Why Converses and Contrapositives Matter

Now that we understand if-then statements, let's explore two important variations:

• The Converse: Reversing the hypothesis and conclusion.
• The Contrapositive: Both reversing and negating the hypothesis and conclusion.

The Converse: $q\to p$ (instead of $p\to q$)

Given an original statement:

the converse is:

In words: - Original: "If $p$, then $q$."
- Converse: "If $q$, then $p$."

Example:
Original: "If it's a goat ($p$), then it's an animal ($q$)."
Converse: "If it's an animal ($q$), then it's a goat ($p$)."

The Contrapositive: $\mathrm{\neg }q\to \mathrm{\neg }p$

The contrapositive of:

is:

In words, we take the original if-then ("If $p$, then $q$"), and form "If not $q$, then not $p$."

Surprisingly, a statement and its contrapositive are logically equivalent — they are always true or false together.

If the original statement about the box is true, its contrapositive is also true (and vice versa). They stand or fall together.

The Inverse: $\mathrm{\neg }p\to \mathrm{\neg }q$

For completeness, consider the inverse of the original statement $p\to q$, which is:

The inverse says: "If not $p$, then not $q$." The inverse is actually the converse of the contrapositive (or equivalently, the negation of both parts of the converse). The truth of the inverse is not guaranteed by the original statement (much like the converse).

Example: Original: "If it's a goat, then it's an animal." (True)
Inverse: "If it's not a goat, then it's not an animal." (False, since not-goat could be a cow, which is still an animal.)

In general, a statement and its inverse are not equivalent. (In fact, the inverse has the same truth value as the converse in all cases.)

Common Misconceptions

• "The converse has the same meaning as the original."
  Reality: They can differ drastically. $p\to q$ and $q\to p$ are usually not equivalent (unless $p$ and $q$ happen to be logically equivalent statements themselves).

• "If $p\to q$ is true, then $q\to p$ must be true."
  Reality: False. The goat/animal example shows $p\to q$ can be true while $q\to p$ is false.

• "The contrapositive is basically the same as the converse."
  Reality: Not at all. The contrapositive of $p\to q$ is logically equivalent to the original statement, whereas the converse is generally not. It's a common error to mix these up.

Exercises

Below are exercises to practice identifying converses and contrapositives, and to test your understanding of their truth values.