Consider the sentence
John will pass his exams if he works hard and is he is clever or he is lucky.
There are two ways of taking it. One could take it as saying that he will pass his exams if either he works hard and is clever or he is lucky. And one could take it as saying that he will pass his exams if he works hard and he is clever or lucky. This sort of ambiguity is a variety of structural ambiguity; it is ambiguity of scope. The scope of the "or" could be either "he works hard and is he is clever or he is lucky" or "he is clever or he is lucky"; correspondingly the scope of the "and" could be either "he works hard and is he is clever" or "he works hard and is he is clever or he is lucky".
One of the merits of our new truth-functors is that they are not prone to this sort of ambiguity. Suppose we use the following abbreviations:
and we could translate (formalise) the second way of understanding it as
In each case the translation is unambiguous. This is guaranteed by the presence of the brackets which are a feature of the 2-place truth-functors. If we left them out, we would have in both cases
So it is important to make sure that each of "Ù", "Ú", "®" and "«" is given its pair of brackets.
We can define the scope of a truth-functor for present purposes thus:
The scope of a truth-functor in a sentence is the smallest constituent sentence in which it occurs. (For convenience we count the entire sentence as a constituent of itself.)
So in "[[HÙ[CÚL]]®P]" the scope of "Ú" (strictly speaking of "[jÚy]") is "[CÚL]"; the scope of "Ù" is "[HÙ[CÚL]]"; the scope of "®" is "[[HÙ[CÚL]]®P]".
When we are dealing with the predicate calculus, we shall extend the use of truth-functors so that they can be used with predicates as well as sentences. So we will have to amend our account of scope.