Converse
The converse of "If P, then Q" is "If Q, then P," which swaps the hypothesis and conclusion and is not always true when the original is true.
Formula
\text{Converse of } P \to Q \text{ is } Q \to P
Definition
The converse of a conditional statement flips the "if" and "then" parts around. Formally, the converse of $P \to Q$ is $Q \to P$; a statement and its converse are logically independent, so one can be true while the other is false, both can be true, or both false, and when both are true we write $P \leftrightarrow Q$ (biconditional). The converse $Q \to P$ has the same truth value as the inverse $\neg P \to \neg Q$ (they are logically equivalent to each other), while the contrapositive $\neg Q \to \neg P$ is logically equivalent to the original $P \to Q$; these four forms and their equivalences are fundamental to propositional logic.
Example
"If it is raining, then the ground is wet" has converse "If the ground is wet, then it is raining," which can be false since the ground might be wet from a sprinkler. "If a number is divisible by $4$, then it is divisible by $2$" is true, but its converse, "If divisible by $2$, then divisible by $4$," is false ($6$ is divisible by $2$ but not $4$). "If $f$ is differentiable, then $f$ is continuous" is true, but its converse, "If $f$ is continuous, then $f$ is differentiable," is false, since $f(x) = |x|$ is continuous but not differentiable at $0$.
Key Insight
The converse sounds related to the original but is logically a completely different statement; do not assume the converse is true just because the original is. Mistakenly assuming the converse is true is called "affirming the consequent," one of the most common logical fallacies in everyday reasoning. Characterization theorems ("if and only if" theorems) prove both a statement and its converse simultaneously, giving the deepest possible understanding of a mathematical concept.