If it rains, the match is cancelled. It rained. Which of the following must be true?

The match was played after a delay.

The match may or may not have been cancelled.

Which of the following means exactly the same as: "If a shape is a square, then it is a rectangle"?

If a shape is a rectangle, then it is a square.

If a shape is not a square, then it is not a rectangle.

A shape is a rectangle only if it is a square.

A person can vote only if the person has registered. Asha voted. Which of the following must be true?

Asha registered on the day of voting.

Asha may or may not have registered.

If the bill is passed, the tax will rise. If the tax rises, prices will go up. Prices did not go up. Which of the following must be true?

The bill was passed but the tax did not rise.

Nothing can be said about the bill.

The tax rose but prices held steady.

Which of the following means the same as: "Unless you pay the fee, you will not be admitted"?

You will be admitted whether or not you pay the fee.

If you are not admitted, then you have not paid the fee.

If you pay the fee, then you will be admitted.

A notice says: "If it is a public holiday, the library is closed." Which of the following, if found, would show the notice to be false?

A public holiday on which the library is closed

A working day on which the library is closed

A working day on which the library is open

Under the lease, a tenant who falls into arrears either pays the arrears within thirty days or is evicted. A tenant fell into arrears and was not evicted. If the lease was followed, which of the following must be true?

The tenant paid after thirty days.

The tenant did not pay the arrears.

Nothing can be said about whether the tenant paid.

Logical Reasoning · Micro-test

Conditional Reasoning: If-Then Statements

An if-then statement supports exactly two certain inferences. Most wrong answers in this area come from drawing one of the two that it does not support.

10 questions · 5 minutes · instant scoring

What this topic actually tests

Take 'If it rains, the match is cancelled.' Rain is the condition and cancellation is the result. Two inferences are valid. If the condition holds, the result follows: it rained, so the match is cancelled. And if the result is absent, the condition must have been absent: the match was not cancelled, so it did not rain. The second is the contrapositive, 'If the match is not cancelled, it did not rain', which always means exactly the same as the original. Two other inferences look similar and are invalid. From the result alone you cannot infer the condition: the match was cancelled, but perhaps for a strike. And from the absence of the condition you cannot infer the absence of the result: it did not rain, but the match may have been cancelled anyway. The other forms translate into this one. 'Only if you register can you vote' means 'If you voted, you registered': 'only if' introduces the necessary condition. 'Unless you pay the fee, you will not be admitted' means 'If you were admitted, you paid the fee.' Conditionals chain: if A then B, and if B then C, give if A then C, and so not-C gives not-A. The one case that disproves 'If A then B' is A being true while B is false. Lastly, 'either P or Q' with one of them ruled out leaves the other.

The common trap on this topic

The two standard errors have names. Affirming the consequent infers the condition from the result: 'the match was cancelled, so it rained'. Denying the antecedent infers no result from no condition: 'it did not rain, so the match was played'. Both feel natural because in daily speech 'if' is often used to mean 'if and only if'. In a reasoning question it never does. A third error is misreading 'only if': 'you can vote only if you register' does not promise that everyone who registers votes. When the correct answer is that nothing certain follows, expect to find that option sitting beside two tempting definite ones.

Conditional Reasoning: If-Then Statements: answers and explanations

If it rains, the match is cancelled. It rained. Which of the following must be true?
Answer: D. The match was cancelled.
The condition is satisfied, so the result follows with certainty. This is the first of the two valid inferences from a conditional.
If it rains, the match is cancelled. The match was not cancelled. Which of the following must be true?
Answer: A. It did not rain.
Had it rained, the match would have been cancelled. It was not, so it cannot have rained. This is the contrapositive.
If it rains, the match is cancelled. The match was cancelled. Which of the following must be true?
Answer: B. Nothing certain can be said about whether it rained.
The statement says what happens if it rains; it does not say rain is the only cause of cancellation. Concluding that it rained would be affirming the consequent.
If it rains, the match is cancelled. It did not rain. Which of the following must be true?
Answer: D. Nothing certain can be said about whether the match was cancelled.
The statement is silent about what happens when it does not rain. Concluding that the match was played would be denying the antecedent.
Which of the following means exactly the same as: "If a shape is a square, then it is a rectangle"?
Answer: D. If a shape is not a rectangle, then it is not a square.
The contrapositive, reversing and negating both parts, is always equivalent to the original. The other three statements are all false of shapes: an oblong rectangle is a rectangle and not a square.
A person can vote only if the person has registered. Asha voted. Which of the following must be true?
Answer: A. Asha had registered.
'Only if' marks registration as necessary for voting, so anyone who voted had registered. It does not say that everyone who registered went on to vote.
If the bill is passed, the tax will rise. If the tax rises, prices will go up. Prices did not go up. Which of the following must be true?
Answer: C. The bill was not passed.
Prices did not go up, so the tax did not rise; the tax did not rise, so the bill was not passed. The contrapositive is applied twice along the chain.
Which of the following means the same as: "Unless you pay the fee, you will not be admitted"?
Answer: C. If you are admitted, then you have paid the fee.
The statement makes paying the fee necessary for admission: no fee, no admission. So admission implies the fee was paid. It does not promise admission to everyone who pays.
A notice says: "If it is a public holiday, the library is closed." Which of the following, if found, would show the notice to be false?
Answer: D. A public holiday on which the library is open
A conditional is false only when the condition holds and the result does not: a public holiday with the library open. A working day with the library closed is consistent with the notice, which says nothing about working days.
Under the lease, a tenant who falls into arrears either pays the arrears within thirty days or is evicted. A tenant fell into arrears and was not evicted. If the lease was followed, which of the following must be true?
Answer: A. The tenant paid the arrears within thirty days.
One of the two outcomes must occur. Eviction is ruled out, so the other, payment within thirty days, must have happened.

FAQ

What is a contrapositive?

The contrapositive of 'If A, then B' is 'If not B, then not A'. It is always logically equivalent to the original statement, and it is the form in which most valid inferences in these questions are drawn.

What does 'only if' mean in logic?

'A only if B' means B is necessary for A, that is, 'If A, then B'. It does not mean that B guarantees A.

How do I disprove an if-then statement?

Find one case in which the condition is true and the result is false. Nothing else disproves it: cases where the condition is false are consistent with the statement whatever the result.

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