MISSION · Data and probability
If-Then Statements
Why reversing a true statement often makes a false one
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The idea in one sentence
“If \(p\) then \(q\)” makes one promise only — that \(q\) follows whenever \(p\) holds — and three related statements can be built from it, of which just one is guaranteed to share its truth.
The four statements
Start with a conditional: if it is raining, then the ground is wet. Call the parts \(p\) (raining) and \(q\) (ground wet).
| Name | Form | Our example |
|---|---|---|
| Conditional | \(p \rightarrow q\) | If raining, ground wet |
| Converse | \(q \rightarrow p\) | If ground wet, raining |
| Inverse | \(\lnot p \rightarrow \lnot q\) | If not raining, ground not wet |
| Contrapositive | \(\lnot q \rightarrow \lnot p\) | If ground not wet, not raining |
The original is true. The converse is false — a burst pipe wets the ground. The inverse is false for the same reason. The contrapositive is true, and this is the key fact: a conditional and its contrapositive are always either both true or both false. They say the same thing.
The converse and inverse are likewise a matched pair, and their truth is independent of the original’s.
Why the contrapositive works
If \(p\) guarantees \(q\), then finding \(q\) absent proves \(p\) was absent too — because if \(p\) had been there, \(q\) would have followed.
That equivalence is not merely a curiosity. It is a proof technique: proof by contrapositive attacks “if \(n^2\) is even then \(n\) is even” by proving the far easier “if \(n\) is odd then \(n^2\) is odd”. Same statement, much less work.
Confusing the converse
Assuming the converse is one of the most common reasoning errors there is, in and out of mathematics.
All squares are rectangles does not make all rectangles squares. Every dog is a mammal does not make every mammal a dog. A medical test that is positive for everyone with a disease does not mean everyone testing positive has it — which is the base rate problem, wearing logic’s clothes.
Whenever you catch yourself reversing an “if-then”, check whether you have evidence for the reversal or have simply assumed it.
Disproving with one example
To prove a conditional false, you need a counterexample — one case where \(p\) holds and \(q\) doesn’t. Just one, and the statement is finished.
All prime numbers are odd dies at 2. If a shape has four equal sides it is a square dies at the rhombus.
Note the asymmetry: a single counterexample disproves, while no number of supporting examples proves. That asymmetry is why proof exists at all.
Biconditionals
When a statement and its converse are both true, they combine into “if and only if”, written \(p \leftrightarrow q\).
A triangle is equilateral if and only if all its angles are \(60°\). Both directions hold, so either property may be used to establish the other. Definitions are usually biconditionals, which is what makes them usable in both directions.
Common mix-ups
- Assuming the converse follows. It doesn’t.
- Mixing up inverse and contrapositive. The contrapositive negates and swaps.
- Trying to prove a general claim by examples.
- Reading “if” as “if and only if”. A conditional promises one direction only.
Check yourself
For if a number is divisible by 6, then it is divisible by 3, write the converse and say whether each is true.
The converse is if divisible by 3, then divisible by 6. The original is true; the converse is false, and 9 is a counterexample.
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