Consequent in logic: meaning and examples

Q: Can I infer the antecedent from the consequent?

No. From A -> B and B nothing follows about A; row 1 of the table, A = FALSE, B = TRUE, makes both true with A false. That is the fallacy of affirming the consequent, and modus tollens, from NOT B to NOT A, is the valid direction.

The consequent is the “then” part of a conditional: in A -> B, B is the consequent. The conditional promises the consequent only when the antecedent holds, and says nothing when it does not.

In “if it rains, the ground is wet”, the consequent is “the ground is wet”. It is the right operand of ->, whatever the sentence order, and it is a necessary condition for the antecedent: if the conditional is true and the ground is dry, it is not raining. A true consequent makes the conditional true whatever the antecedent, which is one of the two cases in which a conditional holds for reasons that have nothing to do with the link between its parts.

The artefact on this page is A -> B, with the consequent B. The 4-row table reads true, true, false, true. On rows 1 and 3, where B = TRUE, the conditional is true regardless of A; on row 2, A = TRUE, B = FALSE, it is false, and that assignment is the tautology check's counterexample. The engine's reading adds the warning that matters for consequents: the condition does not say the reverse, so a true consequent does not establish the antecedent. The satisfiable check holds with the witness A = FALSE, B = FALSE.

Consequents are where rules state their outcome (“then grant access”) and where arguments go wrong: affirming the consequent, concluding A from A -> B and B, is the commonest fallacy in reasoning about rules, and the counterexample page shows it refuted on one row. The term is confused with the conclusion of an argument (a separate statement, not a part of a conditional) and with a consequence in the causal sense; logic records no causation.

Example

f := A -> B · output

Exact checks

Always true (tautology)? No Counterexample: A = T, B = F · Exhaustively checked and solver cross-checked · 3 rows

Always false (contradiction)? No Counterexample: A = F, B = F · Exhaustively checked and solver cross-checked · 1 rows

Can be true? Yes Example assignment: A = F, B = F · Exhaustively checked and solver cross-checked · 1 rows

Truth table (4 rows)

Consequent in logic: meaning and examples
#	A	B	f
0	FALSE	FALSE	TRUE
1	FALSE	TRUE	TRUE
2	TRUE	FALSE	FALSE
3	TRUE	TRUE	TRUE

Plain-language reading

Beginnerf is true exactly when the following holds: if A is true, then B is true.
In every other case, f is false.
About the operators:
- An “if … then” condition only constrains the case where its “if” part holds: when the “if” part is false, the condition is satisfied whatever the “then” part is. It does not say the reverse.

Intermediatef is true exactly when the following holds: if A is true, then B is true.
In every other case, f is false.

Advancedf is true exactly when the following holds: if A is true, then B is true [material implication →].
In every other case, f is false.
Formal notation: A → B

The common slip

Reading A -> B as “B happens because of A”. The conditional is true whenever B is true, including when A is false and unrelated. Truth-functional conditionals record a pattern of values, not a mechanism, which is why FALSE -> B and A -> TRUE are both tautologies.

Questions

Can I infer the antecedent from the consequent?

No. From A -> B and B nothing follows about A; row 1 of the table, A = FALSE, B = TRUE, makes both true with A false. That is the fallacy of affirming the consequent, and modus tollens, from NOT B to NOT A, is the valid direction.

Glossary

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.