The Lottery Paradox

Trying to find an answer to the question when we can say that we have knowledge I found an article by Baron Reed titled “Fallibilism” (Philosophy Compass 7/9 (2012): 585–596). In a section of this article Reed discusses the Lottery Paradox and its relevance for knowledge as justified true belief. The paradox has been described first by Henry E. Kyburg, Jr, in 1961. I must say that, like the Gettier cases debate, I didn’t follow the discussion that developed since then, so maybe my arguments against it are not new (but I didn’t get the impression when I read Reeds article), but it is interesting to treat the paradox here because of its significance for our concept of knowledge.

I start with Reed’s description of the Lottery Paradox (id, pp. 588-589):

“[The lottery paradox] depends on the following two principles:

1. If it is highly probable that p, then it is rational to believe that p.

2. If it is rational to believe that p and it is rational to believe that q, then it is rational to believe that p & q.”

These two principles are quite plausible, so Reed, and if the “two propositions are each rational to believe, then surely it is rational to believe their conjunction. … Nevertheless, the two principles together yield a counterintuitive result. Suppose there is a fair lottery in which 1000 tickets are sold and in which only one ticket will win. For each ticket, there is thus a .999 probability that it will lose. Principle (1) tells us that it is rational to believe of each ticket that it will lose. So, where proposition p_i is the proposition that ticket t_i will lose, it is rational to believe that p₁, that p₂, …, that p₁₀₀₀. Principle (2) tells us that it is rational to believe the conjunction of all these propositions: that p₁ & p₂ & … & p₁₀₀₀. But, because we know it is a fair lottery, it is also rational for us to believe that some one of the tickets will win – i.e., it is rational for us to believe that either not-p₁, or not-p₂, …, or not-p₁₀₀₀. We know (and rationally believe) that this is equivalent to the proposition that not-(p₁ & p₂ & … & p₁₀₀₀). Using principle (2) again, it is rational to believe that p₁ & p₂ & … & p₁₀₀₀ & not-(p₁ & p₂ & … & p₁₀₀₀). But that proposition, of course, is a contradiction.”

At first glance, the Lottery Paradox may seem plausible, indeed, but at second glance? Premise (1) says that it is rational to believe that p if p is highly probable. But why should it? For example, I work as a security officer on an airport where I check the passengers at the gate. It is highly probable that the next passenger is a decent person and not a terrorist. Nevertheless, for me it is rational to believe that he might be an exception. The only thing that is rational to believe is that p is highly probable and not that it certainly will happen, as premisse (1) seems to state. Otherwise we add information without any argument.

Take now premise (2). What Reed doesn’t make clear is what “&” means. Say I throw the dice. Then there is a probability of .83 that 1-5 will come up. If I throw several times, that doesn’t change for each independent (separate) throw. If we interpret the & for separate throws, the probability for each throw remains .83. However, now we interpret & in the way that the first throw p is 1, 2, 3, 4, or 5 and the next throw q is and the next throw r is etc.. Then after four throws the probability is 0.83x0.83x0.83x0.83=0.48 that this will happen, which is already less than a half! In other words, if p, q etc. are dependent on each other in some way and they have probabilities of less than 1 it is not rational to believe that p & q & etc. (i.e. in my example that each time 1-5 will come up).

This brings me to another flaw in the lottery paradox. Even if the premises are true, the world is wider and it contains also other knowledge, for instance what statisticians say about probabilities. Our rational believe is not based on single isolated facts but it is interwoven with other knowledge, like statistical knowledge. And statisticians can tell us that it is simply not true that we have to analyse the probabilities of the 1000 tickets in the lottery as independent from each other: If we know that one lottery ticket will not win, then the probabilities that the other tickets will win increase. Knowing the probabilities of the separate lottery tickets as such tells us nothing about the probabilities when we put some (or all) tickets together, unless we do the relevant statistical computations. I think that more can be said against the Lottery Paradox, but it is clear that neither premise (1) nor premise (2) is correct. The upshot is that there is no Lottery Paradox.

Gettier cases and knowing-how

Knowing how (foto B.bij de Weg)

The Gettier problem calls into question that knowledge is justified true belief. We have seen it in my last blog. The theme has been intensively discussed since Gettier published his article in 1963 and for me it is impossible to follow back the whole debate, but when I look up on the Internet the argumentations put forward there is one thing that strikes me: Although the Gettier problem and its consequences for what knowledge is have been examined in many ways, nonetheless the discussion has been one-sided. Or have I missed something? For when we take a closer look at the discussion we see that the concept of knowledge involved is basically propositional knowledge or knowledge-that. But didn’t already Gilbert Ryle defend in his The Concept of Mind (1949) that there are two kinds of knowledge: knowledge-that and knowledge-how? When we apply this distinction to the Gettier problem, I think we have to reject the conclusion that Gettier cases undermine the idea that in general knowledge is not justified true belief.

Once (in my blog dated June 9, 2008) I have dedicated a blog to Ryle’s distinction, but it’s already more than four years ago, so let me repeat the essence. When we talk about knowledge-that, we mean intellectual knowledge or rationally knowing. Basically, knowing-that is about facts or theories that can be true or false. Betsy, my cow, is in the field or she isn’t. E=mc² or E≠ mc². However, knowing-that is not all knowledge there is. Much of what we know is knowing-how, which does not refer to what we intellectually contain in our minds but to what we can practically do with our bodies (steered by our mind, it’s true, but not only). It refers to the way we do things and are able to do them. How we ride a bike, for instance. Maybe I cannot explain verbally what I do when I am cycling, but nevertheless I can do it and I can teach others how to do it. Also most professional knowledge, skill and craftsmanship fall in this category. A carpenter can excel in his trade, even when he cannot explain in words the details of what he is doing.

I think that it has no sense to talk about the Gettier problem when we talk about knowing-how. For instance, when I know how to ride a bike, something like a Gettier problem cannot happen. Maybe I belief that I can cycle because in the past I could, but if it is no longer so, because I have become too old for it and I would fall over, it is simply a false belief. I don’t see how a kind of Gettier problem can bear on cases like this one or on other cases of knowing-how. It should have to be something like that I have the true knowledge that I can ride a bike, but when I am going to check it, I do something else although I still have the belief that I am cycling. I cannot imagine how that is possible, but maybe you can. If I am right, it can still happen that knowing-how is not a kind of justified true belief but it is not because of Gettier-like objections.

On forgetting

In these blogs I write on subjects that I find interesting and I use this writing for developing my mind. Because I publish my blogs on the Internet, I hope that my readers will profit by it as well. This week’s blog has a different purpose, though, for I have a problem. I read a lot on philosophy and then it’s normal that I often come across the same theories, arguments and cases. Therefore I have developed not only my ideas through the years but also I have acquired also a good knowledge of what is going on in philosophy, at least in the fields I am reading and writing on. Nevertheless, there are some subjects that, how often I have read on them, I always forget what they are about. For instance, I always forgot what Frankfurt cases are. However, since I have written a few blogs on them, they have been printed in my memory. Another example is the so-called Gettier problem. I have read on it several times. Often I have looked up what it is. But what happened today: I stumbled on it in a book on knowledge and despite all my efforts in the past, it had again slipped my memory what it involves. So I got the idea to write a blog on the Gettier problem, for what has helped me to remember what Frankfurt cases are will without a doubt help me keep the Gettier problem in my mind, as well. But I want to make excuses to my readers, if they find this blog boring, for what I actually do is merely repeating some stuff that I have found on the Internet.

A standard definition of knowledge says that knowledge is justified true belief: We belief that something is the case; we have good reasons for this belief; and what is believed is also true. So far, so good, but take this example, which I have adapted from the Wikipedia:

I am a bit worried whether my best cow Betsy hasn’t been stolen from the field where she is supposed to be at pasture. I walk from my farm to the field, where I see a cow in the middle of the herd that exactly looks like Betsy, although I don't find it necessary to walk so near to her that I am 100% sure that she really is Betsy. Back home, I tell my wife that I know that Betsy is safe. My wife wants to check it, too, and goes also to the field. There she sees Betsy somewhere in the back and Jane in the middle of the herd. Because Betsy is often confused with Jane, if you look at her from a distance, she makes herself 100% sure that it is really Betsy there in the back of the field. Betsy hasn’t been stolen, just as I thought.

Now the question is: Did I know that Betsy hadn’t been stolen? For (1) I believed that Betsy was safe; (2) my belief was justified for I had checked it; (3) it was true that Betsy hadn’t been stolen.

In a famous paper published in 1963 Edmund Gettier discussed cases like this one where we seemed to have justified true belief, but where most of us would not say that we “know”; cases that cast doubt upon the definition of knowledge as justified true belief.

What this example and other more refined “Gettier cases” show is that it is possible to have justified true belief without having knowledge. The theory of knowledge that holds that knowledge is justified true belief is therefore false. We need more for being able to say to have knowledge. But what?

But isn’t this exactly what we do in scientific research: looking for something that we belief to be true on good (i.e. methodologically justified) grounds? And yet often it happens that our theories, once believed to be true and justified by experiments and argumentations, are later rejected or revised on the base of new data, experiments and argumentations. But weren’t then the old revised ideas no knowledge it all? And how about the new ideas that replace the old ones and that are the best we have at the moment: Are they knowledge? Seen this way, we can doubt whether we have any knowledge at all; whether knowledge in science can exist anyhow. We simply have justified true belief. And isn’t that also so for most we think to know outside science? It seems better never to say “I know” but “I think I know” at most. A lot more can be said about this, but I only wanted to write on the Gettier problem in order to know it forever.

Some websites on the Gettier problem

http://en.wikipedia.org/wiki/Gettier_problem

http://www.theoryofknowledge.info/what-is-knowledge/the-tripartite-theory-of-knowledge/gettier-cases/

http://www.iep.utm.edu/gettier/

Feeling happy

Sometimes I think that I should take more time for writing my blogs than the afternoon I allot for it now. Then I could write thorough essays without the mistakes or rudimentary thoughts that are so typical for my present scribbles. But would it make sense? Wouldn’t I get simply another type of blogs? Blogs that are articles rather than the present small pieces of writing that – I hope – make the reader think? Anyway, now and then I come across thoughts or phrases that would certainly have given some blogs another turn had I known them before.

Today I happened to reread Montaigne’s essay titled “That Men are Not to Judge of Our Happiness Till After Death” (Book I, 18). It remembered me of what I recently had written about happiness and especially of what I had written about Wittgenstein’s two opinions on his life and my idea that there are two views on happiness: the view of the moment that something happens which makes me happy or unhappy and the view backwards on my life and my thinking how it was. Had I recalled Montaigne’s essay, I would certainly have referred to it in my blogs on happiness, and I would have come to other conclusions.

In this essay Montaigne comments on the statement by the Greek poet and statesman Solon (630 BC - 558 BC) “ ‘That men, however fortune may smile upon them, could never be said to be happy till they had been seen to pass over the last day of their lives,’ by reason of the uncertainty and mutability of human things, which, upon very light and trivial occasions, are subject to be totally changed into a quite contrary condition”. After some discussion, Montaigne concludes that what Solon wanted to say is that because of the precariousness of life “the very felicity of life itself, which depends upon the tranquillity and contentment of a well-descended spirit, and the resolution and assurance of a well-ordered soul, ought never to be attributed to any man till he has first been seen to play the last, and, doubtless, the hardest act of his part.” And that is “Wherefore, at this last, all the other actions of our life ought to be tried and sifted: ’tis the master-day, ’tis the day that is judge of all the rest, ‘’tis the day,’ says one of the ancients, ‘that must be judge of all my foregoing years’.”

When writing down these quotations, I saw that in the last one the Dutch translator of the Essays uses the word “happiness” (geluk) – in agreement with the 1595 French edition, which uses the word “bonheur” – whereas the English edition quoted speaks of “felicity”. Be this as it may, it doesn’t change what I was wondering already when I wrote my blogs on happiness but what I did not wrote down, namely: Do we really mean the same with “happy” when we say “I am happy now”, and when we say “looking backward I am happy” or rather “looking backward I am happy with it”? For instance, did Wittgenstein refer to the same thing when he wrote in his diary “There is no happiness for me; no joy ever” and when he muttered at the end of his life “Tell them I’ve had a wonderful life”? A few weeks ago I thought so, following Groeger, but now I think that what I wrote just after the quotes but what I ignored contains an important truth: “a wonderful life needs not also be a happy life”. It’s just the essay by Montaigne that defends the idea that whether we really were happy can only be judged at the end of our lives that made it clear to me that our happiness now and our feeling happy with our life as a whole are different things. For for the latter Montaigne uses also the word “reputation” and he speaks here of a “judgment” on a life. Although Montaigne talks of the lives of other people in the first place, I think that it is the same for Wittgenstein or for you or me: When I look back on my own life and say that I am happy with it, what in fact I am doing is judging my life, although it may be so, of course, that the judgment leads to a feeling of happiness in me who judges myself. But such a feeling and a judgment are basically two different things. Unlike the latter, a feeling is neither an opinion nor a kind of view. And it is just a feeling that I am referring to when I say “I am happy now” or “basically I feel happy (in the long run)”. This cannot be changed by a dramatic act at the end of my life (as Montaigne thinks), for such an act can change my reputation and the judgment on my life as a whole but not what I felt some time ago, even not for myself.

Quotations are from the online edition of the Essays English version) on

http://ebooks.adelaide.edu.au/m/montaigne/michel/m76e/book1.18.html