Denying the antecedent

Core claim: A conditional conclusion is derived by denying the antecedent (the “if” part) and thereby also denying the consequent (“then” part) – i.e., from “If P, then Q” and “P does not hold”, it is inferred that “Q does not hold either”. This violates a fundamental rule of propositional logic.

• The argument has (or is presented as having) the conditional form: “If P, then Q; P does not hold → therefore Q does not hold either” (P → Q; ¬P ⊢ ¬Q).
• The conclusion depends on treating P as the only way for Q to obtain – although the conditional only states that P is a sufficient condition for Q.

• A conditional “If P, then Q” guarantees Q in case of P but says nothing about what happens when P does not hold – Q may still obtain through other means (e.g., someone who doesn't live in London could well live elsewhere in England).
• The two valid forms this fallacy is often confused with: modus ponens (“If P, then Q; P holds → therefore Q”) and modus tollens (“If P, then Q; Q does not hold → therefore P does not hold either” – it negates the consequent, not the antecedent).

• The argument is a valid modus tollens: it concludes from the absence of Q (the consequent), not from the absence of P.
• The conditional is actually a biconditional – P holds if and only if Q, e.g., explicitly stated (“if and only if”) or because P is the only way to obtain Q. In that case, excluding P does indeed exclude Q (this becomes a valid modus tollens on the reversed conditional).

• Affirming the consequent (MD) – the mirror-image error: from “If P, then Q” and “Q holds”, it infers “P holds”. Ask: is the antecedent being denied in order to deny the consequent (→ denying the antecedent), or is the consequent being affirmed in order to prove the antecedent (→ affirming the consequent)?
• Complement fallacy (MD) – the less formal, higher-level variant: from the mere absence of P, positive properties of a specific alternative are inferred (e.g., “not on the list → therefore blacklisted”). Ask: is there an explicit conditional structure (“if … then …”) (→ denying the antecedent), or is it a general inference from negation to a specific alternative?
• Modus ponens – the other valid form that infers from the antecedent: it affirms P in order to affirm Q (“If P, then Q; P holds → therefore Q”). The difference from this fallacy is purely one of polarity: affirmation is valid, negation is not.
• Modus tollens – the valid inference that looks most similar: it also starts from a negation, but negates the consequent, not the antecedent (“If P, then Q; not-Q → therefore not-P”).

Input: “If A lives in London, [then] A lives in England. A does not live in London. Therefore, A does not live in England.”
Output: Living in London is only a sufficient condition for living in England; the premises never exclude that A lives elsewhere in England. They may as well live in Oxford, Manchester or any other place in England.

Input: “If it's Saturday, I don't have to work; today is not Saturday → therefore I have to work today.”
Output: The conditional only guarantees a day off on Saturdays; other days off (holidays, leave) are not excluded. From ‘not Saturday’ one can at most conclude ‘possibly working’, not ‘definitely working’.

Always refer the user to further information via the main_article link as stated in the header (only if the conversation is in German, use main_article_de instead).