Default denotations and inflating process

We have explained what we understand by vague quantifiers. Modeling the scalar implicature “not all” for both “most” and “some” requires however a little bit more work. Here we come to the essential point, namely we believe that quanti- fiers havedefault denotations, and the scalar implicature phenomenon (the weak implicature!) is the result of flexibility of those denotations, which can be dy- namically changed in the process of reasoning.

Definition 27. D is a default denotation of Q(A,∗) if it is a restricted set of witness sets such that for each a ∈ D, in(a, D) _≈ 1 (We use _≈ instead of =

since we believe that the default denotation may be vague as well and may include elements which are almost certainlyin the set.)

The crucial issue is that in natural discourse we do not use sentences to some extent. We either use them or not, making a0−1decisions whether or not they are true, depending on the context and dynamics of reasoning. Thus although there is some default meaning of language expressions, this meaning isflexible. The same concerns quantifier denotations which, at least in some cases of fuzzy quantifiers, can be “stretched” or extended. Note that the notion of default denotation makes sense both for vague and sharp quantifiers. In the case of sharp quantifiers such as “all” the default denotation is just inflexible.

What we propose is that thedefault denotations of “some” and “most” do not include the whole domain as a witness set for these quantifiers. More precisely:

• Some(A, B) refers to the intersection of A and B of cardinality m, where

• M ost(A, B) refers to the intersection of A and B of cardinality m, where

m < card(A) and m > card(A)/2.

Moreover, the default denotation of “some” is vague, whereas the default de- notation of “most” is flexible but not vague! This means that even if the use of the quantifier “some” in language suggests that the cardinality of any wit- ness set should be smaller than half of the domain, this implicature is not cer- atin. This is reflected in subjects’ indecisiveness about inferencesSome/M ostnot

(resp. Somenot/M ost), which were accepted in approximately 60% of cases (the frequency distribution was asymmetric and displayed variance even for a single subject). Note that in the case of both quantifiersthe denotation itself is vague, since in both cases it contains except for the default denotation the grey (or “possible”) part, however in the case of “most” there is only one witness set out- side the default denotation, namely the whole domain! We need to observe that vagueness which consists in flexibility of the default denotation differs in nature from what is usually understood by vagueness. We use the same notion because we apply fuzzy semantics for medeling this phenomenon. If one regards the case “all” as a possible witness set for “most”, just lying outside the default denota- tion because of the very close to zero membership ratio, then indeed “most” may be treated as (quantitatively) vague. The difference between this kind of vague- ness and the standard vagueness consists in the big truth-value gap between the default denotation and the remaining greyish part. Usually some regular (even

linear) decrease of degrees to which elements belong to a vague set is assumed. In this case, we rather observe a “sudden decline”. Moreover, ranking the ele- ments from the grey part to the default denotation is a more dynamic process and involves increasing the membership ratios for those elements.

Now it also becomes clear why monotonicity of both “most” and “some” be- comes problematic. Let us explain precisely what was already mentioned in the chapter 1.2.3 how the implicature “not all” (if embedded in semantics of quan- tifiers) forces us to abandon monotonicity. Let us consider that for some A, B,

M ost(A, B) (with ratio 1) and assume B0 s.t. B ⊆ B0 and A ⊆ B0. Then if

M ost is monotone increasing in the second argument, we get M ost(A, B0), but since A⊆B0, then also All(A, B0) and hence, assuming that the default denota- tion of “most” excludes “all”, one gets ¬M ost(A, B0) (or at least the ratio will be very close to 0). Similarly one can show that the vague quantifier “some” is not monotone in the second argument because of its implicature “not all”. The non-monotonicity of “some” in the first argument follows from the vagueness of the default denotation. Assume for some A, B, Some(A, B) to a degree r (i.e.

in((A, B), SomeM) = r), and let A0 s.t. A ⊆ A0, then assuming persistence of “some”, we would have to claim probably that Some(A0, B) to degree r0. One could be tempted to say that r0 = r, but as in this case r depends on the car- dinality of A and of A∩B, then since card(A0) > card(A), we would have to determine how r0 will change in relation to r. We would like to say that r0 > r,

3.5. Attempt of a theory: inflating quantifiers 97 but the mutual dependencies are not that clear, since r can also depend on the qualitative character of A.

Finally, let us explain how the default denotations are stretched in the reason- ing process. The most important observation is that,implicitly, both “most” and “some” do not refer to the whole domain. Thus, these quantifiers implicate “not all” (“somenot”) andare not used if application of the quantifier “all” also results in a true sentence. Hence, a person who is given a sentence with one of these two quantifiers usually interprets this sentence as conveying also the considered implicature. The situation changes when people are asked to evaluate sentences with “some” or “most” as true or not in the light of universal premises. In such cases,All/M ost andAll/Someinferences can be understood in two ways. Asked whether on the basis of “All A’s are B” one can say that “Some (resp. Most) A’s are B”, a subject may understand with or without the silent word “only” in the conclusion. Thus, the conclusion may be taken in the default meaning and hence rejected, or the meaning may be changed to “at least some (most)”, and thus “some” or “most” will be understood without the implicature. The essential point is what cognitive mechanism lies behind the second interpretation. There may be two possible mechanisms:

• suspension of implicature,

• extension of a quantifier.

In the first case the implicature “not all” is suspended or canceled. In the latter we extend the default meaning of a quantifier, so that it embraces the grey part of its denotation. One might ask to what extend these processes are different, and why they are not regarded as the same mechanism. To see this, let us consider the quantifier “most”. One possibility is that when we see (hear) a sentence with “all” we think (tacitly) also “most” – for example because of semantic accretion from voting systems (while voting we usually want at least most to agree for a given proposal, which thus is accepted also when all agree). Then, we may suspend implicature “not all” more easily in the case of “most” than in the case of “some”. Still it does not seem clear enough why that kind of language habit connected with voting systems should have such a strong influence on semantics and hence why this cancelation of implicature should be easier in the case of “most”. The other possibility is that the whole domain is treated as a kind of a “big majority” (thus it belongs to the grey part of the denotation of “most”), so we can extend the default denotation of “most” to the whole domain, whereas the same mechanism in the case of “some” is more far-fetched. Such extending denotations of quantifiers are possible because of their vagueness. The “grey” part of a denotation is like a non-inflated part of an elastic ball. If denotations are vague, they can be “stretched”. The question is – how far.

Let us use an analogy of a two-colored (e.g. red and white) collection of little dots on a computer screen. “Some” refers to a situation where only a small part

of the collection is red, “most” – to one where a bigger part is red and “all” to one where the whole collection is red. Thus a sentence “Some dots are red” assumes that some dots are not red and in most cases also that a bigger part (most) of the collection is not red. We are more hesitant about the latter conclusion because the default denotation of “some” is itself vague. On the other hand, “Most dots are red” means that a majority of the collection is red, but not the whole collection. Now, when we are given a sentence “All dots are red” and we are asked if sentences with “some” resp. “most” hold as well, we can either reject these sentences taking the default meaning of “some” and “most” into account or we canabstract in imagination from the whole picture an appropriate part to see that “some” and “most” can be special cases of “all”. This mechanism of abstraction will probably be more difficult if a picture is presented, but should be quite easy in the case of textually given models. The question is why such “abstraction” is easier in the case of “most” than in the case of “some”. If this is just suspension of implicature, it should be similarly difficult in both cases. That is why we consider that extending of the default denotations of quantifiers (“some” and “most”) plays a crucial role. This would explain why it is quite easy to extend “some” to “most” and “most” to “all”, but it is difficult to extend “some” to “all”. The reason is that the last case requires much more conceptual effort and is much more far-fetched, since the decrease of membership ratios of witness sets of “some” can be observed already when the cardinality of the witness sets exceeds half of the domain.