CHAPTER 7 MODALITY, SCOPE, AND QUANTIFICATION
C. Quantification
8. Propositional Quantifier
Many things including mathematics is about statements that are either true or false. Such statements are called propositions. People use logic to describe them, and proof techniques to prove whether they are true or false and it is as same as linguistic, especially for propositional quantifier.
Meat eaters
Corgis M’
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From the picture above, the corgis in the part of the oval that bulges out on the left in the picture are not included in the set of meat eaters. The bulge is the set of corgis minus the (large) subset of meat eaters amongst them: C – M. the meat-eating corgis, as in the picture, are in the intersection of corgis and meat eaters: C ∩ M. for most corgis are meat-eating is simply that the number of meat-eating corgis │C ∩ M│is greater than the number of corgis who do not eat meat │C - M│. Another way of putting this is to say that more than half of the members of the corgi set are meat eaters.
Two more sentences with proportional quantifiers (italicised) are given in (7.31). The symbol ∈ stands for ‘is a subset of ’. The set-theoreti-cal truth condition for both is that corgis be a subset of meat eaters. It is obvious that these sentences do not show the symmetry of the sentences in (7.29): Every meat eater is a corgi and All meat eaters are corgis are both obviously false, whereas the sentences in (7.31) might, just possibly, be true.
(7.31) a. Every corgi is a meat eater. C ∈ M b. All corgis are meat eaters. C ∈ M
How does this involve comparison between an intersection and a remainder set, as seen with the proportionally quantified sentences in (7.30)?
Think of the corgi oval moving to the right in Figure 7.2 until the C – M remainder, labelled M in the figure, has vanished. That is what it means for C to be a subset of M. If all corgis are meat eaters and if the meat eater set is larger than the corgi set, then corgis would be a proper subset of meat eaters, C M (which, incidentally, is the condition for hyponymy; see Chapter 3).
Though the same set-theoretical specification is given for all and every on the right in (7.31), these two quantifiers are not identical in meaning. Every is a distributive quantifier, so that, for instance, Every corgi at the dog show was worth more than £1,000, would mean that if there were ten of them, the total was over £10,000. All, however, is ambiguous between a collective and distributive reading: if it is true that All the corgis at the show were worth more than £1,000, then that figure could be the value per dog or could be the total for all of them.
Each is another distributive quantifier. Like every and all, it is specified in terms of a subset relationship, C M, just like the quantifiers in (7.31). There is thus more to the meanings of these quantifiers than is covered by their set- theoretical properties.
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Earlier in this chapter, in Table 7.1, three quantifiers were used as components in explanations of modal meaning. These quantifiers are repeated in (7.32), in a more general form than the sentences about corgis, with the modality markers that they contributed to shown in parentheses.
(7.32) a. All As are Bs. A B (must, has to, will, should, ought to)
c. At least one A is a B. 0 < | A∩ B | (may, might) b. No As are Bs. | A∩ B | = 0 (can)
Like negation and markers of modality, quantifiers are operators with scope. The quantifiers considered in this chapter are syntactically located in noun phrases, but they have clauses as their scope. Clauses express propositions; so the quantifiers are propositional operators. When two quantifiers, or a quantifier and negation, are present, there can be differences in meaning attributable to relative scope. Consideration of a few examples will give the flavour of this. Think of the sentences in (7.33) as reports back after a problematic visit to the library. Italics have been used for operators with scope. Informal bracketed indications of relative scope are shown to the right of each sentence, along the same general lines as in Section 7.2.
(7.33) a. None of the books was available. As for every book (not (it was available)) b. All the books were not available. Not (as for every book
(it was available))
or As for every book (not (it was available))
c. Not all of the books were available. Not (as for every book (it was available))
Unlike in the corgis examples, we are not talking here about the full set of books in the world. Almost certainly not about the whole stock of a particular library, but about a contextually recoverable set of books. Use of the before books is a pragmatic indicator to the listener: ‘With a minimal amount of thinking you’ll be able to work out which set of books I am referring to’. Perhaps this is a student talking to the university teacher who
136 recommended a set of six books for reading.
It is easy enough to interpret the combination of quantification and negation in (7.33a), because the two operators are pre-packaged in a single word (none) with the quantifier having wider scope than the nega-tion. The majority of people using sentences like (7.33b) give inton-ational prominence to the quantifier all, with a sharp pitch inflection, perhaps greater loudness and maybe slightly more length to the vowel.
This is a signal in English that an item falls within the scope of negation, as shown in the first of the bracketings for this sentence, which is also the meaning of the unambiguous (7.33c). The meaning can be built up by starting with the operator more deeply buried in brackets: think what it means for all, or every one, of the relevant set of books to be available;
consider the effect of negating that situation; if even one of the books was not available, then that is a situation in which it is not so that ‘all the books are available’. Without intonational prominence on all, (7.33b) in prin- ciple has the alternative meaning shown for it. Looking back to the set- theoretic formulation in 7.32a, this is: ‘the entire set of books in question is a subset of the unavailable ones’. However, this meaning of (7.33b) is not often in play because there is an unambiguous way of conveying it, namely (7.33a).
Pragmatically, listeners are likely to reason that a speaker who said (7.33b) could not have meant what is encoded by (7.33a) because that would then have been the obvious way to say it.
In (7.34) sentences are presented that have two quantifiers (italicised) in them.
(7.34) a. Each student borrowed a book
As for every student (there was a book (the student borrowed it))
b. A student borrowed each of the books.
There was a student (as for every book (the student borrowed it))
or As for every book (there was a student (the student borrowed it)) It is an encoded feature of the meaning of each that it is generally taken to have wider scope when it occurs with another quantifier, so that the bracketing shown to the right of (7.34a) is practically the only scope pattern
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for it: start inside the brackets; think of a student borrowing a book; then think of each student in turn doing that, so there were as many books borrowed as the number of student borrowers. However, (7.34b) illustrates another factor regarding the decoding of relative scope: quantifiers in subject noun phrases tend to have wider scope. This favours the first interpretation: just one student borrowing book after book. In (7.34a) the tendency for the subject’s quantifier to have wide scope reinforces the bias favouring wide scope for each. But in (7.34b) each, in the object noun phrase, is pitted against the quantifier on the subject. Neither tendency wins out conclusively and (7.34b) is ambiguous, with an alternative book- focused meaning: book after book was borrowed, each one by a student, and it is not said how many students did the borrowing.
CONCLUSION
Must, should, can’t and similar expressions encode modality. Markers of modality are interpreted either in relation to the demands and preferences of people, or in relation to evidence. With interpretations of the first kind (called deontic), You must communicates that the speaker demands that you …; You can’t … that the speaker disallows it, and soon. Interpreted in a context where the issue is the sender’s degree of certainty about inferences from evidence (epistemic modality), It must … conveys strong conviction about the likelihood of something being true, It should … that the proposition is expected to be true if things unfold in an average sort of way, and so on.
Necessity and possibility are fundamental concepts in modality, and elucidating them involved consideration of quantifiers, such as all and some (the second topic of the chapter), because for example, what is necessarily true pertains all the time; and what holds some of the time is possible. The chapter also covered relative scope: the interactions between modality markers, negation and quantifiers when more than one of them is involved in the meaning of a proposition.
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139 CHAPTER 8