Part 5: The Converse and the Contrapositive
Why Converses and Contrapositives Matter
We're continuing our journey into if-then statements by exploring two ways to modify them:
The Converse: Reversing the hypothesis and conclusion The Contrapositive: Reversing and negating both parts
The Converse: "q → p" Instead of "p → q"
Given an original statement:
the converse is:
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 (\(p\))."
Why the Converse Might Not Have the Same Truth Value
The truth of "If p, then q" does not guarantee that "If q, then p" is also true. They are entirely different statements in logic.
Goat vs. Animal
Original (\(p \to q\)): "If it's a goat, then it's an animal." This is true.
Converse (\(q \to p\)): "If it's an animal, then it's a goat." This is false because there are many animals that are not goats.
Key Fact:
The truth value of the converse cannot be determined just by knowing the truth value of the original if-then statement.
Truth Table Perspective
Let's look at \(p \to q\) 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 lines 2 and 3:
Line 2: \(p \to q\) is false (since \(p=T\), \(q=F\)), but \(q \to p\) is true (since \(q=F\), \(p=T\)).
Line 3: \(p \to q\) is true (\(p=F\), \(q=T\)), but \(q \to p\) is false (\(q=T\), \(p=F\)).
These mismatches illustrate that knowing \(p \to q\) alone tells us nothing certain about \(q \to p\).
The Contrapositive: "¬q → ¬p"
The contrapositive of:
is:
In words, "If \(p\) implies \(q\)," then the contrapositive is "If not \(q\), then not \(p\)."
Surprisingly, \(p \to q\) and its contrapositive \(\neg q \to \neg p\) always have the same truth value.
Box of Colored Balls
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.
Why They Are Equivalent
Intuitive Explanation
If "\(p \to q\)" is false, that means \(p\) was true but \(q\) was false (the only way for an if-then to fail). In that scenario, "\(\neg q\)" is true and "\(\neg p\)" is false, so "\(\neg q \to \neg p\)" is also false. Every time one of them fails, so does the other. Every time one is true, so is the other.
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 |
Look at the columns \(p \to q\) and \(\neg q \to \neg p\): they match in every row. Hence, they are logically equivalent.
Equivalence in Practice
In math proofs (and many real-life arguments), you often prove \(p \to q\) by proving \(\neg q \to \neg p\) instead, because sometimes it's easier to "flip and negate" a statement.
Common Misconceptions
"The converse must have the same truth as the original."
Reality: They can differ drastically.
"If \(p \to q\) is true, then \(q \to p\) must be true."
Reality: This is false. The goat-animal example demonstrates otherwise.
"The contrapositive is the same as the converse."
Reality: They're not the same. The contrapositive is always logically equivalent to the original, but the converse is not.
Exercises
Below are exercises to help you identify converses and contrapositives, and test your understanding of their truth values.
Identify the Converse and Contrapositive
5.1. "If you are wearing a red hat, then the people next to you are wearing blue hats."
5.2. "If the light is red, then the cars will stop."
5.3. "That it is now 9:00 implies that I will be late for class."
5.4. "I will vote for you if you give me 50 dollars."
5.5. "If I am not correct, I will eat my hat!"
5.6. "I won't go with you if you don't show up on time."
For each statement:
* Write its converse: Swap the hypothesis and conclusion.
Write its contrapositive: Swap and* negate both the hypothesis and conclusion.
Determining Truth Values (Birds and Rabbits)
Assume:
* "Birds have feathers" = true
"Birds have teeth" = false
"Rabbits have teeth" = true
"Rabbits have feathers" = false*
Check each statement's truth value. Then decide the truth of its converse and its contrapositive as well.
5.7. "If birds have teeth, then birds have feathers."
5.8. "If rabbits don't have teeth, then birds have teeth."
5.9. "If rabbits have feathers, then birds have feathers."
Determining Truth Values (Jabberwocky Fun)
Assume:
* "The slithy toves did gyre and gimble in the wabe" = true
"All mimsy were the borogroves" = true
"The mome raths were outgrabe" = false
For each statement, decide:
* (a) Is the statement true?
(b) Is its converse true?
(c) Is its contrapositive true?
5.10. "If the slithy toves did gyre and gimble in the wabe, then the mome raths were outgrabe."
5.11. "If the borogroves were all mimsy, then the mome raths were not outgrabe."
5.12. "If the mome raths were outgrabe, then the slithy toves did not gyre and gimble in the wabe."
Existence or Non-Existence of Statements
Find an if-then statement with the given properties or explain why it's impossible:
5.13. The if-then statement is false, and its converse is true.
5.14. The if-then statement and its converse are both true.
5.15. The if-then statement and its converse are both false.
Additional Challenges
5.16. Give an example of a real-world if-then statement whose converse is also true. Why does this happen in your example?
5.17. Prove logically (using truth tables or a step-by-step argument) that a statement and its contrapositive always share the same truth value.
5.18. Create your own example of a statement whose contrapositive is easier to justify or observe than the original statement.
Applications in Real Life
Understanding converses and contrapositives helps us think more clearly in everyday situations:
Legal Reasoning: "If you commit the crime, then you will be punished" does not mean "If you are punished, then you committed the crime." The second statement (the converse) doesn't follow from the first.
Scientific Hypothesis: "If the medicine works, patients will recover" is different from "If patients recover, the medicine works." Patients might recover for other reasons.
Computer Programming: Conditional statements in code frequently use contrapositive logic to handle error cases more elegantly.
Medical Diagnosis: "If you have disease X, then you'll have symptom Y" doesn't mean "If you have symptom Y, then you have disease X." The symptom might be caused by many different conditions.
Final Thoughts
Takeaway:
Changing "\(p \to q\)" to its converse ("\(q \to p\)") can completely change its truth value.
But changing "\(p \to q\)" to its contrapositive ("\(\neg q \to \neg p\)") never changes the truth value.
This distinction matters in mathematical proofs, computer program logic, and even everyday claims. It's essential to recognize when you're simply flipping a statement (and risking losing its truth) versus when you're flipping and negating (which preserves truth).
When evaluating arguments, always be aware of attempts to substitute a statement with its converse, as this is a common logical fallacy. The contrapositive, however, is a valid alternative form of the original statement that can sometimes be easier to understand or prove.