Converse, Inverse, and Contrapositive
Three statements built from one conditional, and only one of them says the same thing. Getting this wrong is the most expensive two seconds on a logic test — and it is entirely mechanical to get right.
The short answer
Start with a conditional p → q. Three related statements come from it:
| Name | Form | What changed |
|---|---|---|
| Original | p → q | — |
| Converse | q → p | Swapped |
| Inverse | ¬p → ¬q | Negated |
| Contrapositive | ¬q → ¬p | Negated and swapped |
That second sentence is not a bonus fact. It falls straight out of the first: the inverse ¬p → ¬q is itself the contrapositive of the converse q → p. Every equivalence on this page is really the same equivalence used twice.
Building each one without slipping
Do it in two separate passes rather than one. Trying to negate and swap in a single motion is how the inverse and the contrapositive get confused.
- Write down p and q separately before touching anything. Name them out loud.
- Swap only → that is the converse.
- Negate only → that is the inverse.
- Do both → that is the contrapositive.
Take the statement: "If n is an even integer greater than 2, then n is the sum of two primes."
Here p is "n is an even integer greater than 2" and q is "n is the sum of two primes."
- Converse: If n is the sum of two primes, then n is an even integer greater than 2.
- Inverse: If n is not an even integer greater than 2, then n is not the sum of two primes.
- Contrapositive: If n is not the sum of two primes, then n is not an even integer greater than 2.
Notice that the converse here is plainly false — 5 = 2 + 3 is a sum of two primes and is neither even nor greater than 2 in the required sense — while the original is a famous open conjecture. One is false, the other is not known to be false. They cannot be the same statement. The contrapositive, meanwhile, is exactly as open as the original, because it is the original.
The truth table that settles it
Four rows, and every claim on this page is visible in them.
| p | q | p → q | q → p | ¬p → ¬q | ¬q → ¬p |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | T | F |
| F | T | T | F | F | T |
| F | F | T | T | T | T |
Read the columns, not the rows. Column 3 (p → q) and column 6 (¬q → ¬p) are identical — that is the contrapositive equivalence. Column 4 (converse) and column 5 (inverse) are identical to each other, and different from the first pair. The two disagreements both live in the middle rows, exactly where p and q have opposite truth values.
This is also why the equivalence is worth memorizing rather than re-deriving: a conditional is false in one row only, and negating-and-swapping moves that row onto itself.
Why the contrapositive is the one that earns its keep
The equivalence is not trivia. It is the license for proof by contraposition: to prove p → q, you may instead assume ¬q and derive ¬p. You are allowed to do that precisely because the two statements are the same statement.
It pays off whenever the negation is the more concrete thing to hold. Consider: if n² is even, then n is even. Starting from "n² is even" gives you n² = 2k, and extracting anything about n from that requires unique factorization. Starting from the contrapositive — assume n is odd — gives you n = 2m + 1 immediately, and then n² = 4m² + 4m + 1 = 2(2m² + 2m) + 1 is odd. Three lines, no machinery.
The vocabulary that causes the swap
Most converse errors do not happen in symbols. They happen in English, one step earlier, when the sentence gets translated. These four phrasings all mean p → q:
| English | Symbolic | Why |
|---|---|---|
| If p, then q | p → q | The base case. |
| p only if q | p → q | The "only if" clause is the conclusion. |
| q whenever p | p → q | Every time p holds, q holds. |
| q is necessary for p | p → q | A necessary condition is the consequent. |
And these two mean the reverse, q → p:
| English | Symbolic | Why |
|---|---|---|
| p if q | q → p | Bare "if" introduces the hypothesis. |
| q is sufficient for p | q → p | A sufficient condition is the hypothesis. |
The pair to burn in is necessary versus sufficient. Sufficient goes in front of the arrow; necessary goes behind it. "Only if" behaves like "necessary", which is why it points the opposite way from plain "if" — a genuinely counterintuitive fact about English, not about logic.
"If and only if" is neither: it asserts both directions at once and is the biconditional p ↔ q. It is strictly stronger than p → q, so a sentence phrased that way is not a correct symbolization of a one-way conditional.
Where these go wrong
- Assuming the converse. The headline error, and it survives outside the classroom — "all fraud is unusual, so anything unusual is fraud" is exactly this mistake wearing a suit. Proving q → p when you were asked for p → q earns no credit even when the work is flawless.
- Negating without swapping. That produces the inverse, not the contrapositive. If your two statements are in the same order, you did not build the contrapositive.
- Negating only one half. ¬q → p and q → ¬p are neither of the three named forms. Both parts change or neither does.
- Negating a compound hypothesis carelessly. The negation of "n is even and n > 2" is "n is odd or n ≤ 2" — De Morgan flips the connective too. Leaving it as "and" is a silent error that makes the rest of the proof unfixable.
- Reading "only if" as "if". It is the reverse. When a sentence uses "only if", "necessary", or "sufficient", stop and translate deliberately instead of by feel.
- Forgetting that a false hypothesis makes the whole conditional true. Rows 3 and 4 of the table are both T. A statement about all elements of an empty set is vacuously true, and that is not a technicality — it decides real exam questions.
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Drill logical equivalences →Frequently asked questions
Is a conditional equivalent to its converse?
No. The conditional p → q and its converse q → p differ in truth value whenever exactly one of p and q is true. A conditional is equivalent only to its contrapositive. Assuming the converse follows from the original is the single most common error in propositional logic.
Why is the contrapositive equivalent to the original conditional?
Both statements are false in exactly one situation and true in the other three. The conditional p → q is false only when p is true and q is false. The contrapositive ¬q → ¬p is false only when ¬q is true and ¬p is false — which is the same situation: q false and p true. Same row false, same rows true, so they are logically equivalent.
What is the difference between the inverse and the contrapositive?
The inverse negates both parts but keeps the order: ¬p → ¬q. The contrapositive negates both parts and reverses the order: ¬q → ¬p. Only the reversal makes it equivalent to the original. The inverse is equivalent to the converse instead, since each is the contrapositive of the other.
Does "p only if q" mean p → q or q → p?
It means p → q. This surprises people because "if" alone points the other way: "p if q" is q → p. In "p only if q" the clause after "only if" is the conclusion, and it states a necessary condition for p. A necessary condition is always the consequent; a sufficient condition is always the hypothesis.