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Many articles say the following:

Statement:

"She is going to school."

Negation statement:

(1)

"She is not going to school."

or

(2)

"It is not the case that she is going to school."


However, I think only (2) is the correct negation statement because (1) is only one of the scenarios (2) implies. For example, the case that another student is going to school also belongs to the (2). And there are many cases belonging to (2) such as the case that I, who is not she, is going to school.

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    In Mathematical logic the term NOT specifically means your number 2 context & no other. You are confusing normal English for Mathematical terminology which does not 100 percent convert from one to another. – Logikal Feb 27 at 15:27
  • Yes, I mean (1) and (2) are very different. The negation of (1) can not be "She is going to school." However the negation of (2) can be. So technically, speaking of logic, only "it is not the case that..." is the way to get a negation statement, right? – vincentlin Feb 27 at 15:54
  • Both can be seen as expressing the same idea in a specific context. Again if the topic of discussion is specific to mathematics then only 2 is allowed period. So you mixing common English within the same scope of math is going to lead to confusion at times. Be clear of the topic of domain so the same words cannot have more than one context. – Logikal Feb 27 at 16:04
  • "It is not the case that she is going to school" is the same as "It is not true that she is going to school" i.e. "It is false that she is going to school." – Mauro ALLEGRANZA Feb 27 at 16:24
  • It is unclear what "belongs to" means logically, if it means "consistent with" then another student going to school "belongs to" (1), (2) and their negations just as well. You can draw a distinction between a negated statement like ~G (1) and a meta-statement like ~T(G) (2), where T is a truth predicate, but (1) and (2) are still logically equivalent. What you are talking about would apply if (1) was "she is skipping school" (she could be absent due to sickness instead), but not under the usual broad interpretation of "not going". – Conifold Feb 27 at 22:42
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  • Sometimes in natural language, negation is understood as the asertion of the contrary. " He's not a theft" could be interpreted as " he's an honest person, a person endowed with moral integrity".

  • So, in that sense, " it is not the case that" prevents us from over-interpreting the negation operator. " He's not a theft" simply says it's not the case he is a theft; but this is not tentamount to asserting that he is honest ; a non-theft can perfectly be, say, a drug dealer.

  • So , though (1) and (2) mean the same thing, it might be a good thing when one learns logic to get used to the " it's not the case" phrasing, in order not to commit the confusion I mentionned above.

  • I think you're wrong when you say that (2) implies assertions such as " another student is going to school".

  • It seems to me that your question has someting to do with pragmatics and the way one can use negation to implicate something that is not expressed literally by a sentence.

Example : "John is not a liar" ( meaning, in some contexts, in virtue of pragmatic rules : "you are a liar")

Note : about pragmatics, see Paul Grice

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  • In my view, (1) and the original statement are mutually exclusive, while (2) and the original statement are collectively exhaustive. However, a negation statement and its original statement just need to have opposite truth values, and both (1) and (2) can do that, so both of them can be a negation statement of "She is going to school". Is my reasoning correct? – vincentlin Feb 27 at 18:49
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The difference between (1) and (2) is subtle. If we say "she is not going to school," we are naming "she" as the subject of the sentence, and this suggests that "she" exists.

"It is not the case that she is going to school" is more neutral on the question of whether "she" exists, and it is for this reason that (2) is the proper negation of "she is going to school."

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