# Regarding usage of "both"

Could you tell me how the following sentences are different?

1. Alice and Bob are not both in the room.
2. Alice and Bob are both not in the room.

Updated:

Could you divide the sentences into a set of grammatical chunks and point me how each chunk affects to another?

• Hi Doofah, welcome to EL&U! Please wait a day or two before selecting an answer. You may get several more answers with other ideas or different advice. But people may not bother to write you another answer if you've already selected one! :-) Commented Jun 29, 2016 at 10:27
• Hi, Araucaria! Thank you for your advice. Okay, I'll wait for other answers to come. :) Commented Jun 29, 2016 at 11:46
• Hi Doofah, if you're learning English, you may find our sister site English Language Learners helpful. Commented Jun 29, 2016 at 12:27
• Hi TrevorD, yes, I'm learning English but the original question was coming from a pure logical/mathematical textbook. I was simply confused by the both sentences so I decided to ask here. ;) Commented Jun 29, 2016 at 13:01

They are logically different as they are asserting different things.

1. Either Alice or Bob, or neither, are in the room, but not both.

2. Neither Alice nor Bob is in the room.

Edit for your additional question (what is the logical negation of each):

1. "Both Alice and Bob are in the room."

2. "Either Alice or Bob, or both, are in the room."

• You could possibly argue that the logical negation of "Alice and Bob are not both in the room." is "Alice and Bob are both in the room." I can argue this only because "logical negation" is actually quite vague in this context. Commented Jun 29, 2016 at 8:15
• Hi, Jeffrey, I think it is better to use "is" instead of "are" in No. 1 and 2 sentences. What is your thought?
– user140086
Commented Jun 29, 2016 at 8:22
• Thank you, Jeffrey! (Unfortunately I've deleted the additional question by accident though...) Commented Jun 29, 2016 at 8:22
• Only #2 needs the singular form. Corrected. Commented Jun 29, 2016 at 8:26
• @Doofah (3) "Both Alice and Bob are not in the room." and (1) "Alice and Bob are not both in the room." are not logically identical. (3) is logically identical to (2) "Alice and Bob are both not in the room." Neither (2) nor (3) sounds natural (to this native British speaker): as the above answer already suggests, I would say "Neither Alice nor Bob is in the room.". Commented Jun 29, 2016 at 12:37

Both Alice and Bob are not in the room means they are together not in the room while Alice and Bob are not both in the room means maybe one of them are in the room.