Inverse of an implication: meaning and examples

The inverse of the conditional A -> B is NOT A -> NOT B: both parts negated, arrow unchanged. Like the converse, it is not equivalent to the original; the inverse and the converse are equivalent to each other.

“If it rains, the ground is wet” has the inverse “if it does not rain, the ground is not wet”, which fails whenever something other than rain wets the ground. The inverse is the contrapositive of the converse, which is why the two always agree with each other and both disagree with the original on the same rows. The four related conditionals form two equivalent pairs: original with contrapositive, converse with inverse. The word has nothing to do with inverse functions or multiplicative inverses.

The artefact on this page compares implication = A -> B with inverse = NOT A -> NOT B. The columns read true, true, false, true and true, false, true, true, differing on rows 1 and 2. The equivalence check is violated with the counterexample A = FALSE, B = TRUE, row 1: the implication is true there because its antecedent is false, while the inverse has NOT A true and NOT B false, so it fails. The inverse's column is identical to the converse's column on the converse page.

The inverse is the form of the fallacy of denying the antecedent: from A -> B and NOT A, concluding NOT B is to use the inverse as if it followed from the original. In rule writing it appears as the assumption that withdrawing a condition withdraws the outcome (“non-members cannot enter”, when guests may). The term is confused with the converse, with negation of the whole conditional, and with the contrapositive, the only one that is equivalent.

Example

implication := A -> B · outputinverse := NOT A -> NOT 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

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

Truth table (4 rows)

Inverse of an implication: meaning and examples
#	A	B	implication	inverse
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

Treating “if not A then not B” as a restatement of “if A then B”. They differ on rows 1 and 2 of the table, and the equivalence check names row 1. The restatement that is actually equivalent negates both parts and also swaps them: the contrapositive.

Questions

Is the inverse ever useful on its own?

Yes, as a statement in its own right: when a rule is meant to run both ways, the inverse is the half that says the outcome needs the condition. Then it should be stated and checked separately, or the biconditional A <-> B should be used, rather than assumed from the original.

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.