Part 5: The Converse and the Contrapositive
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 (\(p\))."
Does the Converse Have the Same Truth?
The truth of "\(p \to q\)" does not guarantee the truth of "\(q \to p\)." They are separate statements.
Goat vs. Animal
Original (\(p \to q\)): "If it's a goat, then it's an animal." This is true (all goats are animals).
Converse (\(q \to p\)): "If it's an animal, then it's a goat." This is false — there are many animals that are not goats (dogs, cats, etc.).
Key Point:
Knowing that \(p \to q\) is true tells us nothing for sure about \(q \to p\). The converse might be true or false independently.
Truth Table Perspective
Let's compare \(p \to q\) and \(q \to p\) side by side:
| \(p\) | \(q\) | \(p \to q\) | \(q \to p\) |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | T | F |
| F | F | T | T |
Notice the middle two lines:
- Line 2: \(p \to q\) is false (T → F is false), but \(q \to p\) is true (F → T is true).
- Line 3: \(p \to q\) is true (F → T is true), but \(q \to p\) is false (T → F is false).
These mismatches show that \(p \to q\) and \(q \to p\) can differ in truth value.
The Contrapositive: \(\neg q \to \neg p\)
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.
Box of Balls Revisited
Original: "If the box has a blue ball (\(p\)), then it has a red ball (\(q\))."
Contrapositive: "If the box does not have a red ball (\(\neg q\)), then it does not have a blue ball (\(\neg p\))."
If the original statement about the box is true, its contrapositive is also true (and vice versa). They stand or fall together.
Why Are They Equivalent?
Intuitive Explanation
If "\(p \to q\)" fails, it means \(p\) happened but \(q\) did not (that's the only failure case). But in that exact scenario, "\(\neg q \to \neg p\)" also fails (because \(\neg q\) is true but \(\neg p\) is false). Whenever one fails, the other fails in the same case. In all other cases, both hold true.
Truth Table Confirmation
| \(p\) | \(q\) | \(p \to q\) | \(\neg q\) | \(\neg p\) | \(\neg q \to \neg p\) |
|---|---|---|---|---|---|
| T | T | T | F | F | T |
| T | F | F | T | F | F |
| F | T | T | F | T | T |
| F | F | T | T | T | T |
Compare the columns for \(p \to q\) and \(\neg q \to \neg p\) — they are identical in every row (T, F, T, T). That means \(p \to q\) and \(\neg q \to \neg p\) are logically equivalent statements.
Equivalence in Practice
In proofs (and many real-life arguments), it's often useful to prove \(p \to q\) by proving \(\neg q \to \neg p\) instead. Sometimes it's easier to assume the outcome doesn't happen (assume \(q\) is false) and show the condition couldn't have happened (\(p\) must be false).
The Inverse: \(\neg p \to \neg q\)
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.
Identify Converse and Contrapositive
5.1. For the statement "If you are wearing a red hat, then the people next to you are smiling," write the converse and the contrapositive.
5.2. Given "If a number is even, then it is divisible by 2," what are the converse and contrapositive? Which of those are true statements in general mathematics?
5.3. Consider the statement "If it is raining, then the ground is wet." Write its converse, inverse, and contrapositive in English sentences.
Truth Check
Determine whether each of the following is true or false (in general). Assume a universe where the statements make sense (e.g., all objects, all people, etc.), and give a brief explanation:
5.4. If the original statement "\(p \to q\)" is true, is the converse "\(q \to p\)" always true? (Provide a counterexample if not.)
5.5. If "\(p \to q\)" is true, is the contrapositive "\(\neg q \to \neg p\)" always true?
5.6. If "\(p \to q\)" is false, what can you say about the contrapositive "\(\neg q \to \neg p\)"? What about the converse "\(q \to p\)"?
Application
5.7. Debugging logic: A student writes, "If a figure is a square, then it has four sides. Therefore if a figure has four sides, it must be a square." Identify the student's error in terms of converse/contrapositive/inverse.
5.8. Think of a real-world conditional statement (not used in the text). State its converse and contrapositive, and comment on their truth in that real context.
Next: In Part 6: The Biconditional, we'll explore "if and only if" statements, which effectively combine an implication and its converse when both are true.