Why does “if A, then B” not mean “if B, then A”?
Quick answer
“If A, then B” says that whenever A is true, B is true. It does not say that A is the only way for B to be true. The reversed statement, “if B, then A,” is called the converse. To disprove it, find one case where B is true but A is false. That counterexample rejects the converse without rejecting the original conditional. [S1]
The question, the details, the next step.
I understand “if A, then B,” but why can’t I just flip it around and say “if B, then A”?
Because the original statement makes a claim only about cases where A is true: in those cases, B must also be true. It does not say that every case with B must have A. The flipped statement is called the converse, and it is a separate claim—not a consequence of the original one. [S1]
For a simple example, let A mean “the number is divisible by 4” and B mean “the number is even.” The original claim is: “If a number is divisible by 4, then it is even.” The converse is: “If a number is even, then it is divisible by 4.” Those are not the same claim.
How can I show that the reversed statement is false without checking every possible number?
Find one counterexample: a case where B is true and A is false. The number 6 is even, so B is true; it is not divisible by 4, so A is false. That single case disproves the converse, “if a number is even, then it is divisible by 4.” A single case with B true and A false is enough to refute the converse. [S1]
It does not disprove the original conditional. Since 6 is not divisible by 4, it is not a case where A is true. The original claim concerns what happens whenever A is true, and this example does not challenge that.
Can you show me a paired-case exercise so I can check both statements separately?
Use two cases, each matched to the claim you are testing:
Case 1 — Check the original: Choose a number divisible by 4, such as 8. Is B true—that is, is it even? Yes. This case fits the original conditional: A is true and B is true.
Case 2 — Check the converse: Choose an even number that is not divisible by 4, such as 6. Here B is true and A is false, so this case breaks the converse.
The general pattern is: to challenge “if A, then B,” look for A true and B false. To challenge “if B, then A,” look for B true and A false. [S1]
What should I write when I explain my answer in a homework problem?
Name the two claims separately, then state what your counterexample shows. For this example, you could write:
“Let A mean ‘the number is divisible by 4’ and B mean ‘the number is even.’ The original claim is ‘if A, then B’; its converse is ‘if B, then A.’ The number 6 is even but not divisible by 4, so B is true and A is false. Therefore, 6 is a counterexample to the converse. It does not refute the original conditional.”
That explanation identifies the case and connects it to the claim it tests. A counterexample shows that a particular claim fails; it does not, by itself, prove a different claim true. [S1]
A practical checklist
Write down what A and B mean in plain language.
Write the original conditional and its converse as separate claims.
To test the converse, look for B true and A false.
Check that your proposed example really makes B true and A false.
State which claim your counterexample challenges.
A few more useful answers
Does one counterexample prove the original conditional is true?
No. A case with B true and A false refutes the converse, but it does not establish that every A case has B. It settles only the claim the counterexample tests. [S1]
What kind of example should I look for when testing the original “if A, then B”?
Look for a case where A is true but B is false. That would contradict the original conditional. A case where A is false does not test its claim about what happens when A is true.
What does “if B, then A” claim that the original does not?
It claims that whenever B is true, A is also true. The original says that A guarantees B; it does not say that A is the only way for B to be true. [S1]
Sources
References used for this guide.
- [S1] 5.2 Logical Statements — Introduction to Philosophy openstax.org · Checked