Transwomen are women. Transmen are men. True or false? Late to this debate, but I take it that despite all the heat of the public debate, this is just an issue in metaphysics.
The public debate - my impression of it anyway - is that it is almost exclusively conducted by those with no training in metaphysics and it shows, for there seem to be two camps, both fairly obviously false. The two views seem to be either that you're a woman if you identify as one (so, identifying as a woman constitutively determines that one is one), or alternatively a biologist determines whether you're a woman. So, it's either you, or a biologist.
Both views are silly. It's true that both are reliably proxies for being a woman. Virtually all people who identify as women are women (just as virtually all people who identify as lawyers are lawyers). And virtually all people who satisfy the biologist's criteria for being a woman are women too. But being a reliable proxy is not the same as being the thing one is a proxy for.
Let's do some entry level metaphysics: first, not every concept can be defined, for that would generate an infinite regress in which it turns out nothing can be defined.
Thus, if there are true definitions, then there are concepts that cannot be defined.
Most people don't realize this and believe - fallaciously - that unless one can provide a definition for a concept, one doesn't understand it or have it. That's demonstrably false. But becausea they believe it, they will not believe they grasp a concept - even one of those basic concepts that are unamenable to definition - unless a definition is provided. And the first one that presents itself or is offered, will normally then be the one they cleave to thereafter, refining it if necessary but not giving it up. It's so common it's got a name: the definist fallacy.
Here's how one might fallaciously arrive at the conclusion that being a woman is constitutively determined by one's own subjective states: virtually everyone who believes they are a woman is a woman, therefore believing you're a woman is what makes you a woman, and thus a woman is just someone who identifies as one.
The other 'side' notices that there are clear counterexamples to this thesis - there are clear cases of men who are identifying as women, yet are not thereby becoming women (for they still seem to answer to the concept of a man, despite their identifying otherwise). And so they offer a different definition: that a woman is someone with immobile gametes, because when biologists look in detail at women's bodies, they find they all have that feature. And biologists - who are not metaphysicians and are just as capable of fallacious reasoning as the next person - reason that as all women they've examined have immobile gamates, then that must be what makes a woman a woman. That's fallacious. All square things have a colour, but that doesn't make the definition of a shape 'coloured'. Plus we can easily imagine someone who answers to the concept of a woman, yet does not have immobile gamates or any at all. So, it's as plainly false upon reflection as the individual subjectivist view about what makes someone a women.
But both sides think understanding comes from definitions and so they just double down on their own and get increasingly angry at the other side (as is typical of the ignorant).
The truth seems to be that we have the concept of a woman without being able to define it. It is in this respect like the concept of a mountain or a tree. Those are not amenable to definition either. In fact, there are loads and loads of concepts like this, or seem to be (we know there have to be some, remember).
We have evidence that we have an indefinable concept - though one that we nevertheless 'have' and are adept at applying - when our best attempts to define it fail. And we know that our best attempts at defining it are failing when there seem to be things that clearly answer to the concept in question, yet do not answer to the definition (and vice versa).
Is there currently a huge debate over the correct definition of a woman? Yes, that's obvious. So, the very existence of the debate - and the fact that both definitions in play are quite plainly false (which is why the debate continues, for each side can correctly highlight the absurdity of the other's defintion) - gives us reason to think that the concept of a woman is indefinable. A woman is someone who answers to the concept of a woman - that, it seems, is as much as can be said. And we already know well enough how to apply the concept - for we judge the credibility of a definition by whether or not it delivers verdicts consistent with the concept. It's just the definist fallacy prevents people from recognizing that they have the concept prior to any attempted definition - and then they feel themselves obliged to substitute their concept for the definition instead.
So, are transwomen women? Well, if a transwoman is someone who identifies as a woman but would not be considered one by a biologist in the grips of the definist fallacy....then some of them might be, and some of them might not be. It depends on whether they answer to the concept of a woman - a concept that is not amenable to definition and that biologists are not authorities about.