Hypothetical syllogism: A → B, B → C, therefore A → C

Q: Does the chain work for longer sequences?

Yes, by repeating the step: A -> B, B -> C, C -> D give A -> D, each link checked the same way. The engine checks any such argument on the full table over its variables, up to the input limit, and reports the first row that breaks the chain if one exists.

Hypothetical syllogism: from A -> B and B -> C, conclude A -> C. The validity check holds on all eight rows, and the premises are consistent; implications chain.

If A then B, and if B then C, so if A then C: two implications joined at B give an implication that skips it. The model has three variables and the premises p1, A -> B, and p2, B -> C, with the conclusion A -> C. The engine's reading of the conclusion is “if A is true, then C is true”. The truth table has eight rows, and the two premise columns are each false on exactly two of them: p1 on rows 4 and 5, where A is true and B false, and p2 on rows 2 and 6, where B is true and C false.

The validity check holds: no admissible assignment makes every premise true and the conclusion false. Both premises are true on rows 0, 1, 3 and 7, from A = FALSE, B = FALSE, C = FALSE to A = TRUE, B = TRUE, C = TRUE, and the conclusion column is true on all four. The premise-consistency check holds with the witness A = FALSE, B = FALSE, C = FALSE, so validity is not vacuous. No counterexample exists; the exhaustive search of the eight rows found none, and the engine reports none.

Chained implications are the backbone of dependency reasoning: if the build is green the tests passed, if the tests passed the contract holds, so a green build means the contract holds. They also drive transitive permission rules and type-subtyping chains. The thing to watch is the direction of each link. The chain only composes when the consequent of one implication is the antecedent of the next; A -> B with C -> B proves nothing about A and C, and the checker would show a counterexample for that version.

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)

Hypothetical syllogism: A → B, B → C, therefore A → C
#	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

Logic diagram

The common slip

Chaining through the wrong end: from A -> B and C -> B concluding A -> C. The shared variable must be the consequent of the first premise and the antecedent of the second. The eight-row table for the correct form has the conclusion true wherever both premises are, which is what makes the chain safe.

Questions

Does the chain work for longer sequences?

Yes, by repeating the step: A -> B, B -> C, C -> D give A -> D, each link checked the same way. The engine checks any such argument on the full table over its variables, up to the input limit, and reports the first row that breaks the chain if one exists.

Guides

Rules of inference

Operators and gates

Implication (→, if–then): truth table and why a false antecedent makes it true

Tools

Worked examples

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.