Conditional Statements and Counterexamples

A conditional statement has the form if P then Q. It only promises that P leads to Q, not the reverse. 'If a shape is a square, then it is a rectangle' is true because every square happens to satisfy the rectangle definition. But flip it around and you get the converse, 'if a shape is a rectangle, then it is a square,' which fails instantly: a 2 cm by 5 cm rectangle is a rectangle but not a square. The arrow in the original statement only runs one direction, so truth going forward tells you nothing about truth going backward.

Every conditional has a contrapositive, formed by swapping and negating both parts: 'if not Q, then not P.' The contrapositive is logically tied to the original statement, so if the original is true, the contrapositive is automatically true too. The converse and the inverse get no such guarantee. That asymmetry is why geometry proofs lean so heavily on contrapositives when a direct argument is awkward, and why a true theorem's converse always needs its own separate proof or its own separate counterexample.

This is also why a single counterexample is so powerful: one shape that satisfies P but fails Q destroys the claim completely, no matter how many other shapes satisfy both. This connects directly to the proof and construction work nearby, where showing a statement false often means building one exact figure rather than writing a general argument.