Questions: Conditional and Biconditional Statements

Question 1

Consider the statement: 'If all rivers flow uphill, then the Pythagorean theorem is false.' What is the truth value of this statement?

• A. False — the conclusion is false, so the conditional must be false
• B. Undefined — conditionals with impossible hypotheses have no truth value
• C. True — the hypothesis is false, making the conditional vacuously true
• D. True — both the hypothesis and conclusion are false, so they match

Answer: (C)

A conditional P→Q is false only when P is true and Q is false. In all other cases it is true — including when P is false. The hypothesis 'all rivers flow uphill' is false, so no promise is broken regardless of the conclusion. This is vacuous truth: the conditional only makes a commitment when P holds. If P never fires, no violation can occur. Option A confuses 'false conclusion' with 'false conditional' — the conclusion's truth value is irrelevant when P is false.

Question 2

A mathematics student wants to prove: 'If n² is even, then n is even.' She finds the direct proof difficult and instead proves: 'If n is odd, then n² is odd.' Which best describes this strategy and its validity?

• A. She proved the converse (Q→P), which is not logically equivalent to the original — the proof is invalid
• B. She proved the inverse (¬P→¬Q), which has the same truth value only in special cases
• C. She proved the contrapositive (¬Q→¬P), which is logically equivalent to the original — the proof is valid
• D. She made an error — proving a statement about odd n cannot establish a claim about even n²

Answer: (C)

The original statement is P→Q where P = 'n² is even' and Q = 'n is even.' The contrapositive is ¬Q→¬P: 'If n is not even (odd), then n² is not even (odd)' — exactly what she proved. The contrapositive and the original conditional have identical truth tables; they are logically equivalent. This is not a trick — it is a legitimate proof strategy. Contrast with the converse (Q→P: 'if n is even, then n² is even'), which is a different statement that is also true but requires a separate proof.

Question 3

The statement 'If P then Q' and its contrapositive 'If not Q then not P' always have the same truth value.

• T. True
• F. False

Answer: True

This is a fundamental logical equivalence, verifiable by truth table: P→Q and ¬Q→¬P are true in exactly the same rows (false only when P is true and Q is false). This equivalence is why proof by contrapositive is a legitimate proof strategy — proving ¬Q→¬P is identical to proving P→Q. Memorizing this equivalence as a rule is less valuable than understanding why it holds: if you know P guarantees Q, then failing to have Q guarantees you didn't have P.

Question 4

The converse of a conditional statement is logically equivalent to the original statement, just as the contrapositive is.

• T. True
• F. False

Answer: False

The converse of P→Q is Q→P, and these are NOT logically equivalent in general. Counterexample: 'If n is divisible by 4, then n is even' is true, but its converse 'If n is even, then n is divisible by 4' is false (n=6 is even but not divisible by 4). The contrapositive (¬Q→¬P) IS equivalent to the original. The inverse (¬P→¬Q) is equivalent to the converse, not to the original. Confusing converse and contrapositive is one of the most common errors in introductory logic.

Question 5

Explain why a conditional statement with a false hypothesis is considered true. Why does this make logical sense, especially in the context of universal mathematical statements like 'For every x in set S, if P(x) then Q(x)'?

Model answer: A conditional makes a promise: 'whenever P holds, Q will hold.' If P never holds, no promise is ever broken — nothing false was implied. In the context of a universal statement over a set, if no element satisfies P(x), the conditional is vacuously satisfied for every element. For example, 'for every integer in the empty set, if it is even then it is greater than 1000' is trivially true because there are no integers to check. This is consistent with classical logic: the conditional's truth table defines it to be false only in the one case where P fires but Q fails.

Vacuous truth is not a loophole — it is a necessary feature of how implication works. If we declared conditionals with false hypotheses to be false, every universal statement 'For all x, P(x)→Q(x)' would be false on any element where P(x) doesn't hold, making virtually all mathematical theorems false. The definition ensures that a theorem makes a claim only about cases where the hypothesis applies.