of disconfirming it.\u00a0 VERY ROUGHLY SPEAKING, we could translate this as saying that you can’t consider a theory to be confirmed unless it could have been falsified by the data (but wasn’t).<\/p>\n Even so, there are a number of important caveats.\u00a0 In some situations in which we can and should believe things which are, in various senses, unfalsifiable.\u00a0 This occurs either because (a) The Reflection principle doesn’t rule them out, or (b) the Reflection principle has an exception and doesn’t apply.\u00a0 Here are all the important caveats I can think of:<\/p>\n To conclude, these are four types of reasonable beliefs which cannot be falsified.\u00a0 It is a separate question to what extent these types of exceptions tend to come up in “Science” as an academic enterprise (as opposed to other fields).\u00a0 But I don’t see any good reason why these exceptions can’t pop up in Science.<\/p>\n (*) Footnote: Some fictitious person (let’s call her Georgina) might say that the Bozo is still falsifiable since nothing stops us from doing the experiment<\/em> over and over again, until the Bozo is either detected or made extremely improbable.\u00a0 Hence, Georgina would argue, the Bozo IS <\/em>falsifiable.<\/p>\n My answer to Georgina is that it actually depends on the situation.\u00a0 Maybe the Bozo experiment can<\/em> only be done once.\u00a0 Maybe (since I’m making this story up, I can say whatever I want) the Bozo can only be detected coming from a particular type of Supernova, and it will be millions of years before the next one.\u00a0 More realistically, maybe the Bozo is detected using its imprint on the Cosmic Microwave Background, and the phenomenon of Cosmic Variance<\/a> means that you can’t repeat the experiment (since there is only one observable universe, and you can’t ask for a new universe).\u00a0 More realistically still, maybe the experiment costs 100 billion dollars and Congress can’t be persuaded to fund it more than once.<\/p>\n Georgina might not like the last example very much, since she might say that all she cares about is that the Bozo is in principle <\/em>falsifiable.\u00a0 Perhaps as a holdover from logical positivism<\/a>, the Georginas of this world often talk as though this makes some kind of profound metaphysical difference.\u00a0 But it’s not clear to me why we should care about falsifiability in principle.\u00a0 The only thing that really helps us is falsifiability in fact.<\/p>\n If a critical experiment testing the Bozo will not be performed until next year, for purposes of deciding what to believe now<\/em>, we should behave in exactly the same way as if the experiment could never be done.\u00a0 Experiments can’t matter until we do them.<\/p>\n","protected":false},"excerpt":{"rendered":" In a previous post, I argued that falsifiability is not the be-all and end-all of Science.\u00a0\u00a0 There are valid scientific beliefs that are not falsifiable. However, there is something to the idea that beliefs should be falsifiable.\u00a0 One way to … Continue reading \n
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\nPropositions like these need not be falsifiable.\u00a0 This does not conflict with Reflection, because that only applies when you need to increase the probability that something is true using new evidence.\u00a0 But these propositions start out with high probability.
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\nThis doesn’t contradict Reflection.\u00a0 The reason is that Reflection tells us that you can’t verify a hypothesis without some chance of lowering its probability.\u00a0 But it doesn’t say that the probability has to be lowered all the way to 0.\u00a0 In the Bozo case, we balanced a small chance of a large probability increase against a large chance of a smaller probability decrease.
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\nThe Ring Hypothesis<\/a> was another example of this effect.\u00a0 We have verified the existence of a planet with a ring.\u00a0 Had we looked at our solar system and not seen a planet with a ring, this would indeed have made the Ring Hypothesis less likely.\u00a0 But not necessarily very much <\/em>less likely.\u00a0 Certainly not enough to consider the Ring Hypothesis falsified.
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\nFor example, no living creature could ever falsify the hypothesis that the universe permits life.\u00a0 Even though it didn’t have to be true.\u00a0 Nor could you (in this life) ever know that you just lost a game of Russian Roulette.
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\nThis type of situation is an exception to the Reflection principle.\u00a0 The arguments for Reflection assume that you exist both before and after the experiment.\u00a0 (You can also construct counterexamples to Reflection involving amnesia, or other such funny business.)<\/li>\n<\/ol>\n