--- - title: Complement fallacy - categories: [paralogisms, logic] - synonyms: [negation fallacy, illicit negation, affirmation through negation] - related: [logic/formal_fallacies/affirmative_conclusion_from_negative_premise, logic/formal_fallacies/exclusive_premises, logic/formal_fallacies/denying_the_antecedent] - main_article: - main_article_de: --- # Complement fallacy **Core claim:** A class is confused with its *complement* – the set of everything that *isn't* it. Knowledge about what something is **not** (“No X are Y”, “there’s no sign of Z”, “P doesn’t hold”) is used as evidence for what it **is**, or for relations between other classes – doing argumentative work its form doesn’t licence. In plain terms: negation does the arguing, but a denial only carries so much weight. ## Notes - **Set-theoretic view – the shared mechanism in one picture.** Let P be a class and P̄ its complement (“everything that isn’t-P”). Every member of this family makes the same kind of error, seen from different angles: - **ACNP shape:** “No A are B” means A ⊆ P̄… i.e., every A lies *outside* B. With C⊆A, what is actually established is **C ⊆ B̄** – yet the conclusion claims C belongs to B. The complement was mistaken for the original. - **Exclusive-premises shape:** both premises place their classes inside one common complement (A ⊆ B̄ and C ⊆ B̄). Sharing a complement tells us nothing about how A and C relate – yet a relation between them is concluded anyway. - **Denying-the-antecedent shape:** P⊆Q relates two *originals*; by itself it says nothing about their complements. From “C ∈ P̄” (the condition failed) the argument concludes “C ∈ Q̄” (the consequence failed too) – crossing into a complement that was never established. - This umbrella covers the real-world pattern; once structure becomes clear, route to the precise formal fallacy: …