Converse
Calculus & Advanced MathThe 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.
Example
Original: "If it is raining, then the ground is wet." Converse: "If the ground is wet, then it is raining." The converse can be false: the ground might be wet from a sprinkler.
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.
Definition
The converse of $P \to Q$ is $Q \to P$. A statement and its converse are logically independent: one can be true while the other is false, both can be true, or both false. When both are true, we write $P \leftrightarrow Q$ (biconditional).
Example
"If a number is divisible by $4$, then it is divisible by $2$." True. Converse: "If divisible by $2$, then divisible by $4$." False ($6$ is divisible by $2$ but not $4$).
Key Insight
Mistakenly assuming the converse is true is called "affirming the consequent," one of the most common logical fallacies in everyday reasoning.
Definition
The converse $Q \to P$ has the same truth value as the inverse $\neg P \to \neg Q$ (both are logically equivalent to each other). The contrapositive $\neg Q \to \neg P$ is logically equivalent to the original $P \to Q$. These four forms (original, converse, inverse, contrapositive) and their equivalences are fundamental to propositional logic.
Example
"If $f$ is differentiable, then $f$ is continuous" is true. Converse: "If $f$ is continuous, then $f$ is differentiable" is false ($f(x) = |x|$ is continuous but not differentiable at $0$).
Key Insight
Characterization theorems (also called "if and only if" theorems) prove both a statement and its converse simultaneously, establishing the equivalence of two conditions and giving the deepest possible understanding of a mathematical concept.