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Thanks for your comment! Very interesting. > Finite discrete probability theory is purely deductive. Under the assumption that a process is stochastic (e.g. th

Thanks for your comment! Very interesting. > Finite discrete probability theory is purely deductive. Under the assumption that a process is stochastic (e.g. that a die is fair when it is thrown), probabilities follow deductively. I’m FAR from an expert here but I read that probability theory in general can be based on fuzzy logic, which is purely deductive. Can you please comment on why it needs to be finite and discrete? And why that stochasticity assumption is needed?

jonathanstrange 4y ago
I'd say that fuzzy logic is neither purely deductive nor does it represent probability theory unless you make some weird constructions and re-interpretations. Fuzzy logics are logics with infinitely many truth values, usually uncountably many in the interval from 0 to 1. Binary connectives are thus functions from two real numbers in [0, 1] to a real number in [0, 1]. Unlike in classical logic where there are 16 such functions, there are infinitely many such functions for fuzzy logic. These are characterized in different ways (called "norms") and it is very hard to justify them or why you'd pick a particular one. For example, there are different competing definitions of conjunction, and different rules for the conditional are particularly problematic. The rules for the conditional don't have much to do with probabilistic conditioning. That's not to say that fuzzy logic and representations based on it like fuzzy sets aren't useful, these are just better regarded as something else, ways of representing certain forms of non-probabilistic uncertainty or vagueness. As for the discrete cases: I was merely talking about the justification of descriptive statistics. If you have a 6-sided die, for example, and you make analytic assumptions that it is fair from the way it is constructed, then describing the outcomes of die throws involve only basic combinatorics. The same for all other finite probability spaces. However, in the philosophy of probability it is widely accepted that you should avoid circular reasoning. So you cannot test empirically whether the die is fair, for instance, when talking about the foundations of probability. You either have to assume it or need some analytic model that justifies why fairness follows from the way the die is constructed. In any case, at some point you need to assume that the underlying process is indeed random. Maybe you're right, though, and probability theory for uncountable domains is equally deductively justified. Since mathematics uses deduction and complete induction in their proofs it seems reasonable to claim that. But I think there are several leaps of faith in the uncountable case that you don't get with descriptive probability over discrete spaces. For example, you cannot "lift" probabilities of complex events from the probabilities of singleton events when the domain is uncountable. The point is that this quickly gets as or more complicated than models of abduction (though arguably in a more "math" way). In contrast, basic descriptive probability theory for finite spaces is IMHO as intuitively plausible as logic. That's why I think it's almost on a par with deduction in terms of justification. Anyway, in both finite and uncountable domains you do make educated guesses when making statistical inferences from a small to a larger population or when making assumptions about an underlying probability distribution, but you can e.g. estimate errors very precisely and justify the methods in much detail. Abduction mostly doesn't even come with error estimates. That being said, there is interesting work on model selection that kind of bridges the gap between statistics and abduction. See Burnham & Anderson: Model Selection and Multimodel Inference, Springer 2002.

jonathanstrange 4y ago

I'd say that fuzzy logic is neither purely deductive nor does it represent probability theory unless you make some weird constructions and re-interpretations. Fuzzy logics are logics with infinitely many truth values, usually uncountably many in the interval from 0 to 1. Binary connectives are thus functions from two real numbers in [0, 1] to a real number in [0, 1]. Unlike in classical logic where there are 16 such functions, there are infinitely many such functions for fuzzy logic. These are characterized in different ways (called "norms") and it is very hard to justify them or why you'd pick a particular one. For example, there are different competing definitions of conjunction, and different rules for the conditional are particularly problematic. The rules for the conditional don't have much to do with probabilistic conditioning. That's not to say that fuzzy logic and representations based on it like fuzzy sets aren't useful, these are just better regarded as something else, ways of representing certain forms of non-probabilistic uncertainty or vagueness. As for the discrete cases: I was merely talking about the justification of descriptive statistics. If you have a 6-sided die, for example, and you make analytic assumptions that it is fair from the way it is constructed, then describing the outcomes of die throws involve only basic combinatorics. The same for all other finite probability spaces. However, in the philosophy of probability it is widely accepted that you should avoid circular reasoning. So you cannot test empirically whether the die is fair, for instance, when talking about the foundations of probability. You either have to assume it or need some analytic model that justifies why fairness follows from the way the die is constructed. In any case, at some point you need to assume that the underlying process is indeed random. Maybe you're right, though, and probability theory for uncountable domains is equally deductively justified. Since mathematics uses deduction and complete induction in their proofs it seems reasonable to claim that. But I think there are several leaps of faith in the uncountable case that you don't get with descriptive probability over discrete spaces. For example, you cannot "lift" probabilities of complex events from the probabilities of singleton events when the domain is uncountable. The point is that this quickly gets as or more complicated than models of abduction (though arguably in a more "math" way). In contrast, basic descriptive probability theory for finite spaces is IMHO as intuitively plausible as logic. That's why I think it's almost on a par with deduction in terms of justification. Anyway, in both finite and uncountable domains you do make educated guesses when making statistical inferences from a small to a larger population or when making assumptions about an underlying probability distribution, but you can e.g. estimate errors very precisely and justify the methods in much detail. Abduction mostly doesn't even come with error estimates. That being said, there is interesting work on model selection that kind of bridges the gap between statistics and abduction. See Burnham & Anderson: Model Selection and Multimodel Inference, Springer 2002.