Under the rule 'If a student passes the exam (P), then they receive credit (C),' a student fails the exam. What can we conclude about whether they receive credit?
ANothing — the implication P → C makes no claim when P is false
BThe student definitely does not receive credit, since they failed
CThe implication has been violated, since the outcome is uncertain
DThe student must receive credit, since the rule still applies
P → C is false only when P is true and C is false. When P is false (the student fails), the implication is vacuously true regardless of C — the conditional promise is simply never triggered. The student failing says nothing about whether they receive credit; perhaps they receive it on other grounds. This vacuous truth surprises beginners but follows directly from the truth table: a conditional that is never triggered cannot be violated.
Question 2 Multiple Choice
Which of the following is logically equivalent to p → q?
A¬p ∨ q
Bp ∧ ¬q
C¬p → ¬q
Dq → p
p → q is false only when p is true and q is false. ¬p ∨ q is false only when ¬p is false (p is true) and q is false — exactly the same condition. They have identical truth tables and are therefore logically equivalent. Option C (¬p → ¬q) is the inverse of p → q — not equivalent. Option D (q → p) is the converse — not equivalent. Option B (p ∧ ¬q) is actually the negation of p → q, not the equivalence.
Question 3 True / False
The statement 'If 2 + 2 = 5, then the moon is made of cheese' is a true proposition according to propositional logic.
TTrue
FFalse
Answer: True
This is true — and deliberately counterintuitive. An implication p → q is false only when p is true and q is false. Here, p ('2 + 2 = 5') is false, so the entire implication is vacuously true regardless of q. The logical meaning of implication does not match the everyday meaning of 'if…then.' In formal logic, a false antecedent renders the conditional automatically true because no promise has been broken.
Question 4 True / False
The contrapositive of p → q is ¬p → ¬q.
TTrue
FFalse
Answer: False
False. The contrapositive of p → q is ¬q → ¬p (flip and negate both sides). It is logically equivalent to the original. The statement ¬p → ¬q is the inverse — a different statement that is not necessarily equivalent to p → q. A common error is to confuse the inverse with the contrapositive. Only the contrapositive preserves logical equivalence.
Question 5 Short Answer
A classmate argues: 'Since p → q is false when p is true and q is false, it should also be false when p is false and q is false — after all, the conclusion still fails.' Explain why this reasoning is wrong.
Think about your answer, then reveal below.
Model answer: The implication p → q is a conditional promise: 'if p holds, then q must hold.' It is only violated when the premise is in force (p is true) and the conclusion fails (q is false). When p is false, the premise is never activated — no promise was made that applies to this situation. The truth value of q is then irrelevant. Logically, p → q is equivalent to ¬p ∨ q, which is true whenever p is false, regardless of q.
Vacuous truth is one of the hardest aspects of formal implication to accept intuitively. Everyday 'if…then' language often implies a causal relationship, but in propositional logic, implication is purely about truth values. The asymmetry — only one row is false — is what makes implication useful in formal proofs: you can freely use an implication whose antecedent is false without checking the conclusion.