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Can "Neither Nor" and "Either Or" appear in the same sentence? If yes, what is your inference regarding the sentence below?

"Neither L nor S lives in either A or B"

The answer my friend thinks is "No one (i.e L,S) lives in any place that has been mentioned (i.e. A,B)", But for me, some logic seems to be missing.

I believe the sentence can be written like this.

"Neither L nor S lives in A"

or

"Neither L nor S lives in B"

What I think is "One (L or S) cannot live in the places that have been mentioned (i.e. A or B) So the other one can live in A or B"

i.e. "If S cannot live in A or B, L lives in A or B" or "If L cannot live in A or B, S lives in A or B".

Can anyone help find the correct answer?

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    -1 "(Neither L nor S) lives in (either A or B)" -- what's your problem here? – Kris Dec 18 '17 at 12:02
  • I infer that it's intended to be ambiguous. – Hot Licks Dec 18 '17 at 13:03
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    I think it sounds like you believe that "neither L nor S" means "(not-L) or (not-S)". This is not true; it means "(not-L) AND (not-S)." – Hellion Dec 18 '17 at 17:17
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    @NarutoofKonaha to answer your comment-question above, NO: there is no possibility that one of them can live in A, because the statement explicitly says they do not. – Hellion Dec 18 '17 at 17:19
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    There are zillions of such ambiguous natural-language questions involving logical connectors and quantifiers. That's why formal logic uses parentheses, precedence, and other scoping formalisms: so you can unambiguously express each such possibility separately. – Drew Dec 19 '17 at 2:56
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In Standard English, even if a sentence is completely negative in sense, we apply negation only once, at the first possible place. So, for example, the following sentences all mean the same thing:

  • None of them ever found any of it anywhere.
  • Never did any of them find any of it anywhere.
  • None of it was ever found anywhere by any of them.
  • Nowhere did any of them ever find any of it.

All four mean NOT(one of them found some of it somewhere at some point).

(Note that some of the above versions are awkward; I list them only to demonstrate the grammar and meaning.)

Your example is similar: it means NOT(one of {L,S} lives in one of {A,B}). This idea can be phrased in various ways:

  • Neither L nor S lives in either A or B. (your version)
  • In neither A nor B does either L or S live.
  • Neither A nor B is where either L or S lives.

In all cases, to correctly interpret the any or either–or or ever or whatnot, you have notice the negation earlier in the sentence.

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Let's start with pointing out that mathematics and formal logic are two things that do not apply to natural language in the sense that one can not follow clear rules to simply translate a statement in a natural language into some logical construct following simple rules. One has to understand the semantics of the natural language, and often, context is everything.

A simple example is the question would you like coffee or tea? which looks to a mathematician like a simple yes/no-question. Try answering yes to your colleague next time you are asked this question...

That being said, let's see what your sentence translates to:

Neither L nor S lives in either A or B

The first part Neither L nor S means that whatever follows does not apply to L and does not apply to S:

NOT (S OR L) = (NOT S) AND (NOT L)

The second part in either A or B means that whatever went before applies equally to A and B.

(P is TRUE for A) AND (P is TRUE for B)

So the proposition that X lives in Y, applied to this sentence, means:

((NOT S) lives in A) AND ((NOT L) lives in A)
AND
((NOT S) lives in B) AND ((NOT L) lives in B)

So your friend is right: S does not live in A, S does not live in B, L does not live in A and L does not live in B.

What I think is "One(L or S) cannot live in the places that has been mentioned(i,e A or B) which makes the other one to live in A or B"

Nothing in the given sentence justifies such a conclusion. If I tell you that Alice does not live in New York, how would that possibly imply that Bob does live in New York? Nobody ever mentioned that either L or S should live in A or B.

It looks like you fell for a false dichotomy. The fact that one thing is not true does not make an unrelated thing true.

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  • Hey sorry! The question is not appropriate."Nothing in the given sentence justifies such a conclusion. If I tell you that Alice does not live in New York, how would that possibly imply that Bob does live in New York? Nobody ever mentioned that either L or S should live in A or B." Thanks for this. I now understand it to a point. See This sentence was part of Logical reasoning question(Circular seating arrangements). So as you said no such conclusion was passed, is there a probability where "L can live in B or A" if "S doesn't live in B or A"? – Naruto of Konaha Dec 18 '17 at 12:44
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    @NarutoofKonaha In the statement you provided, L does not live in A, nor does L live in B. S also does not live in A, nor does S live in B. Both S and L don't live in both A and B. There is no reading where with L or S can live in A or B, no matter what place the other lives in (and neither of then can live in A or B). – Doc Dec 18 '17 at 18:12
  • I feel compelled to link to tvtropes.org/pmwiki/pmwiki.php/Main/MathematiciansAnswer - this is because I enjoy eating hours of everyones' time who reads this. :p – neminem Dec 19 '17 at 0:29
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Yes, "Neither Nor" and "Either or" can appear in the same sentence.

Your friend is correct.

The Neither L nor S prefix means that the rest of the sentence does not apply to (i.e. is not true for) L and also does not apply to S.

Therefore lives in either A or B is not true for L and is also not true for S.

lives in either A or B means lives in A or lives in B.

Therefore, L doesn't live in A, L doesn't live in B, S doesn't live A and S doesn't live in B.

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  • Hey mate, I agree to what you said "The Neither L nor S prefix means that the rest of the sentence does not apply to (i.e. is not true for) L and also does not apply to S." I might be wrong, Just to clear my doubt entirely I am saying this. I believe that Neither nullifies everything on only one side of a statement (i.e (before "is" or "=") or (after "is" or "=")). So am i wrong again :/? – Naruto of Konaha Dec 18 '17 at 10:25
  • @NarutoofKonaha I'm not sure what you mean by "Neither nullifies everything on only one side of a statement". First of all, the correct word is "negates", not "nullifies". Now, what do you consider to be the two sides of the statement in your example? – Eran Dec 18 '17 at 10:29
  • I was confused! What I was going to comment was " I think That neither is only valid before "OR" or after "OR" and not the both ways(As "OR" means One will always complement the other). is there any way you can explain your answer more elaborately? – Naruto of Konaha Dec 18 '17 at 10:46

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