Questions: Conditional Statements (If-Then Formal)

Question 1

In the conditional statement 'If a triangle has three equal sides, then it is equilateral,' what is the hypothesis?

• A. It is equilateral
• B. A triangle has three equal sides
• C. All triangles are equilateral
• D. If a triangle has three equal sides

Answer: (B)

In 'If P, then Q,' the hypothesis (antecedent) is P — the part after 'if' and before 'then.' Here, P = 'a triangle has three equal sides.' The conclusion (consequent) is Q = 'it is equilateral.' The hypothesis is the condition that triggers the conclusion.

Question 2

The conditional statement 'If 5 > 10, then the moon is made of cheese' is logically false.

• T. True
• F. False

Answer: False

This statement is logically true. A conditional P → Q is false only when P is true and Q is false. Here, P ('5 > 10') is false, so the conditional is vacuously true regardless of Q. This feels strange because neither part is factually true, but logical truth of a conditional depends on the relationship between P and Q's truth values, not on their real-world content.

Question 3

Explain in your own words why a conditional statement is considered true when its hypothesis is false, using a real-world analogy.

Model answer: Think of a promise: 'If it snows, I will cancel school.' On a sunny day, the promise has not been broken — the condition was never triggered. The promise is only broken when it snows AND school is not cancelled. Similarly, P → Q is only false when P is true and Q is false; when P is false, the 'promise' is intact.

The promise analogy captures why vacuous truth makes sense: a conditional is a guarantee about what happens when the hypothesis holds. If the hypothesis never holds, the guarantee cannot be violated. You can only break the rule 'If P then Q' by finding a case where P is true and Q is false.