AI Tutor
If it is raining outside, then the sidewalk is wet. Does finding a bone-dry sidewalk guarantee that it is not raining?
Yes, absolutely. Flipping the order and negating both conditions gives us the contrapositive, one of the cleanest and most reliable proof strategies in mathematics.

Conditional and Contrapositive Statements

A conditional statement is an "if-then" proposition written as P  ⟹  QP \implies Q, where PP is the hypothesis (the condition you assume) and QQ is the conclusion (the result that follows).
Its contrapositive is formed by reversing the order and negating both parts, written as ¬Q  ⟹  ¬P\neg Q \implies \neg P, which reads: "If not QQ, then not PP."
In the 3rd century BCE, the Stoic logician Chrysippus analyzed these exact inference patterns to formalize the foundational rules of deductive reasoning.
Why are a statement and its contrapositive guaranteed to share the exact same truth value in every scenario?

Logical Equivalence via Truth Values

Two statements are logically equivalent (written with ≡\equiv) when they produce the exact same truth output for every combination of truth values of their components.
A conditional statement P  ⟹  QP \implies Q is only false when a true hypothesis leads to a false conclusion; otherwise, it is always true.
What happens if you only swap the statements or only negate them without doing both?