Evaluating (p → q) ↔ (q → p)

Intro to Proofs · Axiom Academy

EXAMPLE Evaluating and Its Converse Build a truth table to test whether a conditional and its converse are logically equivalent. The converse of the conditional is the statement with the hypothesis and conclusion swapped. Using a truth table over every combination of p and q , decide whether and its converse are logically equivalent — that is, whether they share the same truth value in every row. Nice work! You built a truth table and used it to settle a question about logical equivalence. Here's what to carry forward: The converse swaps roles: the converse of is — hypothesis and conclusion are exchanged. Equivalence = matching columns: two statements are logically equivalent only when their truth values agree in every row of the table. Result: and differ in 2 of the 4 rows, so they are not logically equivalent. One counterexample is enough: a single disagreeing row (here p = T, q = F) is all it takes to prove two statements are not equivalent. Don't confuse it with the contrapositive: the contrapositive is equivalent to , but the converse is not. Affirming the converse — concluding from — is a classic logical fallacy. Truth tables are how you catch it.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.