Premises - Type 2: Only If A, then B

In this case, A is the necessary but not sufficient condition for the occurrence of B. Occurrence of A does not guarantee the occurrence of B. Imagine A as the door to the treasure B. Now if you open the door, you may or may not take the treasure, but if you took the treasure, then you must have opened the door.

Logical Deduction: 'Only If A, then B' analogy image

Valid Conclusions:

  1. If A has happened, B may or may not happen (because the occurrence of A does not guarantee the occurrence of B).
  2. If A has not happened, B has not happened (because only A leads to B).
  3. If B has happened, A must have happened (again because only A leads to B).
  4. If B has not happened, A may or may not have happened (again because the occurrence of A does not guarantee the occurrence of B).

Let’s look at some examples:

1. I will meet you only if I come to Chandigarh.

From this statement, we can infer that:

  • If I come to Chandigarh, I may or may not meet you.
  • If I don’t come to Chandigarh, I won’t meet you.
  • If I met you, I must have come to Chandigarh.
  • If I didn’t meet you, I may or may not have come to Chandigarh.

2. Only if you study well, you will pass the exam.

From this statement, we can infer that:

  • If you passed the exam, you must have studied well.
  • If you don’t study well, you won’t pass the exam.

Your Turn:

Let’s practice what we have learned. Each question has a main statement, followed by four statements labeled A, B, C, and D. Choose the ordered pair of statements, where the first statement implies the second, and the two statements are logically consistent with the main statement.

Practice Exercise 1: Only if he runs fast, he will catch the bus.

A. He ran fast.

B. He caught the bus.

C. He did not run fast.

D. He did not catch the bus.

From the given statement we can infer that:

1. If he caught the bus, he must have run fast. So, BA is logically correct.

2. Running fast is the necessary condition for him to be able to catch the bus. So, if he did not run fast, he wouldn’t have caught the bus. Hence, CD is also logically correct.

Therefore, the correct sequences are BA and CD.

Practice Exercise 2: I listen to the radio only if I am bored.

A. I listened to the radio.

B. I did not listen to the radio.

C. I am bored.

D. I am not bored.

Here, being bored is the condition that makes me listen to the radio.

From the given statement we can infer that:

1. If I listened to the radio, I must be feeling bored. So, AC is logically correct.

2. If I am not bored, I wouldn’t have listened to the radio. So, DB is also logically correct.

Therefore, the correct sequences are AC and DB.

Practice Exercise 3: Only if you go to medical school can you become a doctor.

A. You became a doctor.

B. You did not go to medical school.

C. You went to medical school.

D. You did not become a doctor.

Clearly, both AC and BD are logically correct.

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