Scope

Notice that the order in which quantifiers appear can make a difference to the meaning of one's formula. For instance, using the interpretation above, ""x$yLxy" translates "In the case of each thing there is something which it likes" (which leaves it open that one thing might like one thing and another might like another); but "$y"xLxy" translates "There is something such that everything likes this thing".

Consider the sentence:

            Everyone like some cow.

This is ambiguous. It could mean that in the case of each person there is a cow they like. Or it could mean that there is a cow which is the subject of universal affection.

This is another example of ambiguity of scope – in this case the scope is the scope of "Everyone" and of "some". Fortunately our predicate language is not prone to this ambiguity. So we formalise the two ways of taking the English in two different ways. Taking it the first way we have:

            "x[Px®$y[CyÙLxy]]

Taking it the second way we have:

            $y[CyÙ"x[Px®Lxy]].

We will now give a more comprehensive definition of "scope".

The scope of a quantifier or truth-functor in a sentence or predicate is the smallest constituent sentence or predicate in which it occurs.

(Since a sentence is a 0-place predicate, we could, of course, have left out "sentence or".)

So, in the first formula the scope of "$y", for instance, is "$y[CyÙLxy]", whereas in the second it is the whole formula.