Conclusion in logic: meaning and examples

The conclusion is the single statement an argument claims to establish from its premises. An argument is valid when the conclusion is true on every row where all the premises are true.

Every argument has exactly one conclusion, marked in prose by words such as “therefore”, “so” and “hence”, and in the engine's argument mode by its own role beside the premises. The conclusion is not asserted on its own: it is asserted conditionally, as following from the premises. A conclusion can itself be a compound formula, as on this page, and it can be true on rows where the premises are false; those rows are irrelevant to validity, which looks only at the rows where every premise holds.

The artefact on this page is the hypothetical syllogism: premises A -> B and B -> C, conclusion A -> C, with three variables and 8 rows. The validity check holds: no admissible assignment makes every premise true and the conclusion false. The premise-consistency check holds with the witness A = FALSE, B = FALSE, C = FALSE, row 0, where both implications are true because their antecedents are false. The conclusion column reads true on six of the eight rows and false on rows 4 and 6, but neither of those rows has both premises true.

In practice the conclusion is the part of a rule that people act on, so it is the part worth stating in words; the engine's plain-language reading renders this one as “if A is true, then C is true”. In a decision model the counterpart of a conclusion is a named output. The term is confused with the consequent of an implication, which is only the “then” part of one formula, and with the result of a computation, which is a value rather than a statement.

Example

p1 — A -> B · premisep2 — B -> C · premiseconclusion — A -> C · conclusion

Exact checks

Does the conclusion follow from the premises? Yes · Exhaustively checked and solver cross-checked · 8 rows

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

Truth table (8 rows)

Conclusion in logic: meaning and examples
#	A	B	C	p1	p2	conclusion
0	FALSE	FALSE	FALSE	TRUE	TRUE	TRUE
1	FALSE	FALSE	TRUE	TRUE	TRUE	TRUE
2	FALSE	TRUE	FALSE	TRUE	FALSE	TRUE
3	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	FALSE	TRUE	FALSE
5	TRUE	FALSE	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE	FALSE	FALSE
7	TRUE	TRUE	TRUE	TRUE	TRUE	TRUE

Plain-language reading

BeginnerConclusion: if A is true, then C is true.
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.

IntermediateConclusion: if A is true, then C is true.

AdvancedConclusion: if A is true, then C is true [material implication →].
Formal notation: A → C

The common slip

Checking the conclusion on every row. Rows where a premise is false cannot be counterexamples, however the conclusion behaves there. On this page the conclusion is false on rows 4 and 6, and the argument is still valid, because a premise is false on each of them.

Questions

Can a valid argument have a false conclusion?

Yes, when at least one premise is false. Validity promises only that the conclusion is true whenever the premises all are. A valid argument whose premises are actually true is sound, and only a sound argument guarantees its conclusion.

Practice

Argument validity exercises with answers

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.