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I have four sentences:

  1. John and Joe are not both in the room.

  2. John and Joe are both not in the room.

  3. Either John or Joe is not in the room.

  4. Neither John nor Joe is in the room.

I want to know the difference in meaning between these sentences. Specifically I want to know the difference between sentences A and B, and C and D.

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1) If John and Joe were in the room at the same time, they would "both be in the room". Since they are not "both in the room", one or both of them is out of the room.

2) John is not in the room. Joe is not in the room. Therefore they are both "not in the room".

3) If John is in the room, Joe is not. If Joe is in the room, John is not. However, one of them is. The either/or construction implies a choice: Either A or B, but not both.

4) John is not in the room. Joe is not in the room. This can also be expressed as, John is not in the room, and neither is Joe.

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The first is true if either John or Joe is not in the room.

The second is true only if both John and Joe are not in the room.

So if Joe is in the room and John is not, the first sentence is true and the second is false.

The third sentence literally means one of them is in the room and the other is not. It is false if they are both in the room and it is false if they are both not in the room. It is common, however, for people to accept this sentence as true if both are not in the room.

The fourth sentence is logically identical to the second. It is true only if both John and Joe are not in the room.

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