Questions: Constructing Counterexamples to Test Arguments

Question 1

Consider: 'All great scientists are creative thinkers. Einstein was a creative thinker. Therefore, Einstein was a great scientist.' The conclusion happens to be true. Is this argument valid?

• A. Yes — the conclusion is true and both premises are plausible, so the argument is sound
• B. No — a true conclusion is not sufficient for validity; the logical form (All A are B; x is B; therefore x is A) permits counterexamples where the conclusion is false
• C. Yes — if premises and conclusion are all true, the argument is valid by definition
• D. Cannot be determined without knowing whether the premises are necessarily true

Answer: (B)

Validity requires that it be impossible for the premises to be true and the conclusion false — not merely that they happen to all be true. The form 'All A are B; x is B; therefore x is A' is invalid. Counterexample: 'All dogs are animals. Cats are animals. Therefore, cats are dogs.' Premises true, conclusion false. The Einstein argument fails for exactly the same structural reason even though its conclusion is coincidentally correct.

Question 2

When constructing a counterexample to test an argument's validity, what must you preserve and what must you change?

• A. Preserve the specific names and events; change the premises to be false
• B. Preserve the logical form (the pattern of relationships); change the specific content to produce true premises and a false conclusion
• C. Preserve the conclusion's truth value; change the premises to different claims
• D. Preserve the argument's subject domain; change only the specific claims within it

Answer: (B)

A counterexample targets the logical form — the abstract inferential pattern — not the subject matter. Strip away the content, identify the structure (e.g., All A are B; x is B; therefore x is A), then substitute different content that makes each premise true while making the conclusion false. This shows the inferential move is unreliable regardless of what the argument is about. Changing only the subject domain while keeping the same form does nothing to test validity.

Question 3

An argument with a true conclusion can still be logically invalid.

• T. True
• F. False

Answer: True

Validity is about the logical relationship between premises and conclusion — whether true premises guarantee a true conclusion. If the argument's logical form permits cases where premises are true and conclusion is false (even if not in this specific instance), the argument is invalid. The conclusion being true by coincidence provides no logical support for the inference.

Question 4

To refute an argument by counterexample, it is sufficient to find any case in which the conclusion is false, regardless of the premises.

• T. True
• F. False

Answer: False

A counterexample to an argument must show the premises being true while the conclusion is false — with the same logical structure but different content. Simply finding a false conclusion in a different context doesn't address the inferential move the argument makes. You must construct a case that uses the same logical pattern but demonstrates the pattern allows false conclusions. Targeting the form is essential; targeting only the claim is not enough.

Question 5

Why does finding a counterexample to an argument's logical form also refute the original argument, even when the original argument's conclusion is true?

Model answer: Validity requires that the premises make it impossible for the conclusion to be false — not merely that they happen to be followed by a true conclusion. If you can find a case with the same logical form where premises are true and the conclusion is false, you have shown the inference pattern doesn't guarantee its conclusion. The original argument relies on the same unreliable pattern, so its conclusion isn't proved by the premises. The premises may all be true and the conclusion may be true, but there is no logical connection between them that the form secures.

This is why counterexample construction tests the argument rather than just the conclusion — validity is a property of the inferential move, not the outcome. A conclusion can be true for reasons completely unrelated to the premises offered, and the counterexample exposes that independence.