Questions: Testing Validity with Counterexamples

Question 1

Consider the argument: 'All mammals are warm-blooded. Whales are warm-blooded. Therefore, whales are mammals.' Both premises are true and the conclusion is also true. Does this show the argument is valid?

• A. Yes — validity requires all premises and the conclusion to be true, which they are
• B. No — validity is about the logical structure, not about whether premises and conclusion happen to be true. The same form ('All A are B; C is B; therefore C is A') has counterexamples using other substitutions
• C. No — but the argument becomes valid if we add the premise 'All warm-blooded things are mammals'
• D. Yes — if there is no scenario in which this specific argument has false premises and a true conclusion, it is valid

Answer: (B)

Validity is a structural property, not a truth-value property. An argument is valid if there is no possible scenario where all premises are true and the conclusion is false — not if the premises and conclusion happen to be true in the actual world. This argument has the form 'All A are B; C is B; therefore C is A,' which is invalid: substitute A = dogs, B = animals, C = cats and you get 'All dogs are animals; cats are animals; therefore cats are dogs' — true premises, false conclusion. The same substitution test applies here: the form is invalid even though this particular instance has a true conclusion. Actual-world truth of the conclusion cannot establish validity.

Question 2

You are trying to test whether an argument is valid. After extensive effort, you cannot construct a counterexample — no scenario comes to mind where all premises are true and the conclusion is false. What can you conclude?

• A. The argument is valid — if no counterexample exists, validity is established
• B. The argument is probably valid, but your failure to find a counterexample could reflect insufficient imagination rather than logical necessity
• C. The argument is invalid — the inability to find a counterexample means you have not tried hard enough
• D. No conclusion is possible without checking the argument in every possible world

Answer: (B)

This question targets the asymmetry of the counterexample method. One counterexample definitively proves invalidity, but the inability to find one proves nothing — you may simply lack the imagination or technique to construct it. The counterexample method is a tool for refutation, not for positive demonstration of validity. To definitively establish validity, you need formal methods: truth tables, Venn diagrams, or natural deduction proofs. The correct answer (B) captures the evidential weight of 'no counterexample found' — it raises your credence in validity but does not establish it.

Question 3

A single counterexample — one possible scenario where all premises are true and the conclusion is false — is sufficient to prove that an argument is invalid.

• T. True
• F. False

Answer: True

This follows directly from the definition of validity. An argument is valid if and only if there is NO possible scenario with all premises true and the conclusion false. So if even one such scenario exists, the 'no possible scenario' condition fails — the argument is invalid. Invalidity is an existential claim (there exists a counterexample), so one witness suffices. This is the power of the counterexample method: a single well-constructed scenario demolishes the argument's claim to validity, regardless of how often the argument's conclusion happens to be true.

Question 4

If an argument is valid, then most its premises is expected to be true in the actual world.

• T. True
• F. False

Answer: False

Validity is entirely independent of the actual truth values of the premises. A valid argument could have false premises: 'All cats are reptiles. Fluffy is a cat. Therefore Fluffy is a reptile.' This argument is valid — if both premises were true, the conclusion would have to be true. But the first premise is false. Validity says: IF the premises were true, the conclusion WOULD BE true. It does not say the premises ARE true. An argument with true premises and a necessarily true conclusion by virtue of structure is both valid AND sound; an argument can be valid without being sound.

Question 5

Explain the asymmetry of the counterexample method: why does finding a counterexample prove invalidity conclusively, while failing to find one does not prove validity?

Model answer: Validity is a universal claim: an argument is valid if there is NO possible scenario where all premises are true and the conclusion is false. This means invalidity is an existential claim: the argument is invalid if there EXISTS at least one such scenario. Existential claims are refuted by finding a single witness — one counterexample suffices to prove existence. But universal claims cannot be confirmed by checking cases, only by exhaustive proof or formal demonstration. Failing to find a counterexample only means you have not yet found a witness to invalidity; it does not show no witness exists. The asymmetry mirrors the asymmetry between falsification and verification in science: one negative case can falsify a universal claim, but finitely many positive cases cannot verify it.

The counterexample method is fundamentally a falsification tool. It works because invalidity requires only the existence of one problematic scenario, which can be demonstrated by construction. Validity requires the non-existence of any problematic scenario, which cannot be demonstrated by checking finitely many cases — you would need to examine all possible scenarios (infinitely many) or use a formal argument that rules them out structurally. This is why formal logic provides truth tables and proof systems: to establish validity positively rather than just failing to refute it.