Premises - Type 5: If not A, then B

In this case, non-occurrence of A is the sufficient but not necessary condition for the occurrence of B. Imagine B as the destination and non-occurrence of A as one of the several paths that lead to B.

Logical Deduction: 'If not A, then B' analogy image

Valid Conclusions:

  1. If A does not happen, B happens (because the non-occurrence of A lead to B).
  2. If B does not happen, A must have happened (because B happens for sure if A does not happen).
  3. If A happens, B may or may not happen (because there might be other conditions that lead to B).
  4. If B happens, A may or may not have happened (again because there might be other conditions that lead to B).

Let’s look at some examples:

1. If I do not take a cup of coffee, I will feel sleepy.

From this statement, we can infer that:

  • If I didn’t take a cup of coffee, I must be feeling sleepy.
  • If I am not feeling sleepy, I must have taken a cup of coffee.
  • If I took a cup of coffee, I may or may not be feeling sleepy (This is because I might feel sleepy because of some other reason).
  • If I am feeling sleepy, I may or may not have taken a cup of coffee.

2. If it does not rain, the crop will die.

Analogy for Practice Problem 2From this statement, we can infer that:

  • If it didn’t rain, the crop must have died.
  • If the crop didn’t die, it must have rained (because no rain ensures that the crop dies).
  • If it rained, the crop may or may not have died (because the crop can die of some other reason as well).
  • If the crop died, it may or may not have rained (again because the crop can die of some other reason as well).

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 the employees are not micromanaged, they perform well.

A. Employees are micromanaged.

B. Employees do not perform well.

C. Employees are not micromanaged.

D. Employees performed well.

Clearly, CD is logically correct.

Also, we can infer that if the employees do not perform well, they must have been micromanaged. So, BA is also logically correct.

Therefore, the correct sequences are CD and BA.

Practice Exercise 2: Samanta will eat the fruit if it is not mango.

A. The fruit is cherry.

B. The fruit is mango.

C. Samanta ate the fruit.

D. Samanta did not eat the fruit.

From the given statement we can infer that:

1. If the fruit is not mango, Samanta must have eaten it. So, AC is logically correct.

2. If Samanta did not eat the fruit, it must be mango. So, DB is also logically correct.

Therefore, the correct sequences are AC and DB.

Practice Exercise 3: If I do not sleep enough, I get a headache.

A. I slept enough.

B. I got a headache.

C. I did not get a headache.

D. I did not sleep enough.

Clearly, DB is logically correct.

Also, we can infer that if I haven’t got a headache, I must have slept enough. So, CA is also logically correct.

Therefore, the correct sequences are DB and CA.

Practice Exercise 4: You will feel cold unless you wear a coat.

A. You wore a coat.

B. You felt cold.

C. You did not feel cold.

D. You did not wear a coat.

We can interpret it as: “You will feel cold if you don’t wear a coat.”

Clearly, DB is logically correct.

Also, we can infer that if you did not feel cold, you must have worn a coat. So, CA is also logically correct.

Therefore, the correct sequences are DB and CA.

Practice Exercise 5: Unless you want to go to bed early, we should go out.

A. You want to go to bed early.

B. We should go out.

C. You don’t want to go to bed early.

D. We should not go out.

We can interpret it as: “If you don’t want to go to bed early, we should go out.”

Clearly, CB is logically correct.

Also, we can infer that if we don’t go out, you must be wanting to go to bed early. So, DA is also logically correct.

Therefore, the correct sequences are CB and DA.