Denying the antecedent: A → B, NOT A, therefore NOT B (invalid)

Denying the antecedent: from A -> B and NOT A, concluding NOT B. The validity check fails with the counterexample A = FALSE, B = TRUE: the implication and the negated antecedent hold, yet B is true.

If A then B; not A; therefore not B. The form mimics modus tollens but negates the wrong side: “if you are a member you get a discount; you are not a member; so you get no discount”, which ignores every other way to get one. The model has the premises p1, A -> B, and p2, NOT A, and the conclusion NOT B; the engine's reading of the conclusion is “B is false”. On the four-row table p2 is true on rows 0 and 1, where A is false, and p1 is true on both of those rows.

The validity check is violated: in the counterexample every premise is true but the conclusion is false. The engine reports A = FALSE, B = TRUE, row 1: A -> B is true because its antecedent is false, NOT A is true, and NOT B is false. On the other row where both premises hold, row 0, A = FALSE, B = FALSE, the conclusion happens to be true, which is what makes the fallacy feel safe. The premise-consistency check holds with the witness A = FALSE, B = FALSE: the premises are possible together and simply do not determine B.

This fallacy is the argument-form version of the inverse slip, and it appears whenever a rule is read as “only if”: a permission rule that grants access to admins is taken to deny it to everyone else. The thing to watch is what the implication leaves open. A -> B says nothing about the rows where A is false, and both of those rows satisfy the premises here. To conclude NOT B from NOT A you need the converse implication, B -> A, as a separate premise, which is a separate claim to verify.

p1 — A -> B · premisep2 — NOT A · premiseconclusion — NOT B · conclusion

Exact checks

Truth table (4 rows)

Plain-language reading

Logic diagram

The common slip

Reading A -> B as “B if and only if A”. With that reading the fallacy would be valid, but the implication in the premise is weaker, and the counterexample row A = FALSE, B = TRUE is consistent with it. If the policy really is “only members get discounts”, write the biconditional and re-run the check.

Questions

How is this different from modus tollens?

Modus tollens negates the consequent, from NOT B to NOT A, and the validity check holds on all four rows. Denying the antecedent negates the antecedent, from NOT A to NOT B, and the row A = FALSE, B = TRUE refutes it. The side you negate decides whether the argument is valid.

Related

Guides

• Rules of inference

Operators and gates

• NOT gate (inverter): truth table, symbol and the NOT operator
• Implication (→, if–then): truth table and why a false antecedent makes it true

Tools

• Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
• Argument validity checker: premises, conclusion, counterexample or proof
• Karnaugh map solver: 2 to 4 variables, groups, SOP and POS, don’t-cares from assumptions

Practice

• Argument validity exercises with answers

See also

• Modus tollens: A → B, NOT B, therefore NOT A (valid)
• The inverse slip: A → B is not NOT A → NOT B
• Affirming the consequent: A → B, B, therefore A (invalid, with counterexample)
• Counterexample in logic: meaning and examples
• Necessary vs sufficient: “only if” is not “if”
• Argument validity checker: premises, conclusion, counterexample or proof

Worked examples

• Entry rule: (member OR invited) AND NOT banned, checked every way
• Entry rule slip: member OR (invited AND NOT banned) and the case that differs
• Majority of three (2-out-of-3 voting): truth table, K-map and NAND form
• Odd parity of three inputs: A XOR B XOR C

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.