Premises - Type 3: If and Only If A, then B

In this case, A is the necessary and sufficient condition for the occurrence of B.  Imagine B as the destination and A as the only path that leads to B.

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

Valid Conclusions:

  1. If A has happened, B has happened (because the occurrence of A guarantees 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 must not have happened (because if A happens, B has to happen).

Let’s look at some examples:

1. If and only if I listen to the radio, I enjoy myself.

From this statement, we can infer that:

  • If I listened to the radio, I must have enjoyed myself.
  • If I did not listen to the radio, I must not have enjoyed myself.
  • If I enjoyed myself, I must have listened to the radio.
  • If I did not enjoy myself, I must not have listened to the radio.

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: If and only if I am sick, I listen to the radio.

A. I was sick.

B. I was not sick.

C. I listened to the radio.

D. I did not listen to the radio.

From the given statement we can infer that:

1. If I was sick, I must have listened to the radio. So, AC is logically correct.

2. If I was not sick, I must not have listened to the radio. So, BD is also logically correct.

3. If I listened to the radio, I must have got sick. So, CA is also logically correct.

4. If I did not listen to the radio, I must not have got sick. So, DB is also logically correct.

Therefore, the correct sequences are AC, CA, BD, and DB.

Practice Exercise 2: Sam will eat the fruit if and only if it is cherry.

A. The fruit is cherry.

B. The fruit is mango.

C. Sam ate the fruit.

D. Sam did not eat the fruit.

From the given statement we can infer that:

1. If Sam ate the fruit, the fruit must be cherry. So, CA is logically correct.

2. If the fruit is cherry, Sam must have eaten it. So, AC is also logically correct.

3. If the fruit is some other than cherry, Sam must not have eaten it. So, BD is also logically correct.

Therefore, the correct sequences are CA, AC, and BD.

Note that DB is not the correct sequence because if Sam did not eat the fruit, then it is not necessary that the fruit is mango. It can be any fruit other than cherry.

Practice Exercise 3: If and only if it rains, the ground gets wet.

A. It rained.

B. The ground got wet,

C. The ground did not get wet.

D. It did not rain.

Clearly, AB, BA, CD, and DC are logically correct.

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