Quantifiers

Sentences (Ss) such as All cats are grey consist of a predicate are grey and a noun phrase (NP) all cats, itself consisting of a noun cats and a determiner (Det), of which the quantifiers all, some, and no are special cases. Semantically we treat both the noun and the predicate as denoting properties of individuals, and we interpret the S as True in a situation s if the individuals with those properties (in s) stand in the relation expressed by the quantifier. Different quantifiers typically denote different relations. ALL (we write denotations in upper case) says that the individuals that have the noun property (CAT) are included in those with the predicate property (GREY). SOME says that the individuals with the CAT property overlap with those that are GREY; NO says there is no overlap. EXACTLY TEN says the overlap has exactly ten members. MOST, in the sense of MORE THAN HALF, expresses a proportion: The overlap between CAT and GREY is larger than that between CAT and NON-GREY; that is, the number of grey cats is larger than the number of non-grey ones. LESS THAN HALF makes the opposite claim.

The role of the noun is crucial. Syntactically it forms an NP constituent (all cats) with the quantifier. We interpret this NP as a function, called a generalized quantifier, that maps the predicate property to True or False (in a situation). So we interpret All cats are grey by (ALL CAT)(GREY), where ALL CAT is a function mapping a property P to True in a situation s if the cats in s are a subset of the objects with P in s. More generally, Ss of the form [[Det+N]+Predicate] denote the truth value given by (D(N))(P), where P is the property denoted by the predicate, N that denoted by the noun, D the denotation of the Det, and D(N) the denotation of Det+noun NP.

Semantically the noun property N serves to restrict the class of things we are quantifying over. The literature captures this intuition with two very general constraints on possible Det denotations. These constraints limit both the logical expressive power of natural languages and the hypotheses children need consider in learning the meanings of the Dets in their language (Clark 1996; see SEMANTICS, ACQUISITION OF).

One condition is extensions (Van Benthem 1984), which says in effect that the truth of Det Ns are Ps cannot depend on which individuals are non-Ns. For example, scouring Old English texts, you will never stumble upon a Det blik that guarantees that Blik cats are grey is true if and only if the number of non-cats that are grey is ten.

The second condition is conservativity (Keenan 1981; Barwise and Cooper 1981; Higginbotham and May 1981; Keenan and Stavi 1986), which says that the truth of Det Ns are Ps cannot depend on Ps that lack N. So Det Ns are Ps must have the same truth value as Det Ns are Ns that are Ps. For instance, Most cats are grey is equivalent to Most cats are cats that are grey. And most can be replaced by any Det, including syntactically complex ones: most but not all, every child's, or more male than female. Despite appearances this semantic equivalence is not trivial. Keenan and Stavi show that in a situation with only two individuals there are 65,536 logically possible Det denotations (functions from pairs of properties to truth values). Only 512 of them are conservative!

The restricting role of the noun property distinguishes natural languages from first-order LOGIC (FOL). FOL essentially limits its quantifiers to (x) every object and (x) some object and forces logical forms to vary considerably and nonuniformly from the English Ss they represent. All cats are grey becomes "For all objects x, ifx is a cat thenx is grey"; Some cats are grey becomes "For some object x, x is a cat andx is grey." Now proportionality quantifiers (most, less than half, a third of the, ten percent of the) are inherently restricted (Keenan 1993): there is no Boolean compound S of cat(x) and grey(x) such that (for most x)S is True if and only if the grey cats outnumber the non-grey ones. Indeed the proper proportionality Dets are not even definable in FOL (see Barwise and Cooper (1981) for most), whence the logical expressive power of natural languages properly extends that of FOL.

English presents subclasses of Dets of both semantic and syntactic interest. We note two such: First, simplex (= single word) Dets satisfy stronger conditions than conservativity and extension. We say that an NP X is increasing () if and only if X is a P (or X are Ps) and all Ps are Qs entails that X is a Q. Proper names are : If all cats are grey and Felix is a cat, then Felix is grey. An NP of the form [Det+N] is when Det is every, some, both, most, more than half, at least ten, infinitely many, or a possessive Det like some boy's whose possessor NP (some boy) is itself increasing. X is decreasing () if all Ps are Qs and X is a Q entails X is a P. [Det+N] is when Det is no, neither, fewer than ten, less than half, not more than five, at most five or NP's, for NP . X is monotonic if it is either increasing or decreasing. [Det+N] is non-monotonic if Det equals exactly five, between five and ten, all but ten, more male than female, at least two and not more than ten. Simplex Dets build monotonic NPs (usually increasing), a very proper subset of the NPs of English.

Syntactically note that NPs license negative polarity items in the predicate, ones do not (Ladusaw 1983). Thus ever is natural in No/Fewer than five students here have ever been to Pinsk but not in Some/More than five students here have ever been to Pinsk. Often, as here, grammatical properties of NPs are determined by their Dets.

Second, many English Dets are intersective, in that we determine the truth of Det Ns are Ps just by checking which Ns are Ps, ignoring Ns that aren't Ps. Most intersective Dets are cardinal, in that they just depend on how many Ns are Ps. Some is cardinal: whether Some Ns are Ps is decided just by checking that the number of Ns that are Ps is greater than 0. Some other cardinal Dets are no, (not) more than ten, fewer than/exactly/at most ten, between five and ten, about twenty, infinitely many and just finitely many.No . . . but John (as in no student but John) is intersective but not cardinal. All and most are not intersective: if we are just given the set of Ns that are Ps we cannot decide if all or most Ns are Ps.

Intersectivity applies to two-place Dets like more . . . than . . . that combine with two Ns to form an NP, as in More boys than girls were drafted. It is intersective in that the truth of the S is determined once we are given the intersection of the predicate property with each of the noun properties. Other such Dets are fewer . . . than . . . , exactly as many . . . as . . . , more than twice as many . . . as . . . , the same number of . . . as . . . . In general these Dets are also not first-order definable (even on finite domains). Moreover, of syntactic interest, it is the intersective Dets that build NPs that occur naturally in existential There contexts: There weren't exactly ten cats/more cats than dogs in the yard is natural but becomes ungrammatical when exactly ten is replaced by most or all.

Finally, we can isolate the purely "quantitative" or "logical" Dets as those whose denotations are invariant under permutations p of the domain of objects under discussion. So they satisfy D(N)(P)) = D(pN)(pP), where p is a permutation and p(A) is {p(x)|x A}. All, most but not all, just finitely many always denote permutation invariant functions, but no student's, every . . . but John, more male than female don't.

Cognitive and logical complexity increases with Ss built from transitive verbs (P2s) and two NPs. For example Some editor reads every manuscript has two interpretations: One, there is at least one editor who has the property that he reads every manuscript; and two, for each manuscript there is at least one editor who reads it (possibly different editors read different manuscripts). Cognitively, language acquisition studies (Lee 1986; Philip 1992) support that children acquire adult-level competence with such Ss years after they are competent on Ss built from one-place predicates (P1s). And mathematically, whether a sentence is logically true is mechanically decidable in first-order languages with just P1s but loses this property once a single P2 is added (Boolos and Jeffrey 1989).

But some Ss built from P2s and two NPs cannot be adequately represented by iterated application of generalized quantifiers (Keenan 1987, 1992; van Benthem 1989): Different people like different things; No two students answered exactly the same questions; John criticized Bill but no one else criticized anyone else. Adequate intrepretations treat the pair of NPs in each S as a function mapping the binary relation denoted by the P2 to a truth value.

Lastly, quantification can also be expressed outside of Dets and NPs: Students rarely/never/always/often/usually take naps after lunch (Lewis 1975; Heim 1982; de Swart 1996). Bach et al. 1995 contains several articles discussing languages in which non-Det quantification is prominent: Eskimo (Bittner), Mayali (Evans), or possibly even absent: Straits Salish (Jelinek) and Asurini do Trocará (Vieira). Recent overviews of Det type quantification are Keenan (1996) and the more technical Keenan and Westerståhl (1997).

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The formal approach to meaning

-- Edward L. Keenan

References

Bach, E., E. Jelinek, A. Kratzer, and B. Partee, Eds. (1995). Quantification in Natural Languages. Dordrecht: Kluwer.

Barwise, J., and R. Cooper. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy 4:159-219.

Boolos, G., and R. Jeffrey. (1989). Computability and Logic. 3rd ed. New York: Cambridge University Press.

Clark, R. (1996). Learning first order quantifier denotations, an essay in semantic learnability. Technical Reports I.R.C.S. - 96-19, University of Pennsylvania.

de Swart, H. (1996). Quantification over time. In J. van der Does and J. van Eijck 1996, pp. 311-336.

Gärdenfors, P., Ed. (1987). Generalized Quantifier: Linguistic and Logical Approaches. Dordrecht: Reidel.

Heim, I. R. (1982). The Semantics of Definite and Indefinite Noun Phrases. Ph.D. diss., University of Massachusetts, Amherst.

Higginbotham, J., and R. May. (1981). Questions, quantifiers and crossing. The Linguistic Review 1:41-79.

Keenan, E. L. (1981). A boolean approach to semantics. In J. Groenendijk et al., Eds., Formal Methods in the Study of Language. Amsterdam: Math Centre, pp. 343-379.

Keenan, E. L. (1987). Unreducible n-ary quantifiers in natural language. In P. Gärdenfors (1987).

Keenan, E. L. (1992). Beyond the Frege boundary. Linguistics and Philosophy 15:199-221. (Augmented and reprinted in van der Does and van Eijck 1996).

Keenan, E. L. (1993). Natural language, sortal reducibility, and generalized quantifiers. J. Symbolic Logic 58:314-325.

Keenan, E. L. (1996). The semantics of determiners. In S. Lappin, Ed., The Handbook of Contemporary Semantic Theory. Oxford: Blackwell, pp. 41-63.

Keenan, E. L., and J. Stavi. (1986). A semantic characterization of natural language determiners. Linguistics and Philosophy 9:253-326.

Keenan, E. L., and D. Westerståhl. (1997). Generalized quantifiers in linguistics and logic. In J. van Benthem and A. ter Meulen (1997), pp. 837-893.

Ladusaw, W. (1983). Logical form and conditions on grammaticality. Linguistics and Philosophy 6:389-422.

Lee, T. (1986). Acquisition of quantificational scope in Mandarin Chinese. Papers and Reports on Child Language Development No. 25. Stanford University.

Lewis, D. (1975). Adverbs of quantification. In E. L. Keenan, Ed., Formal Semantics of Natural Language. Cambridge: Cambridge University Press, pp. 3-15.

Philip, W. (1992). Distributivity and logical form in the emergence of universal quantification. In Proceedings of the Second Conference on Semantics and Linguistic Theory. Ohio State University Dept. of Linguistics, pp. 327-345.

Szabolcsi, A., Ed. (1997). Ways of Scope Taking. Dordrecht: Kluwer.

Van Benthem, J. (1984). Questions about quantifiers. J. Symbolic Logic 49:443-466.

Van Benthem, J. (1989). Polyadic quantifiers. Linguistics and Philosophy 12:437-465.

Van Benthem, J., and A. ter Meulen, Eds. (1997). The Handbook of Logic and Language. Amsterdam: Elsevier.

Van der Does, J., and J. van Eijck. (1996). Quantifiers, logic, and language. CSLI Lecture Notes. Stanford.

Quantifiers

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