Modus ponens: A → B, A, therefore B (valid)

Modus ponens: from A -> B and A, conclude B. The validity check holds on all four rows, the premises are consistent, and the table shows the single row where both premises are true.

This is the oldest valid argument form: if A then B; A; therefore B. In argument mode the model carries two premises, p1 is A -> B and p2 is A, and a conclusion, B. An argument is valid when no assignment makes every premise true and the conclusion false. The engine's reading of the conclusion is simply “B is true”. The four-row table shows the premise columns and the conclusion column side by side.

The validity check holds: the conclusion follows, because no admissible assignment makes both premises true and the conclusion false. Reading down the table, p1 is false only on row 2, A = TRUE, B = FALSE, and p2 is true only on rows 2 and 3, so both premises hold together only on row 3, A = TRUE, B = TRUE, where the conclusion is true. The premise-consistency check also holds, with the witness A = TRUE, B = TRUE: the premises can be true together, so the validity is not vacuous. There is no counterexample to report; the engine shows none because the exhaustive search found none.

Modus ponens is how rules fire in a rule engine, how a type checker applies an implication from a signature, and how every chained guard in code works: the condition holds, so the body runs. The thing to watch is the second premise. A rule engine that has A -> B but never establishes A concludes nothing, and the vacuous rows 0 and 1 of the table, where A is false and p1 is true, are exactly the situations in which the implication alone proves nothing about B.

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

Exact checks

Does the conclusion follow from the premises? Yes · Exhaustively checked and solver cross-checked · 4 rows

Can all premises be true together? Yes Example assignment: A = T, B = T · Exhaustively checked and solver cross-checked · 4 rows

Truth table (4 rows)

Modus ponens: A → B, A, therefore B (valid)
#	A	B	p1	p2	conclusion
0	FALSE	FALSE	TRUE	FALSE	FALSE
1	FALSE	TRUE	TRUE	FALSE	TRUE
2	TRUE	FALSE	FALSE	TRUE	FALSE
3	TRUE	TRUE	TRUE	TRUE	TRUE

Plain-language reading

BeginnerConclusion: B is true.

IntermediateConclusion: B is true.

AdvancedConclusion: B is true.
Formal notation: B

Logic diagram

The common slip

Running the argument backwards: from A -> B and B, concluding A. That is affirming the consequent, and the table refutes it on the row A = FALSE, B = TRUE, where p1 and B are true and A is false. Modus ponens needs the antecedent, not the consequent, as its second premise.

Questions

How does the engine decide validity?

It builds the full truth table over the premise and conclusion variables and searches for a row where every premise is true and the conclusion is false. If it finds one it replays that row through the independent evaluator and reports it as the counterexample; if the search completes without one, the verdict is that the conclusion follows.

Is a valid argument also a sound one?

Only if its premises are actually true in the situation at hand. Validity is about form and is what the table decides; soundness adds the truth of the premises, which no table can check. The glossary pages on valid and sound arguments draw the line.

Guides

Rules of inference

Operators and gates

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

Tools

Practice

Argument validity exercises with answers

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.