The statement 'If it is raining, then the ground is wet' is given. You observe that the ground is wet. What can you validly conclude?
AIt is raining, because wet ground implies rain in this context
BIt is not raining, because the conditional only permits wet ground when it rains
CNothing definite about whether it is raining — wet ground can have other causes
DThe conditional is false, since wet ground does not guarantee rain
Observing Q (wet ground) does not let you conclude P (raining) — this is the fallacy of affirming the consequent. P → Q says only that rain guarantees wet ground, not that wet ground guarantees rain. Wet ground can have other causes. You can only validly move from P to Q (forward), or from ¬Q back to ¬P (contrapositive). Not the reverse.
Question 2 Multiple Choice
Which of the following is logically equivalent to 'If n is divisible by 4, then n is even'?
AIf n is even, then n is divisible by 4
BIf n is not divisible by 4, then n is not even
CIf n is not even, then n is not divisible by 4
Dn is divisible by 4 if and only if n is even
A conditional P → Q is logically equivalent only to its contrapositive ¬Q → ¬P. Here P = 'n is divisible by 4' and Q = 'n is even,' so the contrapositive is 'If n is not even, then n is not divisible by 4' (option C). Option A is the converse; option B is the inverse — neither is equivalent to the original. Option D is a biconditional and is actually false (12 is even but not divisible by 4).
Question 3 True / False
The conditional statement 'If 2 + 2 = 5, then elephants can fly' is logically false.
TTrue
FFalse
Answer: False
This is an example of vacuous truth. The only way P → Q can be false is when P is true and Q is false. Here the hypothesis '2 + 2 = 5' is false, so the conditional cannot be violated — it is vacuously true. Think of the conditional as a promise: 'if it rains, I'll carry an umbrella.' The promise is only broken if it rains and you don't have one. If it never rains, the promise was never tested and you cannot be accused of breaking it.
Question 4 True / False
The conditional P → Q and its converse Q → P are logically equivalent — if one is true, the other is expected to be true.
TTrue
FFalse
Answer: False
P → Q and Q → P are independent statements. 'If it is a square, then it is a rectangle' is true, but its converse 'If it is a rectangle, then it is a square' is false. They are equivalent only in the special case of a biconditional (P ↔ Q). Confusing a conditional with its converse is one of the most common logical errors in mathematical reasoning.
Question 5 Short Answer
Why is a conditional statement with a false hypothesis considered 'vacuously true' rather than simply undefined or meaningless?
Think about your answer, then reveal below.
Model answer: A conditional P → Q claims that whenever P is true, Q must also be true. The only violation is P being true and Q being false. If P is false, the hypothesis never fires — the claim is never put at risk and cannot be violated. Declaring it 'true' in this case maintains logical consistency: in a complete truth table, every row must have a definite truth value, and the false-hypothesis rows must be true to avoid falsely labeling statements as violated when their hypothesis never applies.
Vacuous truth is not just a convention — it is necessary for mathematical logic to work cleanly. Universal statements like 'for all x in the empty set, P(x) holds' rely on vacuous truth: there is no counterexample, so the statement is true. Proof by contradiction also relies on this: deriving a vacuously true conclusion from a false assumption is the mechanism that identifies the assumption as false.