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denying_a_conjunct
Denying a conjunct is a formal fallacy that occurs when, from the premise that a conjunction is false (not both A and B), and the premise that one conjunct is false, it is concluded that the other conjunct must be true. This confuses the logical conjunction (AND) with the exclusive disjunction (XOR). If 'not both A and B' is true, denying A only tells us the conjunction fails — it does not tell us anything about B, which could be either true or false.
"You can't be both rich and happy. You're not rich. Therefore, you must be happy."
A fitness coach tells a client: 'You can't be both consistent at the gym and eating junk food every day. And I know you're not consistent at the gym. So you must be eating well.' — The client could simply be doing neither.
A political pundit argues: 'A candidate can't be both electable and truly progressive. Senator Harris isn't electable. So she must be truly progressive.' — The fallacy ignores that she might be neither, or that the original premise is flawed.
Binary (yes/no) questions an LLM must answer to identify this aspect:
Does the argument state that two things cannot both be true simultaneously (negation of a conjunction)?
Type: binaryDoes the argument deny one of the conjuncts?
Type: binaryDoes it conclude that the other conjunct must therefore be true?
Type: binaryDenying a conjunct is a formal fallacy that occurs when, from the premise that a conjunction is false (not both A and B), and the premise that one conjunct is false, it is concluded that the other conjunct must be true. This confuses the logical conjunction (AND) with the exclusive disjunction (XOR). If 'not both A and B' is true, denying A only tells us the conjunction fails — it does not tell us anything about B, which could be either true or false.
People intuitively interpret 'not both' as 'one or the other,' treating AND as if it were exclusive OR. The binary framing of the conjunction obscures the possibility that neither conjunct is true.
Point out that 'not both A and B' is compatible with three scenarios: A but not B, B but not A, or neither A nor B. Denying one conjunct does not establish the other.
Appears in everyday reasoning about incompatible-seeming properties, in personality typing ('you can't be both analytical and creative'), and in political discourse where complex identities are reduced to false dichotomies.
Presenting only two options when many more exist.
From a disjunction (A or B) and the truth of one disjunct (A), concluding that the other must be false (not B). Invalid because both disjuncts can be true in an inclusive 'or'.
If A then B; not A; therefore not B. (Invalid modus tollens reversal).
If A then B; B; therefore A. (Invalid modus ponens reversal).
Use these tools to detect, analyze, or train this aspect.