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I have done a simple exercise, but now I am still a bit confused. What is the difference between not both and both not? And the difference between either or and neither nor?

For example, in the following example:

  1. Messi and Ronaldo are not both in the match.
  2. Messi and Ronaldo are both not in the match.
  3. Either Messi or Ronaldo is not in the match.
  4. Neither Messi nor Ronaldo is in the match.

I would translate them with:

  1. $\neg(M \land R)$
  2. $\neg M \land \neg R$
  3. $\neg(M \lor R)$
  4. $\neg M \lor \neg R$

Where:

M = Messi is in the match
R = Ronaldo is in the match

I am mostly confused about the last 2 sentences.

1 Answers1

2

You are correct about 'not both' and 'both not'. Do you need any further clarification about what they mean?

"Either A or B" is actually quite tricky. I think it may depend on the exact context sometimes whether it means "A or B but not both" (that is, exclusive or) or just "A or B" (which includes the possibility of both A and B being true). Have a look at this question on the English Language & Usage stackexchange.
If this is an exercise for a course, then perhaps you have been told which usage is intended?

My interpretation of the third sentence before I thought about it too much was that it meant exclusive or, but now I'm not absolutely sure. In any case, NOT(M v R) is definitely not correct, because this says that M and R are both not true (think about a Venn diagram if it helps).

Your answer for 4. is also incorrect. It says that Messi is not in the match or Ronaldo is not in the match (including the possibility that they might both be not in the match). Therefore it allows the possibility that one of the two might be in the match, whereas "Neither Messi nor Ronaldo is in the match" means "Messi is not in the match and Ronaldo is not in the match".

Notice that 2. and 4. both mean the same thing!

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