A scientist reasons: 'If this compound is toxic, lab animals will show symptoms. The animals showed symptoms. Therefore, the compound is toxic.' Which inference pattern is being applied?
AModus ponens — the scientist has P and P→Q, so concludes Q
BModus tollens — the scientist has ¬Q and P→Q, so concludes ¬P
CAffirming the consequent — the scientist has P→Q and Q, so concludes P (invalid)
DDisjunctive syllogism — the scientist is eliminating one disjunct
This is the classic invalid inference pattern: affirming the consequent. The form is P→Q, Q ∴ P. Even if the conditional is true ('toxicity causes symptoms'), the converse is not guaranteed — the animals might show symptoms for many reasons other than this compound. This looks superficially like modus ponens but runs in the wrong direction: modus ponens goes from antecedent to consequent (P, P→Q ∴ Q), while this goes from consequent back to antecedent, which is not truth-preserving.
Question 2 Multiple Choice
You know 'If Alice is present, the meeting goes forward (A→M)' and 'If the meeting goes forward, the report is finalized (M→R).' Which rule lets you directly conclude 'If Alice is present, the report is finalized (A→R)'?
AModus ponens — applying the first conditional to a given premise
BModus tollens — taking the contrapositive of the chain
CHypothetical syllogism — chaining two conditionals P→Q and Q→R into P→R
DDisjunctive syllogism — eliminating one option from a disjunction
Hypothetical syllogism is the transitivity rule for conditionals: from P→Q and Q→R, conclude P→R. It underlies multi-step proofs where you build a path from hypothesis to conclusion through intermediate steps — exactly what this scenario does with A→M and M→R to conclude A→R.
Question 3 True / False
Modus tollens is essentially the contrapositive of modus ponens applied as an inference rule.
TTrue
FFalse
Answer: True
Exactly right. Modus ponens says: P→Q, P ∴ Q. The contrapositive of P→Q is ¬Q→¬P, which is logically equivalent. Modus tollens applies that contrapositive: P→Q, ¬Q ∴ ¬P. If you know the conditional and you know the consequent is false, you can infer the antecedent is also false. The two rules are mirror images — both valid because contraposition preserves truth.
Question 4 True / False
From 'If it rains, the game is cancelled (R→C)' and 'The game was cancelled (C)', you can validly conclude 'It rained (R)'.
TTrue
FFalse
Answer: False
This is affirming the consequent, an invalid inference form. The game could have been cancelled for many reasons other than rain. P→Q being true does not make Q→P true. The valid inference from R→C and ¬C would be ¬R (modus tollens). But from R→C and C, no conclusion about R follows. The pattern looks persuasive because in everyday speech we often treat conditionals as biconditionals — but in logic, they are not.
Question 5 Short Answer
Why is 'affirming the consequent' an invalid inference rule, even though it superficially resembles modus ponens?
Think about your answer, then reveal below.
Model answer: Affirming the consequent has the form P→Q, Q ∴ P, which runs from consequent back to antecedent. This is invalid because P→Q allows Q to be true for reasons other than P — multiple different antecedents can produce the same consequent. Modus ponens (P→Q, P ∴ Q) is valid because starting from the sufficient condition and moving to its consequence preserves truth. Affirming the consequent reverses this direction through a one-way gate.
A conditional P→Q says P is sufficient for Q, not that P is the only way to get Q. Knowing Q is true doesn't tell you whether P caused it or something else did. Modus ponens works because it starts from the antecedent — the sufficient condition. Affirming the consequent tries to run the inference backwards, losing truth preservation in the process.