Implication vs biconditional

Q: Is the biconditional just equality?

On two Boolean values, yes: A B is TRUE when they are equal, which is why == is rejected by the parser with a repair to . As a claim about two formulas, “equivalent” means F G is a tautology.

Q: Which is the contrapositive of the biconditional?

NOT A NOT B, which is equivalent to A B itself, since agreeing values still agree when both are negated. For the implication, the contrapositive NOT B -> NOT A is equivalent and the converse is not.

A -> B is FALSE on one row, A TRUE and B FALSE. A <-> B is FALSE on that row and on A FALSE, B TRUE as well. The biconditional is the implication plus its converse, and the one extra FALSE row is where “if” and “if and only if” part.

The engine's comparison reports the two not equivalent with the witness A = FALSE, B = TRUE, and marks that single row: the implication holds there (A never happened) and the biconditional fails (A and B differ). The implication is TRUE on rows 0, 1 and 3, Σm(0, 1, 3), with the single maxterm ΠM(2); its CNF is NOT A OR B, one clause, which is the implication elimination law, and two NAND gates build it.

The biconditional adds the converse B -> A, so it is (A -> B) AND (B -> A), TRUE exactly when A and B agree, the same function as the XNOR gate; it eliminates to (A AND B) OR (NOT A AND NOT B). Where the implication says B is necessary for A, the biconditional says B is necessary and sufficient. The necessary-and-sufficient lesson drills the words; this page is the two tables side by side.

In rules the difference is enforcement. “The discount applies if the card is valid” forbids a valid card without a discount and says nothing about a discount without a valid card; “if and only if” forbids both. Requirements that mean the second are often written as the first, and the extra FALSE row is the test that tells them apart.

Example

implication := A -> B · outputbiconditional := A <-> 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 biconditional equivalent? No Counterexample: A = F, B = T · Exhaustively checked and solver cross-checked · 2 rows

Truth table (4 rows)

Implication vs biconditional
#	A	B	implication	biconditional
0	FALSE	FALSE	TRUE	TRUE
1	FALSE	TRUE	TRUE	FALSE
2	TRUE	FALSE	FALSE	FALSE
3	TRUE	TRUE	TRUE	TRUE

Logic diagram

Karnaugh map

A \ B

1m0

1m1

0m2

1m3

cells m0, m1NOT A
cells m1, m3B

NOT A OR B · Derived NOT A OR B. Exactly equivalent on every assignment.

Minterms and maxterms

implication(A, B) = Σm(0, 1, 3)implication(A, B) = ΠM(2)

Variable order: A, B · 4 rows

Normal forms

Conjunctive normal form (CNF)

NOT A OR B

1 clauses, 2 literals · complete · Implication elimination ×1

Disjunctive normal form (DNF)

NOT A OR B

2 terms, 2 literals · complete · Implication elimination ×1

NAND-only and NOR-only

NAND-only form

NAND(A, NAND(B, B))

2 gates · verified equivalent on every row

NOR-only form

NOR(NOR(NOR(A, A), B), NOR(NOR(A, A), B))

5 gates · verified equivalent on every row

The common slip

Writing <-> to mean “implies”, or -> to mean “exactly when”. Each is a one-row mistake, and it is always the same row: A FALSE, B TRUE, where the implication is content and the biconditional is not.

Questions

Is the biconditional just equality?

On two Boolean values, yes: A <-> B is TRUE when they are equal, which is why == is rejected by the parser with a repair to <->. As a claim about two formulas, “equivalent” means F <-> G is a tautology.

Which is the contrapositive of the biconditional?

NOT A <-> NOT B, which is equivalent to A <-> B itself, since agreeing values still agree when both are negated. For the implication, the contrapositive NOT B -> NOT A is equivalent and the converse is not.

Compare

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.