Let P = 'It is raining' (false) and Q = 'The streets are wet' (false). What is the truth value of P → Q?
AFalse, because neither P nor Q is true
BFalse, because the consequent Q is false
CTrue, because the antecedent P is false
DUndefined, because P and Q are both false
Material implication P → Q is false in exactly one case: when P is true and Q is false. In all other combinations — including when P is false regardless of Q — the conditional is true. When P is false, the conditional makes no false promise: it says 'if it is raining, then streets are wet,' and since it is not raining, the claim is never tested and cannot be violated. This vacuous truth is the most counterintuitive feature of material implication and the source of most student errors.
Question 2 Multiple Choice
How is the formula P ∨ Q ∧ R parsed according to standard precedence conventions?
AAs (P ∨ Q) ∧ R, because operators are evaluated left to right
BAs P ∨ (Q ∧ R), because ∧ binds more tightly than ∨
CAs (P ∨ Q) ∧ R, because ∨ appears first in the formula
DAs P ∨ (Q ∧ R) only if explicit parentheses are added
Standard precedence in propositional logic (tightest to loosest): ¬, then ∧, then ∨, then →, then ↔. Because ∧ binds more tightly than ∨, Q ∧ R is grouped first, giving P ∨ (Q ∧ R). This is analogous to multiplication binding before addition in arithmetic: 2 + 3 × 4 = 2 + (3 × 4) = 14, not (2 + 3) × 4 = 20. Precedence is a notational convention that reduces parentheses; when in doubt, add them explicitly.
Question 3 True / False
The disjunction P ∨ Q is true even when both P and Q are true.
TTrue
FFalse
Answer: True
Logical disjunction (∨) is *inclusive* OR: it is true when at least one disjunct is true, which includes the case where both are true. This differs from everyday English 'or,' which often carries an exclusive sense ('soup or salad' usually means one or the other). The inclusive interpretation is a deliberate choice in formal logic because it has cleaner algebraic properties. Exclusive OR (XOR) is a separate connective, defined as true when exactly one disjunct is true.
Question 4 True / False
The conditional P → Q is false whenever Q is false.
TTrue
FFalse
Answer: False
P → Q is false only when P is true *and* Q is false simultaneously. When Q is false but P is also false, the conditional is vacuously true. The formula can be read as 'it is not the case that P is true while Q is false.' So Q being false is necessary but not sufficient for P → Q to be false — P must also be true. Equivalently, P → Q is logically equivalent to ¬P ∨ Q, which is true whenever P is false (regardless of Q).
Question 5 Short Answer
Why is material implication (P → Q) defined to be true when P is false, even if Q is also false? Explain using the truth-functional definition.
Think about your answer, then reveal below.
Model answer: Material implication is defined purely by its truth table: P → Q is false only when P is true and Q is false — the one case where a true premise leads to a false conclusion. When P is false, the conditional makes no claim that can be falsified. There is no instance where the promise 'if P then Q' was broken, because P never held. Formally, P → Q is equivalent to ¬P ∨ Q, which is true whenever P is false.
This is called vacuous truth: the conditional is trivially satisfied because its hypothesis is never fulfilled. The definition captures the logical idea that an implication is only violated by a counterexample — a case where the hypothesis holds but the conclusion does not. With no counterexample possible (because the hypothesis is false), the implication is satisfied. The cost is that material implication cannot capture causation or relevance — two unrelated facts can form a valid 'if-then' — but this keeps the logic purely truth-functional and algebraically tractable.