...1.1 Generalization of quantifiers
First of all, one can see in the table above that it was composed in the frame of the logic of the 1960s, when the generalized quantifiers were a new and not widely applicable branch of the logical studies. Indeed, the very idea of generalized quantifiers is going back to Andrzej Mostowski’s (1957) seminal paper “On Generalization of Quantifiers”, and, in its present form, to Per Lindström (1966). The application of (Fregean, not generalized) quantifiers to the structures of natural languages is the famous idea of Richard Montague (1970). The recent development in the field was provoked, in a great extent, by needs of natural language studies; see now (Peters, Westerståhl 2006).
First of all, the generalized quantifiers are considered as free from the so-called existential import. This is important from the philosophical viewpoint, too, because, philosophically, the procedure of quantification over the objects has a transparent (Fregean) sense, but the sense of the quantification in modalities is somewhat obscure, or, as Quine (1953) coined this problem, such quantification is “ontologically opaque”: in modalities, it is some states of objects rather than to the objects themselves which are quantified. Quine believed that such quantification is applied to intensionals and that it misses denotates. This difficulty could be avoided in the semantics of possible worlds by means of presupposition that the domain of quantification is limited to the worlds where the appropriate objects do really exist. Dealing with the generalized quantifiers, we do not need, however, any reference to the semantics of possible worlds. Instead, we exempt the quantifiers from any ontological commitment at all.
This condition means that the quantifier “all” (") could be defined on the empty set as well as on a non-empty one. Thus, if an existential import takes place, all (A, B) means that all As are B and there are some As. However, if our all is a generalized quantifier, then, all (A, B) means only that all As are B, without any supposition about the existence of As. Thus, taking an example from a medieval logician (Paul of Venice’s Logica Magna (c. 1400)), the phrase “Some man who is a donkey is not a donkey” is true since the subject term is empty (Peters, Westerståhl 2006, 25).
For our practical purpose, we need take into account only three kinds of the generalized quantifiers. The corresponding types are designated, in Lindström’s terms, as ‹1›, ‹1, 1›, and ‹1, 1, 1›.
The quantifiers of type ‹1› define sets within a given universe, e.g., a quantifier Q defines set A. The quantifiers of types ‹1, 1› and ‹1, 1, 1› define the relations between two or three sets, correspondingly, e.g., Q (A, B) and Q (A, B, C). Everywhere in these designations number “1” means “first order” in a generalized sense. This “first order” is not excluding the functions, as it is the case in the logic of predicates. Instead, it includes any kind of objects whose internal structure is irrelevant, that is, any kind of set considered as lacking any internal structure of relations. Such a generalization of the Fregean notion of quantifier was proposed by Mostowski. Mostowski’s generalization of the notion of quantifier was first applied to the natural language by David Lewis (in his 1975 paper “Adverbs of quantification”), while without knowing neither Mostowski’s nor Lindström’s works. Lewis observed that such adverbs as “often,” “usually,” “seldom” etc. are quantifying over neither time nor events but something called by Lewis “cases.” These “cases” are analogous to Mostowski’s sets and can conclude times, events, and many other things.
Lindström proposed a further step of generalization allowing quantification over relations. Thus, if the quantifier is applicable to the relations between sets, its order is “2”; then, if it is applicable to the relations between relations, its order is “3,” and so on. E.g., quantifier each other is a denotate of the ‹1, 2› type quantifier.
The difference between types ‹1› and ‹1, 1› could be illustrated by difference between two possible denotates of most. The ‹1› denotate of most is the so-called Rescher quantifier QR “more than half of the elements of the universe” (as it was introduced in (Rescher 1962)). The ‹1, 1› denotate is “more than half of” that could be applied to the relation between two different subsets A and B of the universe (Peters, Westerståhl 2006, 62), that is, A is the most (“more than half”) of B, not of the whole corresponding universe.
In contrast with the first-order logic of predicates where the most used quantifiers (especially $ and ") are those of type ‹1›, the natural language is operating mostly by the quantifiers of type ‹1, 1› and, not seldom, of type ‹1, 1, 1›. The quantifiers of type ‹1› determine the denotates of the noun phrases, while the verb phrases’ quantifiers are of type ‹1, 1› or higher.
As to the narratology, the quantifiers in Doležel’s table are all of type ‹1, 1›, Q (A, B). They are defined over relations between two sets in the universe of narrative, where the set B is the set of states appropriate to a given universe, and the set A is the set of the elements of the same universe which are able to change their states according to a given modality. Indeed, modality is, by definition, a kind of relation, and so, its corresponding operators must be of types higher than ‹1›.
1.2 Negative operations on the quantifiers
Already in Doležel’s table the quantifiers are presented via the operations of negation. They are almost self evident in the case of operators of type ‹1› and could be not so evident in other cases.
Thus, the operations of negation on the quantifiers of type ‹1› form the square of this quantifier Q(A), where square (Q) is conjunction of four elements, namely, Q, ¬Q, Q¬, and Qd. The latter three elements mean the following:
¬Q is external negation = complement set of A = M – A, where M is universe,
Q¬ is internal negation = Q(M – A) = Q (complement set of A),
Qd = ¬Q¬ is dual of Q = external negation of internal negation.
Example: the square of the quantifier some is none (external negation), not all (internal negation), all (dual).
For the quantifiers of type ‹1, 1› Q(A, B) the square is basically the same but with another definition of internal negation: Q¬ = Q(A, M – B).
Thus, in our examples from Doležel’s table: Q (A, B) means that the elements of the subset A of the universe of narrative M are possible or permitted or good or known. Its external negation means that, in the set A, everything is either impossible or prohibited or bad or unknown. Its internal negation (Q¬) (A, B) means that the elements of A are, correspondingly, not necessary or not obligatory or not axiologically neutral or “not believed” (in the sense that they are not to be believed, being either known or unknown). Finally, its dual means that the elements of A are either necessary or obligatory or axiologically neutral (neither good nor bad) or believed (epistemically neutral, that is, neither known nor unknown).
It is obviously that the interpretation of the internal negation, when applied to modalities, can be problematic. At first glance, it looks at odd with the fact that a derivation of the internal negation, dual, is one of the fundamental quantifiers in any logical system. We will turn to this problem in the next section.