An argument is valid if the conclusion must be true whenever the premises are true. To test it, try to build a case where the premises hold and the conclusion fails.
This is the last lesson in the logical reasoning chapter. It uses statements, negation and implications, so revisit writing implications and their converse if those feel shaky.
How do I test whether an argument is valid?
Assume both premises are true. Then ask whether you can imagine a situation where the conclusion is false. If you can, that situation is a counterexample and the argument is invalid.
Three forms come up most often, and each is valid when written correctly.
- All A are B. C is A. Therefore C is B.
- If p, then q. p is true. Therefore q is true.
- If p, then q. q is false. Therefore p is false.
Worked example: two arguments, one valid and one not
Argument 1. Premise 1: All multiples of 4 are even. Premise 2: 28 is a multiple of 4. Conclusion: 28 is even.
Try to break it. For the conclusion to be false, 28 would need to be odd.
But premise 1 says any multiple of 4 is even, and premise 2 says 28 is one. No counterexample exists, so the argument is valid. (Check: 28 = 4 × 7, which is even.)
Argument 2. Premise 1: All multiples of 4 are even. Premise 2: 18 is even. Conclusion: 18 is a multiple of 4.
Try to break it. Both premises are true. The conclusion is false, because 18 ÷ 4 = 4.5. So 18 is itself a counterexample, and the argument is invalid.
The two arguments use the same words, yet only one is valid. The direction of the reasoning decides which: Argument 1 goes from “multiple of 4” to “even”, and Argument 2 tries to go back.
The mistake that costs marks
The common slip is to accept an argument because the conclusion is true. Consider: “All multiples of 4 are even. 12 is even. Therefore 12 is a multiple of 4.”
The conclusion is true, since 12 = 4 × 3. The argument is still invalid.
| Step | Wrong | Right |
|---|---|---|
| Check the conclusion | 12 is a multiple of 4, so valid | true here, but that is not the test |
| Look for a counterexample | not attempted | replace 12 with 10 |
| Result with 10 | not seen | premises true, conclusion false |
| Decision | valid | invalid |
Replacing 12 with 10 keeps the pattern and breaks the conclusion. A valid argument cannot survive that swap.
Check yourself
Decide whether this argument is valid. Premise 1: If a number is divisible by 6, then it is even. Premise 2: 20 is even. Conclusion: 20 is divisible by 6.
Answer
The argument is invalid. Both premises are true, since 20 is even and every number divisible by 6 is even. But 20 ÷ 6 is not a whole number, so the conclusion is false.
The mistake in the argument is using “q is true” to conclude “p is true”. A valid form needs either “p is true” or “q is false”.
What to study next
Apply the test to a claim from everyday life with proving or disproving an everyday mathematical claim. Then try the logical reasoning practice set.
The claim evidence revision desk gives you a structure for checking a claim against evidence. For lessons with a teacher, see online one-to-one Mathematics tuition.