But there seems to be something very different about each of the two situations. In the first, we would say that the “brush your teeth” abstraction is composed of the subtasks, but we wouldn’t say that “animal” is composed of humans, dogs and cats in the second.
Actually, from an extensive point of view, that is exactly how you would define “animal”: as the set of all things that are animals. So it is in fact composed of humans, dogs and cats—but only partly, as there are lots of other things that are animals.
This is just a pedantic point since it doesn’t cut to the heart of the problem. As johnswentworth noted, man-made categories are fuzzy, they are not associated with true or false but with probabilities, so “animal” is more like a test, i.e. a function or association between some set of possible things and [0,1] . So “animals” isn’t a set, the sets would be “things that are animals with probability p” for every p∈[0.1].
Moving from animals to another example: If you are a half-bald person you do not belong to the set of bald people with probability 0.5. Probability is a epistemic concept, but the vagueness (fuzzyness) of the concept of baldness is not epistemic, but semantic. No amount of information makes you more or less bald. Therefore, for fuzzy concepts, there is no probability of membership of a set, but a degree of membership of a set. Which is again a number between 0 and 1, but it is not a probability. There is actually a very unpopular logic which is based on this notion of fuzzy sets: Fuzzy logic. It’s logical constants behave different from their equivalents in probability theory. E.g. commonly: A and B = MIN(A, B); A or B = MAX(A, B).
Just a quick, pedantic note.
Actually, from an extensive point of view, that is exactly how you would define “animal”: as the set of all things that are animals. So it is in fact composed of humans, dogs and cats—but only partly, as there are lots of other things that are animals.
This is just a pedantic point since it doesn’t cut to the heart of the problem. As johnswentworth noted, man-made categories are fuzzy, they are not associated with true or false but with probabilities, so “animal” is more like a test, i.e. a function or association between some set of possible things and [0,1] . So “animals” isn’t a set, the sets would be “things that are animals with probability p” for every p∈[0.1].
Moving from animals to another example: If you are a half-bald person you do not belong to the set of bald people with probability 0.5. Probability is a epistemic concept, but the vagueness (fuzzyness) of the concept of baldness is not epistemic, but semantic. No amount of information makes you more or less bald. Therefore, for fuzzy concepts, there is no probability of membership of a set, but a degree of membership of a set. Which is again a number between 0 and 1, but it is not a probability. There is actually a very unpopular logic which is based on this notion of fuzzy sets: Fuzzy logic. It’s logical constants behave different from their equivalents in probability theory. E.g. commonly: A and B = MIN(A, B); A or B = MAX(A, B).