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More generally, we can show that the attitude that regards any unproven statement as having a 50% probability of being true is trivially self-contradictory. Sup

More generally, we can show that the attitude that regards any unproven statement as having a 50% probability of being true is trivially self-contradictory. Suppose X is an unknown number, and all we know is that it's between 0 and 100. We can make the following statements about X, none of which is provable with the information we currently have: 1. X < 1 2. 49 < X < 51 3. X > 99 If we assume that each of these statements has a 50% probability of being true, then we trivially get a contradiction, because the probability of the union of these 3 options becomes 150%, because mutually exclusive probabilities are additive. So the idea that a 50% chance of truth represents a state of no knowledge about the truth of a statement is just wrong. If we actually knew that any one of the above statements had a 50% probability of being true, we would clearly have more information about X than we would otherwise. There is no sense in which a 50% probability of truth is a useful default.

newen 9y ago
You are misapplying probabilities here. If you assume no knowledge, and you have two options for X, true or false, then it's a 50% probability of X being true. Similarly, if there are N possibilities for X, and no prior knowledge for X, then it's 1/N probability of X being on of those possibilities. The key is what prior knowledge you bring to the table.
- rcthompson 9y ago
  The random variable X has an infinite number of possible values (real number between 0 and 100). But each of the statements I made has two options: true or false. Despite each having two options, there's no way that all 3 statements can have a 50% probability of truth. The whole point of my comment was that "no prior knowledge" cannot possibly be equivalent to "equal probability of all options".
  - skissane 9y ago
    > The whole point of my comment was that "no prior knowledge" cannot possibly be equivalent to "equal probability of all options". But your scenario isn't "no prior knowledge". In your scenario, you know X is a real number between 0 and 100, which is prior knowledge. And, so the correct thing to do, is use that prior knowledge to assign probabilities, which gives you 1%, 2% and 1% for each of your propositions, not 50%.
    - rcthompson 9y ago
      You don't have any knowledge of X beyond the fact that it's between 0 and 100. It could be uniformly distributed, it could be the number of heads in 100 coin flips, or it could be the average number of yards that a kickoff was returned for in all NFL games last season. It could even just be 73. Assuming a uniform distribution is an arbitrary prior assumption that you're making with no support from any data, and it's no more or less reasonable than any other prior assumption you could make about the distribution. There's nothing special about the uniform distribution that makes it some kind of natural or default prior for a random variable on a bounded interval.
      - skissane 9y ago
        > There's nothing special about the uniform distribution that makes it some kind of natural or default prior for a random variable on a bounded interval. I don't agree. Suppose I carry out this process: Step 1) Generate a random histogram over the bounded interval Step 2) Pick a random number according to the distribution that histogram represents If I repeat that process a large number of times (say a few million), what will be the distribution of the result? It will be approximately uniform, and the more times I repeat the process the closer to uniform I get. So the uniform distribution actually is special, as the average case of randomly chosen distributions, and hence why it should act as the default when the only thing we know about the distribution is an upper and lower bound.
        rcthompson 9y ago
        If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't see how you can justify that. "Generate a random distribution on an interval" is not an obviously well-defined procedure, so I don't think you can say anything about what the distribution produced by "averaging" every possible distribution would look like. Which matches the fact that you have no information on what the distribution of X looks like, other than that it is zero outside of the interval [0,100]. Edit: In any case, the process you've described is just begging the question (i.e. it is circular reasoning). In order to prove that the uniform distribution is a natural choice for X, you assume that choosing uniformly from the set of all possible distributions is natural. Even if there was a finite number of possible distributions to choose from such that choosing randomly from that set according to a uniform distribution was a well-defined operation, you're still assuming that it's natural to assign equal probability to all options in order to prove that it's natural to assign equal probability to all options.

newen 9y ago

You are misapplying probabilities here. If you assume no knowledge, and you have two options for X, true or false, then it's a 50% probability of X being true. Similarly, if there are N possibilities for X, and no prior knowledge for X, then it's 1/N probability of X being on of those possibilities. The key is what prior knowledge you bring to the table.

rcthompson 9y ago
The random variable X has an infinite number of possible values (real number between 0 and 100). But each of the statements I made has two options: true or false. Despite each having two options, there's no way that all 3 statements can have a 50% probability of truth. The whole point of my comment was that "no prior knowledge" cannot possibly be equivalent to "equal probability of all options".
- skissane 9y ago
  > The whole point of my comment was that "no prior knowledge" cannot possibly be equivalent to "equal probability of all options". But your scenario isn't "no prior knowledge". In your scenario, you know X is a real number between 0 and 100, which is prior knowledge. And, so the correct thing to do, is use that prior knowledge to assign probabilities, which gives you 1%, 2% and 1% for each of your propositions, not 50%.
  - rcthompson 9y ago
    You don't have any knowledge of X beyond the fact that it's between 0 and 100. It could be uniformly distributed, it could be the number of heads in 100 coin flips, or it could be the average number of yards that a kickoff was returned for in all NFL games last season. It could even just be 73. Assuming a uniform distribution is an arbitrary prior assumption that you're making with no support from any data, and it's no more or less reasonable than any other prior assumption you could make about the distribution. There's nothing special about the uniform distribution that makes it some kind of natural or default prior for a random variable on a bounded interval.
    - skissane 9y ago
      > There's nothing special about the uniform distribution that makes it some kind of natural or default prior for a random variable on a bounded interval. I don't agree. Suppose I carry out this process: Step 1) Generate a random histogram over the bounded interval Step 2) Pick a random number according to the distribution that histogram represents If I repeat that process a large number of times (say a few million), what will be the distribution of the result? It will be approximately uniform, and the more times I repeat the process the closer to uniform I get. So the uniform distribution actually is special, as the average case of randomly chosen distributions, and hence why it should act as the default when the only thing we know about the distribution is an upper and lower bound.
      - rcthompson 9y ago
        If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't see how you can justify that. "Generate a random distribution on an interval" is not an obviously well-defined procedure, so I don't think you can say anything about what the distribution produced by "averaging" every possible distribution would look like. Which matches the fact that you have no information on what the distribution of X looks like, other than that it is zero outside of the interval [0,100]. Edit: In any case, the process you've described is just begging the question (i.e. it is circular reasoning). In order to prove that the uniform distribution is a natural choice for X, you assume that choosing uniformly from the set of all possible distributions is natural. Even if there was a finite number of possible distributions to choose from such that choosing randomly from that set according to a uniform distribution was a well-defined operation, you're still assuming that it's natural to assign equal probability to all options in order to prove that it's natural to assign equal probability to all options.