Graham Linehan (Father Ted, Black Books) has been pinning his colours to the mast for some time when it comes to trans rights, ie. he is not entirely on board. A major plank of his argument seems to be that the logic of “a woman is someone who identifies as a woman” is circular.
But language has always evolved in part through circular reasoning. We call something ‘X’ because it seems very similar to another thing we call ‘X’. The definition applied afterwards is never absolute, never perfect, and the more a word encompasses, the less likely it is that its meaning is fixed to one particular feature, let alone a concrete feature. The word ‘woman’ has a sprawl of associations, and it’s not surprising that during a period of societal change, we should have to have a serious think about what it means to all of us going forward. Nor is it surprising that for different people it is strongly associated with different sets of experiences, or aspects of their identity.
Obviously, there’s a lot more to this subject, but for now I think the point about language is self-contained, and in itself defeats the complaint that as a matter of logic women cannot self-identify.