Deductive arguments do rely on the meanings of words, and unless we are doing math, those meanings are derived from experience. However, I think that there are set of agreed-upon meanings and presuppositions shared by essentially everyone who is in favor of letting trans women play women’s sports. I also think that those meanings will lead them into this dilemma. You are right that we could re-define the word “woman” by using criteria like testosterone levels, and it’s important information that this would not create a fair situation. I was considering that possibility until I read your article. But the position of almost all advocates for trans rights is that there is only one criterion for determining gender: the deeply held belief of the person in question. People who believe otherwise are called “trans-medicalists”, and they have been ferociously attacked by Twitter mobs. Natalie Wynn was attacked for letting a trans-medicalist say one line of dialogue in one of her videos, and Abigail Thorne was attacked just for being in that video. As long as trans rights advocates say “I am a woman if and only if I say I am a woman” they will be stuck with this dilemma.

You could make the argument that when Ronda Rousey beats another woman in competition, she beats them fairly because both Miss Rousey and her competitor had to start with a woman’s body. Someone who won the same contest through training a man’s body could be seen as starting with an unfair advantage. That’s certainly the underlying justification for segregated sports competitions, and for affirmative action laws as well.