There are various definitions of feminism, but let us adopt the following for present purposes: Feminism is a social movement that affirms a certain view F, and works towards the goal of causing F to become false. A feminist, then, is a person who is part of the feminism movement. The view affirmed by feminism is this:
F: Women (or some women, or at least one woman) is/are oppressed, in society in various ways, by one or more men.
More precisely, let:
w: the gender woman/female
m: the gender man/male
Px: x is a living person.
D: { x | Px }
Gxy: x has the gender y.
Oxy: x is oppressed in society by y.
Then we define F:
F: ∃xy∈D(Gxw ∧ Gym ∧ Oxy)
(Note that F is a weak sense of feminism, and many “feminists” adopt a stronger form; thus, since we’re using this very weak sense, our argument is strengthened).
Furthermore, the goal of feminism is to eliminate the oppression of women by men, that is, its goal is to cause F to be false:
¬F
or:
¬∃xy∈D(Gxw ∧ Gym ∧ Oxy)
So, if F is false, then the goal of feminism will be satisfied and, well, feminism would become unnecessary (because women would no longer be oppressed by men).
Let us now define gender identity theory (GIT) as the following proposition:
Q: a person’s gender is determined by their sense or feelings about what their gender is, that is, a person is some gender if and only if they sense or feel as if they are that gender.
More precisely, let:
Hx: x is a gender.
E: { x | Hx }
Sxy: x feels or senses that they have the gender y.
Q: (∀x∈D)(∀y∈E)(Sxy ↔ Gxy)
Then, a gender identity theorist is a person who believes that Q is true.
Of course, many feminists deny Q (or reject GIT), but many also affirm Q; and our focus is on the latter, so let’s call a feminist who affirms Q a “Q-feminist”.
Now, the Q-feminist faces a difficulty: the conjunction of F (or feminism) and Q (or gender identity theory) leads to a surprising, humorous, and awkward logical conclusion: feminism’s goal can be satisfied, or feminism can become obsolete, simply if nobody identifies as a woman (or if nobody identifies as a man). For example, suppose that nobody identifies as a woman:
¬∃x∈D(Sxw)
then, according to Q, no woman exists:
(1)
(∀x∈D)(∀y∈E)(Sxy ↔ Gxy)
Premise (Q/GIT)
(2)
∀x∈D(Sxw ↔ Gxw)
1, ∀-Instantiation
(3)
∀x∈D((Sxw → Gxw) ∧ (Gxw → Sxw))
2, Definition of Biconditional
(4)
∀x∈D(Gxw → Sxw)
3, ∧-Elimination
(5)
¬∃x∈D(Sxw)
Assumption
(6)
∀x∈D(¬Sxw)
5, Quantifier Shift
(7)
∀x∈D(¬Gxw)
4, 6, Modus Tollens ∎
And this, in turn, implies that F is false (or that the goal of feminism has been satisfied):
(1)
∀x∈D(¬Gxw)
Premise
(2)
∀xy∈D(¬Gxw ∨ ¬Gym ∨ ¬Oxy)
1, ∨-Introduction
(3)
∀xy∈D(¬(Gxw ∧ Gym ∧ Oxy))
2, De Morgan’s Law
(4)
¬∃xy∈D(Gxw ∧ Gym ∧ Oxy)
3, Quantifier Shift ∎
This is obvious because, if there are no women, then it is false that women (or some women, or at least one woman) is/are oppressed, in society in various ways, by one or more men. We reach the same conclusion if nobody identifies as a man (since there would then be no men to oppress women). Therefore, a logical solution to feminism’s goal is if nobody identifies as a woman – let’s call this solution No Women Solution (NWS).
At this point, the Q-feminist should object that NWS is impractical because it is highly improbable that society will reach a state in which nobody identifies as a woman. I agree, especially given how unpopular GIT is; nevertheless, this objection is beside the point. Logically, given Q, NWS is a solution to feminism’s goal; however, it is plain that something is intuitively wrong with this – intuitively, it is obvious that NWS is not a solution to feminism’s goal. Why is this? Well, because merely changing one’s view of one’s gender does not imply that one will no longer be oppressed. Suppose some woman P1 is oppressed by another person P2. Then, will this oppression simply cease if P1 identifies as a man? Will P2 suddenly stop oppressing P1 in this case? Not necessarily, and most probably not. Therefore, it appears that feminism should, rather, define a “woman/girl/female” as a person who has the set of XX chromosomes, and this implies that the feminist should reject gender identity theory.
Where does this leave us? Well, the Q-feminist has several options:
1. Affirm the awkward, unintuitive view that NWS is a solution to feminism’s goal.
2. Try to patch up feminism, gender identity theory, or both, somehow to avoid awkward logical conclusions (but how to do this?).
3. Reject or deny feminism (stop being a feminist).
4. Reject or deny gender identity theory (stop being a gender identity theorist).
5. Reject both feminism and gender identity theory.