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The correspondence between sets and predicates as shows up in formal logic. Applied to the example of the whales, this might be something like "What if the tanners guild wants to say whales dont have hairs?". Basically, it is not the case that there are some propositions that are "facts" that you just have to believe, and some that are "categorisations" where you can pick how you want to do them. You face the same basic situation wrt all of them, and obviously theyre not all up to you to decide - because on what basis could you decide, that is not itself a proposition?
I'm afraid I'm only getting more lost - you seem to be referring to some very specific (philosophical?) discussion that you assume I'll recall if you hint about it, but I'm drawing blanks. When you say "correspondence between sets and predicates (...)", this makes me think you are talking about predicates in extension (is-whale := the set of all things that you want to call a whale) vs. predicates in intension (is-whale := ), but I'm not sure how that would relate to the rest of your post.
Do you want to do something like drawing a distinction between predicates that are "more naturally" expressed extensionally vs. intensionally? So you would for example consider a notion of "nice number" that actually amounts to "is a Fibonacci number" as "factual", whereas a notion of "nice number" that amounts to "gives off good vibes to Lykurg" is "arbitrary".
I don't understand what that would have to do with the "tanners' guild" example, though - that sounds more like a setup where two different entities use different categories under the same label and want to push the respective other to adopt theirs (why? to reduce cognitive load for themselves when they are interacting with each other?). For your example, how do you envision the tanners' guild using that assertion of theirs? Is it (1) if someone gives them a piece of whale skin with hair follicles, they will say "whales don't have hairs, so I will pretend these are not there and not smooth out these before tanning it"? (2) if someone gives them a piece of -"-, they will say "this is hairy, so it is not whale skin and I will not put it in the whale processing pipeline"? (3) nothing changes about how they process whale skin with hair follicles, but they will dispatch a guy to argue all day if anyone anywhere claims that whales have hair?
You can define sets in terms of predicates (x \elem Fish \iff: fish(x)) or the other way (fish(x) \iff: x \elem Fish). So while you might intuitively say that x \elem Fish is a categorisation, it has a brother thats intuitively has the form of a fact, and you cant change one without the other.
There is not a logical distinction between intensional and extensional definitions, except
in modal logics, where it exists but depends entirely on details in your semantic setup that are in no way constrained by evidence, or
in model theory, there are intensions and extensions - but still no intensional or extensinal definitions in the underlying.
This is another manifestation of the problem.
I had (1) in mind when making up the example.
No. Its just an example of how one of the things he lists as "facts" might be redefined with just as good a justification as "fish". The problem is that when you decide what the optimal most convenient way to categorise is, you need to have some facts based on which to make that decision. But if theres no difference between facts and categorisations, then you dont.
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