I mean, it would be interesting to see if Bryson would take Peterson up on his offer to have a debate specifically about what gender is, since he says he would stack his knowledge of the peer reviewed literature up against hers any day.
This is a debate I would like to see. After all, we are codifying into law ethereal concepts like "gender identity" and "gender expression," without any clear definition of these. To illustrate, here are the definitions from the Ontario Human Rights Commission:
Gender identity is each person’s internal and individual experience of gender. It is their sense of being a woman, a man, both, neither, or anywhere along the gender spectrum....
Gender expression is how a person publicly presents their gender. This can include behaviour and outward appearance such as dress, hair, make-up, body language and voice. A person’s chosen name and pronoun are also common ways of expressing gender.
Unlike other protected grounds under Human Rights legislation, we are now protecting a person's *sense* of what they are, rather than what they actually are. Some of you will protest, but a person's gender identity *is* who they really are. So a man who says he feels like a woman inside automatically loses his male privilege, should be called "she," and be given the keys to the women's locker room.
If a man says he "feels like a woman," how does he know? If a woman says she "feels like a man," how does she know? Fact is, there is not a shred of evidence for a biological basis for wanting to present oneself in accordance with the stereotypes of behaviour and dress for the opposite sex.