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What is the exact meaning of scope?

In the following sentence, what is the scopal relation of negation and quantifier?

And how could I know if there is a wide or narrow scope between them?

  • She does not focus on some points.
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The sentence formed by combining an element with others is the scope of that element. (Sometimes the element which is said to have a scope is itself excluded from that scope, but including it comes closer to the original account given in Hans Reichenbach's Elements of Symbolic Logic. It doesn't generally matter which policy one follows,)

For instance, in sentence logic, when negation is combined with a sentence to form a negated sentence, the scope of the negation is everything within that negated sentence. Similarly, when a quantifier is combined with a sentence containing a free variable to form a closed sentence, everything within that closed sentence is said to be in the scope of that quantifier.

The term "scope" is often handy in discussing the logic of language expressions, but it does not have an exclusively logical sense. Really, it just gives a convenient way of discussing the sentence structure of a complex expression. For example, in "((If ((John leaves early) or (Mary stays late))), then (Henry will be angry))", where I've parenthesized the sentences, "or" is in the scope of "if ... then", but not vice versa.

Statements about "command" relations (in Langacker's sense) can be formulated using the term "scope", and vice versa. An element A is said to "command" an element B when the smallest sentence containing A also contains B, or equivalently, B is in the scope of A. For instance, a condition on the reflexivization of a pronoun with antecedent NP could be given as either that the pronoun and the NP must command each other or that the pronoun and the NP must be in each other's scope.

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To take the questions in order, ...

Scope comes originally from a Greek word skopoi, 'a target to aim at'. The Greek root skop- is an ablaut variant of the verb σκέπτομαι skeptomai '(I) look at/examine/consider/think' (that's where both scope and skeptic come from). The earliest senses of scope in the OED have that same meaning -- a mark or target to aim at.

In the context of logic, semantics, and syntax, however, scope has no exact meaning, except in a
formal system in which such matters can be stated precisely, like Lambda Calculus.
That is to say, logical scope is a metaphor, and a visual metaphor at that. The question is which operator can "see" the other one. Generally the metaphor talks about one Operator (Modal, Neg, or Quantifier) being "outside the scope" of another, or of one being "higher" than the other.

Note that whichever operator is "higher" or "outside" can "see" the "lower" or "inside" one,
so linear logical notation is suitable for describing the inevitable ambiguity of multiple operators.

  • & She does not focus on some points is ambiguous,
    as between
  • ¬ (∃x: Point (x)) Focus (She, x)
    'There are no points that she focusses on'
    and
  • (∃x: Point (x)) ¬ (Focus (She, x)
    'There are points that she doesn't focus on'

The negative operator not and the quantificational operator some have two possible scopal relationships. That is to say, the sentence can have either meaning. It's called a scope ambiguity.

And it happens because, while it's always possible to be unambiguous, the rules of syntax
aren't the same as the rules of logic, so the Q and the Neg don't always wind up in the
right order, because of syntax. In speech, intonation would disambiguate them;
but in writing, there's no intonation.

Finally, the only way you (i.e, an addressee) can tell whether Q-Neg or Neg-Q order is
intended in a sentence like the example is to figure it out in context. Because it's ambiguous.

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    You started well, but 'There are no points...' is wrong! You could say 'There are not some points...', but that doesn't mean that 'There are not any points...' – amI Jan 20 '16 at 18:13

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