"A woman is a woman, anything else is not a woman."

Your question is politically charged, calling those who define women as the adult female of the human species "anti-trans". This is the correct definition. Males who identify as women can do so but their self-perceived identify is not a material reality. I can identify as being fabulous and smart. I shouldn't impose that self-perceived image onto others. So if they don't refer to me as fabulous and smart they are being anti-me.

Woman has a specific definition. All non-human animals have words to refer to the male and female of their species (mare-stallion for horses, for instance; women and men for humans). We wouldn't say that there are trans-mares that are natally stallions.

All members of Group A are members of Group A. Therefore, anyone who is not a member of Group A is not a member of Group A. Logically, this is correct, but not particularly meaningful. Technically, if your definition of woman included trans women, this would still hold up.

The underlying subtext, though, is that trans women are not women. Written out explicitly, the argument would change to "A cis woman is a woman. Therefore, anything else is not a woman." This is not logically correct. If I was to say "A Chihuahua is a dog, therefore anything else is not a dog," I would be drawing an illogical conclusion from a true premise. Chihuahuas are a subset within the group of dogs, but do not compromise the entire group. I believe this would be denying the antecedent. You would need to separately establish that all women are cis women or that all dogs are Chihuahuas to make this logically valid. For the former, though, you'd have a hard time doing this without using the definist fallacy .

1. Merriam Webster: “having a gender identity that is the opposite of male”. This is an ideological newly introduced definition. Woman is the female of the human species. What that does even mean to have a “gender identity”? It is cartesian dualism. You believe you are a different sex than you are? What is this gender identity, to believe your personality and behaviour match a rigid idea of what a female human is? A woman can behave and be whatever she wants to be. “Gender identity” is a rancid concept, really backwards, and those who embrace it really don’t even realise how retrograde it is. Like a little boy has a gender identity of a girl… what does that mean? I can assure you it means that the boy “behaves” like a girl, likes “girl” toys, etc. Their parents rather have a “trans girl” than an effeminate boy.

2. “here in the US, the insistence that only anatomical females are women is the basis for a lot of anti-trans sentiment, which has materialized into laws and policies denying trans people access to a number of things including gender affirming medical care.” “Affirming care” is a linguistic trick: it does not include psychotherapy to address why somebody may feel they are born in the wrong body (they call that “conversion therapy”. It refers to surgeries and life-long medicalisation of children above all. Blocking puberty with permanent damage to bone density. Puberty is not an optional thing. You must go through it to develop as a fully adult with a fully developed brain, which matures in this transition to adulthood into 25 years of age. This is child abuse. Trans people deserve therapies to understand the underlying causes of their distress, not surgery and big Pharma. We are pro trans people’s well being.

3. An anorexic patient could say the same: that they are denying her affirmative care: stomach reduction surgery, to be her true self: a very thin woman. If you don’t succumb to her wishes, she will commit suicide.

4. “worms are among the animals that are naturally hermaphroditic” So what? I don’t care about other species. Humans are mammals, not insect. Even insects have only two sexes, like 99,999% of animals of any kind. Even if some few species are hermaphroditic… that means that they can change between TWO sexes. Not more. You bring up bearded dragons as if it was relevant to Homo sapiens species. And then I am the one making wrong comparisons. Humans are not bearded dragons. We know very well that humans are not hermaphroditic or change sex. A woman is not a man who cut off his penis.

"Anything that is not a woman is not a woman" is a tautology, and "anything else" is equivalent to "anything that is not a woman" in this case?