Consider the argument: 'If the economy grows, unemployment falls. Unemployment has fallen. Therefore, the economy has grown.' Which inference pattern does this use, and is it valid?
AModus ponens — valid, because we confirmed the antecedent
BAffirming the consequent — invalid, because other causes could have reduced unemployment
CModus tollens — valid, because we denied the consequent
DDenying the antecedent — invalid, because the antecedent was left unexamined
The argument has the form: if A then B; B; therefore A. This is affirming the consequent, which is invalid. Unemployment could have fallen for many reasons — policy changes, demographic shifts, new industries — without the economy growing. The argument feels compelling because the premises are plausible and the conclusion is desirable, but validity depends on structure, not plausibility.
Question 2 Multiple Choice
'If a student passes the final exam, they pass the course. Alex did not pass the final exam. Therefore, Alex did not pass the course.' What is wrong with this argument?
ANothing — modus tollens applies here correctly
BIt denies the antecedent: failing the exam doesn't rule out passing the course by other means
CIt confuses the antecedent and consequent in the original conditional
DThe argument is valid but unsound because the premise could be false
The argument has the form: if A then B; not-A; therefore not-B. This is denying the antecedent, which is invalid. The original conditional says passing the exam is sufficient for passing the course, not necessary. Alex might pass via extra credit, a makeup policy, or a grade appeal. Modus tollens would be valid: if A then B; not-B; therefore not-A. Here, not-B (didn't pass the course) would let us conclude not-A (didn't pass the exam).
Question 3 True / False
The conditional statement 'If A then B' is false only when A is true and B is false.
TTrue
FFalse
Answer: True
This is the truth table definition of material implication. When A is false, the conditional is vacuously true regardless of B — there is no counterexample to the claim 'whenever A, B.' When A is true and B is true, the conditional holds. Only when A is true and B is false do we have a genuine violation: A occurred but B didn't follow. This explains why 'if pigs fly, I'll eat my hat' is technically true — the false antecedent prevents any falsification.
Question 4 True / False
'If A then B' logically implies 'If not-A then not-B.'
TTrue
FFalse
Answer: False
This is the fallacy of denying the antecedent. 'If not-A then not-B' (the inverse) is not logically equivalent to 'if A then B.' The valid contrapositive is 'if not-B then not-A.' Consider: 'If it is raining, the ground is wet' does not imply 'if it is not raining, the ground is not wet' — a sprinkler could wet the ground. The inverse and the original are independent claims with separate truth values.
Question 5 Short Answer
Why is 'affirming the consequent' an invalid inference pattern? Give an example that shows why it fails even when the premises seem strongly related.
Think about your answer, then reveal below.
Model answer: Affirming the consequent (if A then B; B; therefore A) fails because the consequent B can be true for reasons other than A. The conditional only guarantees B follows from A, not that A is the unique cause of B. Example: 'If it rained, the grass is wet. The grass is wet. Therefore it rained.' Invalid — the sprinkler could explain the wet grass. Even in science, observing a predicted effect (B) does not confirm the hypothesis (A), because rival hypotheses may predict the same effect.
The key is distinguishing sufficient from necessary conditions. 'If A then B' says A is sufficient for B, not that A is necessary. Affirming the consequent treats A as if it were the only sufficient cause of B. This error underlies many fallacies in everyday reasoning, from medical diagnosis ('that symptom means you have X') to political argument ('only bad people oppose Y'). Valid reasoning from B back to A requires the stronger claim 'B if and only if A.'