Someone argues: 'If you are a US Senator, you must be at least 30 years old. Professor Williams is 45 years old. Therefore, Professor Williams is a US Senator.' This argument is:
AValid, because the conclusion is consistent with the premises
BInvalid — this is affirming the consequent: the conditional tells us senators are at least 30, not that everyone 30 or older is a senator
CValid — this is a correct application of modus ponens
DInvalid — this is denying the antecedent
The form is: If P then Q; Q; therefore P. This is affirming the consequent — a formal fallacy. The conditional 'if Senator then at least 30' tells you what follows from being a senator; it says nothing about the converse. Being 45 is consistent with being a senator but does not prove it — there are millions of people over 30 who are not senators. The argument's form is invalid even if the conclusion happened to be true.
Question 2 Multiple Choice
Which of the following is the clearest example of equivocation?
A'If it rains, the streets are wet. The streets are wet. Therefore it is raining.'
B'All laws can be broken. The law of gravity is a law. Therefore the law of gravity can be broken.' (where 'law' shifts from legal statute to natural law)
C'If you study hard, you will pass. You did not study hard. Therefore you will not pass.'
D'All men are mortal. Socrates is a man. Therefore Socrates is mortal.'
Equivocation occurs when a key term shifts meaning between premises, breaking the logical chain. In option B, 'law' means a legal statute that humans can choose to break in the first premise, and a natural regularity that physically cannot be violated in the second. Once disambiguated, the premises don't connect. Option A is affirming the consequent; option C is denying the antecedent; option D is valid (modus barbara).
Question 3 True / False
A formally valid argument is one where the conclusion must be true if the premises are true — regardless of what the argument is actually about.
TTrue
FFalse
Answer: True
Validity is a structural property, not a content property. An argument form like modus ponens — 'If P then Q; P; therefore Q' — is valid for any propositions substituted for P and Q, regardless of subject matter. You can verify validity by inspecting the form in abstraction from content. This is what makes it a formal property: it belongs to the logical skeleton, not to what the sentences actually mean.
Question 4 True / False
If an argument commits a formal fallacy, its conclusion is expected to be false.
TTrue
FFalse
Answer: False
A formally fallacious argument proves nothing — but that is different from proving the conclusion is false. The conclusion might be true for entirely independent reasons. If someone argues 'If it rains the streets are wet; the streets are wet; therefore it rained,' the streets might actually be wet from rain. The fallacy is about the argument's failure to provide proof, not about the conclusion's truth value. This is one of the most important insights in critical thinking: finding a fallacy attacks the argument, not necessarily the claim.
Question 5 Short Answer
Why can a formally fallacious argument have a true conclusion? What does this reveal about the relationship between argument validity and truth?
Think about your answer, then reveal below.
Model answer: A formal fallacy means the argument's logical structure fails to guarantee the conclusion — the premises do not force the conclusion to be true. But this says nothing about whether the conclusion actually is true. The conclusion could be true for reasons entirely unrelated to the argument given. Validity and truth are orthogonal properties: a valid argument from false premises can yield a false conclusion; an invalid argument can land on a true conclusion by accident. What validity guarantees is the inferential connection — if premises are true, conclusion must follow. Fallacious arguments break that connection without determining the conclusion's truth value.
Students often assume that finding a fallacy 'disproves' the conclusion. It does not — it shows only that this particular argument does not prove it. An opponent who commits a fallacy may still be right about the conclusion; you need independent evidence against the conclusion to show they are wrong. Separating 'this argument fails' from 'this claim is false' is fundamental to rigorous critical thinking.