Validity is about the reasoning, and truth is about the facts. A conclusion can be true even when the argument for it is invalid.
This lesson is part of proving or disproving an everyday mathematical claim. It builds on testing deductive arguments with counterexamples.
What is the difference between valid and true?
An argument is valid if the conclusion must be true when the premises are true. A conclusion is true if it matches the facts. The two ideas are independent, so four combinations are possible.
| True conclusion | False conclusion | |
|---|---|---|
| Valid argument | possible, when the premises are true | possible, when a premise is false |
| Invalid argument | possible, by luck | possible |
The top-right cell is the surprising one. Perfect reasoning from a false premise gives a false conclusion.
Worked example: two routes to the same answer
Question: is 12 a multiple of 6? The answer is yes, because 12 = 6 × 2. Here are two arguments that reach it.
Argument A. All numbers of the form 6k are multiples of 6. 12 = 6 × 2, which has the form 6k. So 12 is a multiple of 6.
This is valid, and the premises are true.
Argument B. All multiples of 6 are even. 12 is even.
So 12 is a multiple of 6. The conclusion is true, but the argument is invalid.
To see why B fails, swap 12 for 8. The premises stay true, since 8 is even, yet 8 is not a multiple of 6. The same reasoning produces a false conclusion, so it cannot be trusted for 12 either.
The example shows two arguments with the same conclusion and opposite quality. Only A would earn full marks.
A valid argument with a false premise
Consider: “All even numbers are divisible by 4. 6 is even. So 6 is divisible by 4.” The form is the same as argument A, so the argument is valid.
Premise 1 is false, because 6 is even and 6 ÷ 4 = 1.5. The conclusion is false as well. Valid reasoning does not repair a false premise.
The mistake that costs marks
The common slip is to treat a correct final answer as proof that the working is correct.
| Step | Wrong | Right |
|---|---|---|
| Final answer | 12 is a multiple of 6, correct | correct |
| Reasoning | it is even, so it must be | it equals 6 × 2 |
| Test with 8 | not tried | premises true, conclusion false |
| Verdict | full marks | reasoning invalid |
Always swap in a second value to see if the reasoning still holds.
Check yourself
“All multiples of 10 end in 0. 30 ends in 0. So 30 is a multiple of 10.” Is this argument valid? Give a counterexample to the pattern of reasoning.
Answer
The argument is invalid. It reasons from “ends in 0” back to “multiple of 10”, which the first premise does not support.
Keep the same pattern and change the property: “All multiples of 10 are divisible by 5. 25 is divisible by 5. So 25 is a multiple of 10.” Both premises are true and the conclusion is false, so the pattern is unreliable even though the conclusion about 30 happens to be true.
What to study next
Go on to building an implication from a pair of set relationships. Then test the pattern in the integrated practice set.
The claim evidence revision desk gives a way to check a claim against its reasons. For a teacher to read your working with you, see online one-to-one Mathematics tuition.