We've been working on abduction at our institute for a while. My take on it is that there is no abduction as a logical inference. Although he portrayed it as a third type of inference in some of his writings, C.S. Peirce thought the same, he emphasized many times that abduction is indeed an art and ultimately a psychological phenomenon.
The problem with abduction is that it doesn't have a justificatory value on a par with those of deduction and induction. Deduction forms the basis of mathematics and is basic enough to be intuitively acceptable when you learn logic. In turn, induction is mostly justified deductively. 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. Probability theory for infinite and uncountable domains are straightforward generalizations. There are only few leaps of faith when doing inductive inference, (1) when you transfer conclusions from a smaller to a larger population, (2) apply the law of large numbers, or (3) make reasonable guesses about an underlying probability distribution.
There is no such justification for abduction. Even just hypothesis generation, the narrow sense of "abduction", is way more complicated than induction and deduction. Theory selection, the broader aspect of "abduction" that has been labeled "inference to the best explanation" since Harman's horrible 1965 paper, is even more complicated. Both IBE and abduction require and involve deductive and inductive reasoning, but there is no indication that they can be fully justified deductively and inductively. The best you may get is an account that describes some form of theory discovery as a logical or probabilistic description of a supposed cognitive mechanism, and then the discovered theory might be proved deductively or confirmed inductively.
Defenders of abduction have claimed that we select hypotheses not just on the basis of their likelihood but also based on other properties like theory virtues ("loveliness" in Lipton's terms). That might be true but doesn't concern the justificatory value of the inference. In the end, the only hypotheses we're interested in (and should be interested in, from an epistemic point of view) are the ones that are most likely to be true. Hence, theory virtues, loveliness, plausibility, etc., all just have a justificatory value insofar as they increase the subjective probability that a hypothesis is true, and in that respect they are really just heuristics.
Some Bayesian takes on abduction and IBE get this right and I have nothing against them. I just wanted to mention that many, if not the majority of philosophers of science do not consider abduction a third type of inference and fairly skeptical about it, for the above reasons and a few more.
> 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?
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.
The problem with abduction is that it doesn't have a justificatory value on a par with those of deduction and induction. Deduction forms the basis of mathematics and is basic enough to be intuitively acceptable when you learn logic. In turn, induction is mostly justified deductively. 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. Probability theory for infinite and uncountable domains are straightforward generalizations. There are only few leaps of faith when doing inductive inference, (1) when you transfer conclusions from a smaller to a larger population, (2) apply the law of large numbers, or (3) make reasonable guesses about an underlying probability distribution.
There is no such justification for abduction. Even just hypothesis generation, the narrow sense of "abduction", is way more complicated than induction and deduction. Theory selection, the broader aspect of "abduction" that has been labeled "inference to the best explanation" since Harman's horrible 1965 paper, is even more complicated. Both IBE and abduction require and involve deductive and inductive reasoning, but there is no indication that they can be fully justified deductively and inductively. The best you may get is an account that describes some form of theory discovery as a logical or probabilistic description of a supposed cognitive mechanism, and then the discovered theory might be proved deductively or confirmed inductively.
Defenders of abduction have claimed that we select hypotheses not just on the basis of their likelihood but also based on other properties like theory virtues ("loveliness" in Lipton's terms). That might be true but doesn't concern the justificatory value of the inference. In the end, the only hypotheses we're interested in (and should be interested in, from an epistemic point of view) are the ones that are most likely to be true. Hence, theory virtues, loveliness, plausibility, etc., all just have a justificatory value insofar as they increase the subjective probability that a hypothesis is true, and in that respect they are really just heuristics.
Some Bayesian takes on abduction and IBE get this right and I have nothing against them. I just wanted to mention that many, if not the majority of philosophers of science do not consider abduction a third type of inference and fairly skeptical about it, for the above reasons and a few more.