If I understand you correctly, you’re saying that reasoning “non-probabilistically” about possibilities and impossibilities can be viewed as a subset of Bayesian reasoning, where instead of numerical probabilities, one uses qualitative probabilities—zero corresponds to impossible, nonzero corresponds to possible and so on.
I agree with you, that you can view it that way; but probably there are ways to view any sort of reasoning as a “subset” of some other domain of reasoning.
The best analogy I have is with classical and intuitionistic logic. The implication/negation fragment of intuitionistic propositional logic can be viewed as a subset of classical logic, in that the theorems provable in intuitionistic logic are a strict subset of classical logic theorems (if you map classical implication to intuitionistic implication, and classical negation to intuitionistic negation).
However, there’s a slightly trickier mapping (http://en.wikipedia.org/wiki/G%C3%B6del%E2%80%93Gentzen_negative_translation) which introduces a lot of double negations, and if you use this mapping, classical propositional logic fits entirely inside of intuitionistic logic, and intuitionistic logic can be viewed as introducing a new “constructive” implication while keeping all of the old truths.
What I’m saying is that effective techniques for reasoning non-probablistically derive their effectiveness from their similarity to Bayesian reasoning. In some special cases, they are identical to Bayesian reasoning, but in other cases, Bayes is superior. This is because Bayes is proved, by theorem, to be Right. Not Unbiased or Robust, but Right.
So I’m not saying it’s a subset, I’m saying its similar to. One way to be similar to something is to copy some but not all of it, which is taking a subset plus adding other stuff. This is almost the same concept, obviously.
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Using the two logics I know best:
Classical propositional logic is Bayesian logic for probabilities of 1 and 0.
All results of Bayesian logic can be derived by propositional logic and certain additional assumptions.
But these are qualitatively different. I don’t know how to describe this difference—which one is fundamental? I think both are, in different ways.
What’s missing is how you use it in real-world problems, I think. Because it is often the case that one formal system is contained in another, and the other is contained in the one. But formal systems are only useful if they model reality, or serve some other function. Classical logic is used more than intuitionistic logic—is this because it’s more useful? (Classical logic is more useful than Bayesian in the world of mathematics, but less in the real world, I think.)
If I understand you correctly, you’re saying that reasoning “non-probabilistically” about possibilities and impossibilities can be viewed as a subset of Bayesian reasoning, where instead of numerical probabilities, one uses qualitative probabilities—zero corresponds to impossible, nonzero corresponds to possible and so on.
I agree with you, that you can view it that way; but probably there are ways to view any sort of reasoning as a “subset” of some other domain of reasoning.
The best analogy I have is with classical and intuitionistic logic. The implication/negation fragment of intuitionistic propositional logic can be viewed as a subset of classical logic, in that the theorems provable in intuitionistic logic are a strict subset of classical logic theorems (if you map classical implication to intuitionistic implication, and classical negation to intuitionistic negation).
However, there’s a slightly trickier mapping (http://en.wikipedia.org/wiki/G%C3%B6del%E2%80%93Gentzen_negative_translation) which introduces a lot of double negations, and if you use this mapping, classical propositional logic fits entirely inside of intuitionistic logic, and intuitionistic logic can be viewed as introducing a new “constructive” implication while keeping all of the old truths.
What I’m saying is that effective techniques for reasoning non-probablistically derive their effectiveness from their similarity to Bayesian reasoning. In some special cases, they are identical to Bayesian reasoning, but in other cases, Bayes is superior. This is because Bayes is proved, by theorem, to be Right. Not Unbiased or Robust, but Right.
http://lesswrong.com/lw/mt/beautiful_probability/ seems like the appropriate lesswrong post.
So I’m not saying it’s a subset, I’m saying its similar to. One way to be similar to something is to copy some but not all of it, which is taking a subset plus adding other stuff. This is almost the same concept, obviously. ` Using the two logics I know best: Classical propositional logic is Bayesian logic for probabilities of 1 and 0. All results of Bayesian logic can be derived by propositional logic and certain additional assumptions.
But these are qualitatively different. I don’t know how to describe this difference—which one is fundamental? I think both are, in different ways.
What’s missing is how you use it in real-world problems, I think. Because it is often the case that one formal system is contained in another, and the other is contained in the one. But formal systems are only useful if they model reality, or serve some other function. Classical logic is used more than intuitionistic logic—is this because it’s more useful? (Classical logic is more useful than Bayesian in the world of mathematics, but less in the real world, I think.)
I need to think more about this...