Converse in logic: meaning and examples

The converse of the conditional A -> B is B -> A: the same two parts with the arrow reversed. A conditional and its converse are not equivalent, and assuming they are is the most common error with implications.

Forming the converse swaps antecedent and consequent. “If it rains, the ground is wet” has the converse “if the ground is wet, it rains”, which can be false when the first is true, because sprinklers exist. The converse is logically independent of the original: either, both or neither can hold. Related forms are the inverse, NOT A -> NOT B, which is equivalent to the converse, and the contrapositive, NOT B -> NOT A, which is equivalent to the original.

The artefact on this page compares implication = A -> B with converse = B -> A. The columns read true, true, false, true and true, false, true, true, and the differing-rows list is rows 1 and 2: on row 1, A = FALSE, B = TRUE, the implication is true and the converse false; on row 2 the reverse. The equivalence check is violated with the counterexample A = FALSE, B = TRUE, the summary being that the two formulas give different results there. They agree only where A and B have the same value.

Confusing a conditional with its converse turns “members may enter” into “whoever enters is a member”, and turns a necessary condition into a sufficient one; the pitfall page on necessary versus sufficient is about exactly this. When both directions are intended, the biconditional A <-> B says so, and it is equivalent to the conjunction of a conditional and its converse. The term is confused with the inverse and, more loosely, with any “reverse” of a statement.

Example

implication := A -> B · outputconverse := B -> A · 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

Are implication and converse equivalent? No Counterexample: A = F, B = T · Exhaustively checked and solver cross-checked · 2 rows

Truth table (4 rows)

Converse in logic: meaning and examples
#	A	B	implication	converse
0	FALSE	FALSE	TRUE	TRUE
1	FALSE	TRUE	TRUE	FALSE
2	TRUE	FALSE	FALSE	TRUE
3	TRUE	TRUE	TRUE	TRUE

Plain-language reading

Beginnerimplication is true exactly when the following holds: if A is true, then B is true.
In every other case, implication 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.

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

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

The common slip

Assuming the converse because the original is true. A -> B holds on row 1 and B -> A fails there, so the one does not carry the other; the equivalence check names that row. A rule stated in one direction licenses nothing in the other.

Questions

When are a conditional and its converse both true?

Exactly when A and B always have the same value, which is what the biconditional A <-> B asserts. On this page that is rows 0 and 3, where the two columns agree; on rows 1 and 2 exactly one of the pair holds.

Guides

Converse, inverse and contrapositive

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.