Converse, Inverse, Contrapositive — The Rotation Wheel That Shows Every Swap

"If P then Q" has three cousins: the converse (swap), the inverse (negate), and the contrapositive (swap and negate). The names sound interchangeable. They are not. One of the three is logically equivalent to the original; the other two are strangers that happen to share some syntax.

The wheel below lets you see what each operation actually does. Rotate it and every quadrant lights up to show which swap or negation produced it — and which one brings you home to an equivalent statement.

The four-form wheel

rotation:

Rotate the pointer through four positions: original ($P \Rightarrow Q$), converse ($Q \Rightarrow P$), inverse ($\lnot P \Rightarrow \lnot Q$), and contrapositive ($\lnot Q \Rightarrow \lnot P$). Only the two green-bordered quadrants — top-left and bottom-right — represent statements equivalent to the original. The other two are different theorems entirely.

What each rotation does

Reading the four positions in order makes the three operations — swap, negate, and swap-and-negate — stare you in the face.

Position 1 (top-left, P \Rightarrow Q). The starting point. The statement you were given to prove.
Position 2 (top-right, Q \Rightarrow P). Swap only. Hypothesis and conclusion trade places. No negations. This is the converse — a different claim.
Position 3 (bottom-left, \lnot P \Rightarrow \lnot Q). Negate only. The hypothesis becomes \lnot P and the conclusion becomes \lnot Q, but they stay in the same positions. This is the inverse — also a different claim.
Position 4 (bottom-right, \lnot Q \Rightarrow \lnot P). Swap and negate. Conclusion moves to the front, hypothesis to the back, and both get negated. This is the contrapositive — logically equivalent to the original.

Why swap-alone and negate-alone fail but the combination works: a truth table has four rows. The original P \Rightarrow Q produces the pattern T, F, T, T. Swap alone (converse) produces T, T, F, T. Negate alone (inverse) also produces T, T, F, T. Both single operations shift the "false" row from (T,F) to (F,T) — the wrong position. But swap-and-negate moves it twice, once to (F,T) and then back to (T,F). Two wrongs do make a right, in this one narrow logical sense.

The diagonal pairs

Notice the green highlight on the diagonal — top-left and bottom-right. The wheel encodes a deeper fact: opposite quadrants are logically equivalent.

The diagonal from "Original" to "Contrapositive" holds one pair of equivalent forms.
The diagonal from "Converse" to "Inverse" holds the other pair — the converse and the inverse are equivalent to each other, but not to the original.

So the four forms collapse into two truth-equivalence classes. The original and its contrapositive are one class. The converse and its inverse are the other. Proving anything in one class proves everything in that class. Proving anything in one class says nothing about whether the other class is true or false.

How the wheel helps you avoid the classic swap error

The most common mistake in proof writing is to set out to prove a statement by contrapositive and accidentally prove the converse instead (see Contrapositive vs Converse — The Swap That Secretly Changes the Theorem). The wheel makes the error diagnosable: if your proof moved from position 1 to position 2 (swap only, no negation), you wrote a converse proof.

Before writing a proof by contrapositive, visualise the wheel:

Start at position 1.
Rotate diagonally across to position 4. Not to position 2. Not to position 3.
Both operations — swap and negate — must happen.

If you catch yourself negating without swapping (inverse) or swapping without negating (converse), you are at the wrong quadrant. Only the diagonal partner is a safe substitute for the original.

Quick check on a real example

Original: "If n = 6, then n is even." (True.)

Converse (swap): "If n is even, then n = 6." False — n = 4 breaks it.
Inverse (negate): "If n \neq 6, then n is odd." False — n = 4 also breaks it.
Contrapositive (swap and negate): "If n is odd, then n \neq 6." True — 6 is even, so no odd number equals 6.

The diagonal holds. The off-diagonal forms are different theorems, and both happen to be false here. Proving the original does not prove them; proving them would not prove the original.