The Lottery Paradox
Offline The Lottery Paradox (LP) is a paradox one runs into when working in epistemology. The Lottery Paradox challenges the common sense belief that fallible justifiability only requires high probability. The typical lottery paradox runs as follows: Suppose there is a fair lottery in which 1,000,000 tickets are sold. Moreover, suppose that only one ticket can be the winner. If we select any one ticket, we will see that the probability of that ticket being the winning ticket is .000001%. Since it is highly probable that this ticket will not win, and by assuming the traditional fallible justifiability, we know that the ticket we choose will not win. However, since our choice was random, and since the same probability would have applied to any ticket we selected, we can use the principle of universal generalization to apply our result to every single ticket. The immediate result that we get is that we know that every single ticket will not be the winning ticket. However, as part of the game, one ticket must win. Ergo, we get a contradiction: we know that one ticket will win, but we also know that no ticket will win.
The truly menacing nature of the lottery paradox is that it can be constructed with any situation where high probability is involved. What is the way out of this paradox? Before I answer this, we must conceed to the skeptic that he has presented us with a real head scratcher. One possibility, which I am sympathetic to, is that the concept of knowledge is simply incoherent, and the Lottery Paradox demonstrates this. However, I think it demonstrates something more important: That evidence, over and beyond mere probability is warrented in order to claim knowledge. We can use the example of racial profiling to eulicitate this point. It is a statistical fact, that in an african american community, african americans will commit more crimes than whites. However, does this justify randomly pulling over a black person merely because it is highly probable that he commited a crime? No. What is needed is FURTHER evidence suggesting that he commited the crime. Hence, evidence of the particular black person is needed. The same is true for the Lottery Paradox. One is unjustified in saying the ticket will lose, because even though it is highly probable that it will lose, there is no further evidence suggesting that YOUR particular ticket will lose.
Thoughts?
PS: Yes, it just happens that I have become an expert on paradoxes. I sorta fell into it, oddly enough.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
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no one is going to comment on this?
You probably won't like it(or it will start a whole different discussion completely), but that's not really a paradox. The following conclusion: "Ergo, we get a contradiction: we know that one ticket will win, but we also know that no ticket will win." is fundamentally flawed. The truth is that if we know there are 1 million tickets sold, and only one winner can occur, we know that one ticket will win, and 99,999 tickets will lose.
A proper example of a paradox is time travel. You go back to kill Hitler. You kill him. Time goes on, until you are born. You grow up, and never hear about Hitler. So you don't go back to kill him. But you did. But you can't. Etc.
I need coffee. > >
Enlightened Atheist, Gaming God.
There's no paradox. Every inductive statement is probablistic. When we say that we know that our ticket is not a winner, we are just practicing some understandable laziness: we should be saying "it's very likely that this ticket is not a winner'.
It's never literally true that every ticket is definately a non winner.
By the way, the 'logic' in this 'paradox' is the same 'logic' Douglas Adams used in "Hitchhiker's guide to the galaxy' to prove that the universe doesn't exist.
"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'
This doesn't solve the paradox. The above statement does nothing to resolve the paradox. This is a self-evident truth. This is not in dispute. Perhaps there has been some confusion. What is the real nature of this paradox? Perhaps it is best seen as a reductio ad absurdum against the idea that fallible knowledge ONLY requires high probability. Alright, so now assume you partake in this lottery. Moreover, let's assume that you subscribe the the idea that high probability is sufficient to warrent knowledge of a given proposition.
Given the above assumptions, you know that your ticket will not win. I mean, it has less that 1% chance of winning. Hence, it is highly probable that your ticket will lose. Therefore, given the above assumptions, you know your ticket will lose. However, all of the other tickets have the exact same probability. Therefore, by universal generalization...a valid inference rule in classical predicate logic, you know all 1 million tickets will lose, even though one of them must win.
Therefore, it is concluded, that if there is such a thing as fallible knowledge (which I don't think there is...im a skeptic myself, but thats another matter), then it must require more than high probability.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
You are absolutely correct. However, this doesn't solve the paradox. This isn't a paradox about inductive statements. Its a paradox about how we KNOW that an inductive statement is either true or false. It is highly probable that my car is still parked outside. This proposition is probably true. However, in virtue of what can I KNOW that the above proposition is true? Mere high probability? No.
Go on...
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
Your logic is false. I do NOT know that my ticket will not win. I DO know that it's not likely to. I also know that it CAN. So there is no paradox.
Enlightened Atheist, Gaming God.
Classical Logic? This needs alot of justifying. If you claim that CL is false, you need to demonstrate this.
Why? In virtue of what, can you claim this? If your an epistemic skeptic, like myself, then your claim makes sense...sense you wouldn't subscribe to fallibalist accounts of knowledg. However, if you subscribe to a fallibalist theory of knowledge, then one what grounds can you say that you do not know your ticket will win?
This was never under dispute, and is therefore totally irrelevant.
Sorry, merely declairing that its not a paradox doesn't make the paradox go away. Sweeping dirt under the rug doesn't make the dirt go away.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
I didn't say classical logic. I said your logic(or more specifically, the logic within the above statement), which is not classical.
I subscribe to materialism, and taking known facts and applying them. I know for a fact that one of the tickets will win, so I also know that I might hold that particular ticket. If I didn't know for a fact that one ticket would win, I wouldn't have bought the ticket to begin with.
Granted, but I felt obliged to include it as part of the summary, or risk not making sense.
No, but there has to be a paradox for there to be a paradox. Theres a paradox for you!
Enlightened Atheist, Gaming God.
Sure it does, because it reveals that there is no paradox in the first place. The 'paradox' is built on the error of assuming that knowledge claims must be certain to be knowledge claims. This is false. Inductive claims are probabilistic. It is never true that a ticket is a loser. It is only very likely to be a loser.Therefore, there is no contradiction, because it's never true that a lottery ticket is a certain loser. No contradiction, no paradox.
We know that it is LIKELY to be so, based on probability. This is the justification for the claim, making it a justified belief. Knowledge does not require certainty.
And therefore, there is no paradox. Probable truth is all that is meant by any inductive statement.
Adams proves all sorts of absurdities based on the logic used in this 'paradox'. For example: since the universe' existence is unlikely, it doesn't exist. Or, in a similar example, he holds that since any finite divided by an infinite is 'zero', this means that nothing exists within the infinite universe....
and so on....
"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'
Perhaps you could give us a more formal definition of the generalisation rule? I tried Wiki but I didn't find it helpful.
My current guess is that the generalisation rule cannot be applied to probabilities that depend on each other. For instance, in the lottery of 100 tickets, the probability of ticket 1 losing is 99/100 but this entails one of the 99 other tickets winning. This probability says something about the set of remaining tickets as well as the ticket in question. The conjunction of all the tickets losing is a contradiction.
If you played a 100 different lotteries then I think the generalisation could be applied to the tickets of the independent lotteries.
There are different ways to define probability and I think that is part of the problem here.
Should we give Philosophos a call?
Why? He rarely gives a straight answer.Incorrect. However, it does assume that all claims to knowledge must be true. Any fallibablist would agree that while knowledge doesn't require 100% certianty, it does require truth. This is where the paradox comes in. Because given the fallibalist assumptions, one KNOWS that every ticket will loose, but yet he knows that one ticket must win. In a way, however, you are correct. What this paradox suggests is that knowledge claims require certianty...which I happen to favor. One interpretation of the paradox, is skepticism.
It's not about inductive claims, as expained eariler...
Now you are, what my one Philosophy professor once said in joking engaging in "the root of all evil": Conflating epistemology and metaphysics. It must be the case that some ticket is the loser...however, we can only know that with all probability it won't be some ticket t. Ontologically, it must be true...if we stick within the framework of Classical Logic.
Given bivalence, yes it is.
This is not what a fallibalist claims. A fallibalist thinks that a given inductive proposition is likely to be true. The fallibalist makes a stronger claim. He claims to KNOW...not know with all probability...but simply KNOW that given proposition.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
The generalization rule, as used in logic and mathematics, is stated as follows: If for any random x, Px (where P is some predicate), then for all x, Px. This might seem dubious, so let me explain via an analogy. So, take the following proposition "all alligators are agressive." How could we test this? Well, we could take all the alligators, and yank one out at random and see if it is agressive...and it will be. Moreover, if we do this to every alligator, we will see that it will be agressive. Therefore, since we picked the alligator at random...and there was nothing special about the alligator we picked (this is important) we can generalize.
Universal Generalization is a deductively valid inference rule.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
I thought it might be something like that.
So here's my answer to the paradox:
The 'population' of lottery tickets in that particular lottery, the case of their winning is logically dependent on the result of the others.
The event of ticket 1 losing entails one of the other tickets winning.
The conjunction of all the tickets in a single lottery losing is a contradiction.
In the example you gave me, the aggressiveness of the alligator chosen wasn't logically dependent on the aggressiveness of other alligators in the same way that the result of a ticket was logically dependent on the results of the others in the lottery.
So perhaps a good answer is that Universal Generalisation cannot be applied to a population that are logically dependent on each other.
If you had tickets from independent lotteries then that would be a valid use of universal generalisation.
btw, is Universal Generalisation just a weaker form of induction?
Probability seems to be his thing at the minute.
If the different types of probability was a big issue then he'd have some useful input (or maybe just propaganda for Bayesian methods!
I have something of his here, on probability:
http://candleinthedark.com/inductive.html
"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'
Actually, it appears to be correct my good friend. You even seem to agree here:
Well, there you go! The dictum that it "must be true to be true" equates with "truth claims must be certain!" How else would we be able to say that "X must be true" unless we are certain that X is true?
So that IS the problem here. The paradox implies that a true inductive claim somehow leads to a certain claim. But this is false. A true inductive claim is only probably true. We can have probable truth. I think that's the bone of contention here.
Well, to say that it 'requires truth'' is where the problem lies.... If truth is 'required' then it's a necessary condition of a truth claim. But this goes against what induction is: induction is probable truth. We say that inductive statements are probably true.
So correcting this error disolves the paradox.
Well, yes....
The only certainty in an inductive claim would come from the deduced system it relied upon. For example, we can create a set of probabilities: If we have 100 tickets, and one winner, we know, with deductive certainty, that the odds of a winner are 1 out of a 100. This deductive system gives us our odds, our range of possibilities, our frequency, which we know with certainty to be true. From this, we know that it is probably true that any given ticket is probably a loser. Here we have a probable truth, based on the set of odds that are deductively true.
If one insists that the inductive statements themselves must be 'certainly true" then you're violating induction itself. (Unless you want to say that they are certainly probably true!)
But it is about inductive claims because the definition of the induction tells us how we decide whether inductive claims are true: it tells us that inductive claims are only probably true, given a particular frequency/probability. So any paradox that has inductive statements making certain claims is making a basic error
By giving a basic description of probability?!
How does this differ from what I just said?
It's never true in any way you can possibly imagine.
If it's an inductive claim, as you yourself just said, then by definition the knowledge claim is probable. One is saying "x is likely to be true". Certainty never enters the equation.
The paradox implies that one can have certain knowledge that a ticket is a loser based on equating a .999999 probability with a probability of 1. But .999999 is not 1.
"Hitler burned people like Anne Frank, for that we call him evil.
"God" burns Anne Frank eternally. For that, theists call him 'good.'
Because truth is not dependent upon how certian we are of a given proposition. Take the proposition, "God doesn't exist." As we all know, this proposition is true. However, Jerry Falwell, thinks of the above proposition that it is false. In fact, he is certian of this proposition: "The Christian God exists." I am willing to bet that Falwell is 100% certian that his God exists. Obviously, this doesn't make it so. Nowhere did I say that truth claims must be cerian. There are infinitely many propositions which one can construct where are true...but are impossible to be certian of.
Since I know you, I doubt you were trying to say that truth is dependent upon the cognitive status of a given proposition. However, it is hard to find a charitable interpretation that says otherwise. Perhaps you could explain?
Knowledge requires truth, period. Assume the opposite. This would mean that Rene Descartes knew that the mind and brain are distinct...even though this is false. While some epistemologists are argue that knowledge doesn't require truth, I am not one. It strickes me as self-evidently false (damn Raijo).
So you can know a false proposition? Rene Descartes knew the mind wasn't the brain...even though this is false?
Knowledge and inductive are different. fallible knowledge, if there is such a thing, requires induction. Induction, however, does not require knowledge.
This is not a paradox of induction. This is a paradox of saying that we KNOW something based on induction. This paradox demostrates that something more is needed to justify an inductive proposition. Merely having high probability leads to a paradox. Thus, I proposed my solution. That one of the tickets must stand out from the rest...there must be further evidence justifying the your ticket to be either the winner or loser.
"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions
My answer at this point is going to call this some sort of philisophical paradox that doesn't fit into the traditional definition of unsolvable or impossible.
Enlightened Atheist, Gaming God.
What you have here, it seems to me, is a justified belief that your ticket will lose. But if it wins, then the belief will be false. In order to have knowledge, you must have at least a justified true belief. Hence finding out that you lost means that you had the knowledge that the ticket would lose; finding out that you won means that you did not, in fact, have this knowledge. But before the drawing takes place you cannot know whether or not you know.