11.4 Scope rules

Introduction

Nesting gives rise to something called scope rules. Scope refers to the area of a graph in which something holds true, e.g., a universal quantifier, an existential quantifier, or a coreference link.

Definition

That is, a quantifier or a coreference link need not apply to all parts of a graph. Let us say that a quantifier or coreference link is defined in the referent of a certain concept C. Then that quantifier or coreference link only applies to:

  • The concept C.
  • The graphs nested inside the concept C (to any depth).
  • The concepts which are co-nested with the concept C.
  • The graphs which are nested (to any depth) inside the concepts which are co-nested with C.

Example

Consider the following graph:

   [If:
       [If:
           [Rhino:  *x]
           [Then:
               [Mammal: ?x]
           ] 
       ]
       [If:
           [Mammal:  *y]
           [Then:
               [Animal: ?y]
           ]
       ]
       [Then:
           [If:
               [Rhino:  *x]
               [Then:
                   [Animal: ?x]
               ]
           ]
       ]
   ]

Same scope

The defining label "*y" is in the same scope as the bound label "?y". This is because the bound label "?y" is in the referent of a concept ([Animal]) which is nested inside a concept [Then] which is co-nested with the concept ([Mammal]) in which the defining label "*y" occurs.

Likewise, the first occurrence of "*x" is in the same scope as the first occurrence of "?x" because the defining label "*x" occurs in the referent of a concept ([Rhino]) which is co-nested with a concept ([Then]) inside which we find the nested concept ([Mammal]) in whose referent we find the bound label "?x".

Different scope

Because of the scope rules, the first occurrence of "*x" is distinct from the second occurrence of "*x". That is, the two do not refer to the same coreference link. This is the same as saying that they occur in different scopes.

This is because the first "*x" occurs inside a concept ([Rhino]) which is not co-nested with the concept in which the second "*x" occurs.

Instead, there is a so-called export-barrier, namely the concept [If] (first occurrence) outside of which the first defining label "*x" does not have scope.

This amounts to saying that quantifiers and coreference-links have scope over everything in the immediately surrounding context (i.e., over those concepts that are co-nested with the concept in which the quantifier or defining label occurs), but not outside the immediately surrounding context.

Two types of quantifier

There are two types of quantifiers: The universal quantifier, and the existential quantifier. The scope rules apply to both.


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