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Can you give me some more detail about the difficulty that you are seeing? I didn't think that it is hard to draw in any way that is particularly relevant to the trans question - the only problem that really pertains to it is that people tend to become very coy about why they want to engage in various aspects of the male-female distinction. The reason people care about facts is that facts determine the action->outcome function they are facing as agents; the reason they create categories is that the (facts \times actions -> outcomes) function is hard to evaluate and has a large domain that you would need to search if you seek to optimise. Lost time and effort also affects the outcome negatively, so all else equal it is better if you can approximately factor the function through a smaller domain (facts -> categories, categories \times actions -> outcomes) without skewing the valuation of each resulting outcome much. If you don't understand what actions you are considering and what outcomes you find desirable, though, this is a hopeless or at least hard undertaking.
Scott's King Solomon gives a whole array of good reasons why he wants to categorise whales with fish, given that his outcomes are valued by "edible biomass captured" and his actions are in the class of "allocate money to biomass-capturing institutions". If you cluster whales and fish and your second factor just gets "dag sighted" as its first parameter, the expected outcomes of each available action ("pay the fishing ministry") are about the same as if you evaluated the full function with every little detail of the whale. His psychiatrist avatar does so as well, given that he evaluates on his patients' subjective wellbeing and has actions consisting of talking and prescribing various FDA-approved drugs. What Scott misses in his discussion is that the characterisations the king and psychiatrist use, too, are grounded in facts - just different ones, which are more relevant to how their available actions affect their valuated outcomes. It is just as much of a fact that whales spend all their time in water, have fins and no particularly flexible limbs or neck, and that the transwoman patient will be unhappy if they are called a man to their face.
Aggregating on these factual criteria is useful for these people - but that doesn't give them any standing to suggest or impose categorisations on other people with completely different goals. King Solomon's fishing goals are irrelevant to the geneticist, and the psychiatrist's patient ratings are not similar to the objectives of almost everyone interacting with trans people on a day to day basis. For example, in my academic environment, my actions are basically talk and sometimes putting the thumb on the scale in some hiring decision, while the outcomes I want are about a peaceful social environment that is conducive to doing research. If trans people cluster with their birth gender as far as these are concerned (topic for another discussion thread), then whatever the mechanism is, that is the fact I would want to build my categories around.
All of this is irrelevant, though, because I think granting a human right to have bizarre and impractical categories if one so wishes is necessary for a society that is worth living in.
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?
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