A problem for rejecting knowledge of lottery propositions

November 15, 2008

So, lots of people find it plausible to deny knowledge in the following type of case.  Let there be a fair lottery with n tickets (where n is very high, and where one and only one ticket will win).  You receive ticket i, and reason that because 1/n is so small, you believe you will lose.   Lets say you are right, and ticket is in a loser.  Most people find it emminently reasonable to reject that you know that you will lose, for various (seemingly good) reasons.

But if they do, they will also have to deny the following rather plausible principle about the value of knowledge :

Knowledge is Valuable (KIV): A rational person should never prefer to merely truly believe that p, rather than to know that p.* Read the rest of this entry »

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Solving the Problem of Old Evidence : Part 2

November 2, 2008

Ok, so we now know how to solve the problem of old evidence as it is traditionally stated : just refuse to allow the probability of our evidence to go to 1.  But, then Earman comes along and points out a quantitative problem of old evidence.  Sure, sure, he says, doing what I just recommended works.  But now consider old evidence for which we are quite certain about, even if we don’t assign it a probability of 1.  Lets say our probability in the evidence is equal to .999.  Then confirmation goes like this, again for a hypothesis that entails the evidence.

Pr(H|E)=Pr(E|H)*Pr(H)/Pr(E)=1*Pr(H)/.999.  This is going to be very nearly Pr(H).  And so we might think that the evidence barely confirms the hypothesis.  This will be bad, because we do have cases of old evidence where we were quite confident in that old evidence, and where that old evidence strongly confirmed the hypothesis (again, this is taken to be the case for Mercury and GTR).  And, on some measures of confirmation – measures that define the degree of confirmation of a hypothesis by some evidence – this will be true.  So now we have a quantitative problem of old evidence : how can old evidence that we are quite sure about strongly confirm a hypothesis?

Read the rest of this entry »

Solving the Problem of Old Evidence : Part 1

November 2, 2008

Well, I’ve decided to start doing philosophy blogging here again, so here is the first post since I’ve been at Princeton.  I’m splitting this up into two parts, and the posts will show how to solve the problem of old evidence with the log-likelihood ratio measure.  This post shows how I like to solve the traditional problem of old evidence.

Read the rest of this entry »

Dutch Books

November 13, 2007

So, dutch books are collections of bets (the synchronic kind) or sequences of bets (the diachronic kind) that result in a sure loss.  As an example, consider paying .51 for a bet on A with unit payoff, and paying .51 on a bet against A with unit payoff.  If you make both bets (if you accept the book that consists in both bets), you will gain 1, and pay 1.02, resulting in a sure loss of .02.  The idea is that a rational agent should never accept such a bet.

On the betting interpretation of our degrees of belief, the presence of such Dutch Books present a reason for our degrees of belief (alternatively, credences) to be probabilities.  If our credences in a proposition are defined as the prices we would be willing to pay for a (fair) bet on the truth of that proposition, or if our credences are measured by those prices, then a theorem due to De Finetti shows that if our credences are not (at least finitely-additive) probabilities, then a dutch book can be constructed against us.  A converse theorem due to (I believe) Skyrms shows that if our credences are probabilities, then no such dutch book can be constructed against us.  That is, if our credences are the prices for the bets we would view as fair given our credences, and were we to bet on those prices then dutch books can be constructed against us if our credences are not probabilities and we can avoid dutch books if they are probabilities.  The betting interpretation of credences thus provides an easy way for establishing our credences as probabilities, allowing us to demonstrate various results in formal epistemology about our credences, and motivating the subjective, personalist interpretation of probability.

However, the betting interpretation itself runs into familiar problems.  First, there are agents who may have fundamental reasons not to make bets of any kind.  A religious person whose doctrine forbids gambling may refuse to make any bets whatsoever.  The betting interpretation will thus deliver the result that the person assigns a credence of 0 to every proposition, despite the obvious possibility that the religious person may well have positive credences for many propositions.  Second, there is no obvious reason that a rational person should be willing to bet on any and all propositions for which they have credences.  Something is clearly right about the betting interpretation, but it can’t be that our credences are, or are measured by, our actual betting behavior (or actual dispositions to bet, or whatever).

In light of these difficulties, Alan Hajek and Lina Eriksson propose taking the concept of degree of belief to be a primitive concept that is not in need of, or perhaps cannot be, analyzed into some more basic notion.  While their discussion has a certain appeal, I think a better approach is simply to move from analyzing our credences as our actual betting behavior to merely what we would view as a fair bet.  A fair bet is defined as one where each side of the bet (for A, or against A) is equivalent in terms of expected value.  We view a bet as a fair bet simply if we would be indifferent to either side of the bet.  If we do this, then the objections raised above to the betting interpretation of degrees of belief fall away.  After all, even if a person is religious, or risk averse, or for whatever other reason does not want to engage in various bets, there is no reason to suppose they may not still view bets as fair or not.  They could view a bet as fair, without any corresponding behavior or disposition to bet on their part.

If we move to what we would view as a fair bet, rather than defining an agent as actually disposed to making bets on what they view as fair, then it might appear as if we lose our dutch book justification for having our credences be probabilities.  After all, if we give up the traditional betting interpretation’s assumption that we would take any bet we view as fair, then even if a dutch book could be constructed against us there is no longer the result that we would end up with a sure loss (precisely because we would no longer automatically make this bet).

However, there still might be a dutch-book related argument for requiring our credences to be probabilities.  Henry Kyburg argued that dutch books would never present a problem for a rational agent, because the rational agent could deduce from the dutch book alone that it resulted in a sure loss, and so would never accept that bet.  For the purposes of this post, the rational agent could deduce that the bet was not fair, because it resulted in a sure loss.  However, the nature of a dutch book is that every individual bet is one that the agent would view as fair.  As a result, if a dutch book could be constructed against an agent, then the rational agent would think both that the dutch book was fair (because it was a combination of individual bets, each of which was fair) and that the dutch book was not fair (because the agent could deduce that accepting one side resulted in a sure loss, while accepting the other side resulted in a sure gain).

Given de Finetti’s theorem, then, if our credences are not probabilities then the agent would be in a position to believe a contradiction (that the bet was fair and was not fair).  Given Skyrms’, if our credences are probabilities, then the agent would be safe from this contradiction.  Given the paucity of reasons for our credences to not be probabilities – there may be reasons not to reveal our credences as probabilities, and to represent them non-probabilistically, but no reasons I know of for them not to be probabilities – these considerations should be sufficient to establish that our credences should be probabilities even without the sure-loss result of traditional dutch book arguments.  So we can still have that our credences should be probabilities, even given the failures of the traditional betting interpretation of probabilities so long as our credences are defined as/measured by what we would view as fair bets.

Pessimistic Inductions

June 25, 2007

So a nice example of “the pessimistic induction” is from Larry Laudan’s “A Confutation of Convergent Realism.” Briefly, it claims that the history of science is a pretty sad story in terms of scientific theories getting it right about what exists (because most (all?) theories in the history of science are strictly false on subsequent theories) and this repeated past failure should make us skeptical that current scientific theories refer to real entities.

Now, whether or not it is a fallacy – as Peter Lewis (apparently falsely) claims – the argument form itself is kind of interesting. Let’s just call an argument which infers from repeated past failures to current (and perhaps future) skepticism of success a “pessimistic induction.” As Errol and Adam might recognize, Timothy Williamson makes a somewhat similar pessimistic induction (although he doesn’t call it this) in Knowledge and its Limits regarding an analysis of knowledge in terms of truth, belief and any other condition(s). This partly motivates his claim that no such analysis is possible – knowledge instead is semantically unanalyzeable. John Norton, in “Causation as Folk Science,” makes another pessimistic induction on our past failures at explicating any adequate notion of causation. This motivates his seemingly correct relegation of (scientific) causation to the status of a folk science without any “fundamental” reality not derivative from acausal scientific theories (much as newtonian gravitational force can be recovered from general relatavity although gravity is not a newtonian force in general relativity).

I’m sure there are many other such examples of a “pessimistic induction” out there, even if the one against scientific realism is the one that is associated with the actual phrase. So, this leads to some different questions. First, what degree of support should such arguments lend to their conclusion? Presumably they cannot be decisive or else the history of philosophy should probably lead us to abandon philosophical inquiry about any given topic. But in terms of establishing the burden of proof (for instance), they seem like they work. Second, what other examples of such “pessimistic inductions” can everyone think of? Any particularly good ones? Finally, if anyone has read those articles or Knowledge and its Limits did you think the pessimistic inductions that were employed in them worked for their respective aims?

The Laudan article is from Philosophy of Science, Vol. 48, No. 1 (Mar., 1981), pp. 19-49. The Peter Lewis article is from Synthese 129, pp. 371-380, 2001. The Norton article is from Philosophers Imprint, Vol. 3, No. 4 (Nov., 2003) and is available here

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May 22, 2007

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