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S Two the sense is "exactly two students ", which is non-monotonic. So monotonic NPs are special cases of continuous ones. But many NPs are not continuous. Typical examples are disjunctions of increasing with decreasing NPs either fewer than six students or else more than ten students or disjunctions of properly continuous ones either exactly two dogs or exactly four cats.
Also NPs like more male than female students are not continuous. Thus in analogy with the distinction between lexical vs complex NPs we also see that there are functions denotable by complex Dets which are not denotable by lexical ones, examples being the functions denotable by more male than female and either fewer than ten or else more than a hundred.
Some further monotonicity generalizations are given by the ways the monotonicity of complex NPs and Dets depends on that of the expressions they are built from: 21 a. Conjunctions and disjunctions of increasing decreasing NPs are increasing decreasing. The corresponding claims hold for Dets. Thus John and every student is increasing since both conjuncts are.
Negation reverses monotonicity. Thus not[more than six cats] is decreasing since the NP more than six cats is increasing. And not more than six, as in at least two and not more than six, builds decreasing NPs since more than six builds increasing ones. The monotonicity value of partitives is determined by the Det preceding of. Thus less than five of the students is decreasing since less than five builds decreasing NPs.
Possessive Dets, e. Thus no student's builds decreasing NPs no student's doctor since no student is decreasing. A more surprising monotonicity generalization concerns negative polarity items npi's.
To characterize the set of English expressions judged grammatical by native speakers, we must distinguish 22a and 23a from their ungrammatical counterparts 22b and 23b. John hasn't ever been to Moscow b.
But observe: 24 a. No student here has ever been to Moscow b. Neither John nor Mary saw any birds on the walk b. None of John's students has ever been to Moscow b. But the pairs differ with respect to their initial NPs, not the presence vs. A syntactic attempt to kill both birds with one stone is to say that just as n't is a "reduced" form of not so neither The presence of n- in the reduced forms is thus explained as a remnant of the original not.
So on this view the licensing NPs above "really" have a not in their representation, and that is what such NPs have in common with n't.
Moreover NPs built from not do license npi's: 27 a. Not a single student here has ever been to Moscow b. Not more than five students here have ever been to Moscow But Ladusaw, this solution is insufficiently general: The initial NPs in the a- sentences below license npi's; those in the b- sentences do not.
But neither present reduced forms of not. Fewer than five students here have ever been to Moscow b. At most four students here have ever been to Moscow b.
Less than half the students here have ever been to Moscow b. Clearly the NPs in 24 S 28 which license npi's are decreasing, and those which do not are not.
Also, drawing on 21d and 21c we see that the LFG yields correct results for 30 and 31 , NP types not considered by Ladusaw or Fauconnier. F is increasing iff for all x,y 0 A, if x y then F x F y. That is, if elements x,y of A stand in the order relation in A then their values F x and F y stand in the order relation in B. So F preserves the order.
F is decreasing iff whenever x y in A then F y F x in B. So decreasing functions are ones which reverse the order. Now it is overwhelmingly the case that the sets in which expressions of an arbitrary category denote are domains of partial orders. P1s denote elements of P E , the collection of subsets of E, and the ordering relation is just the subset relation, f.
One verifies that f is reflexive, antisymmetric and transitive. Again one verifies directly that this relation is a partial order relation. And one sees from our earlier definition 15 that increasing NPs are just the ones which denote order preserving maps from the P1 order into the implication order.
And from 16 we see that decreasing NPs are just the ones that reverse the order relations. So is classical logical negation which maps T to F and F to T. In this way then we find a semantic property that "negative" NPs and negation have in common: They all denote decreasing functions. For more extended and refined discussion see Zwarts and Nam to appear. We turn now to a second type of generalization. It serves to limit the domain of objects we use the predicate to say something about.
This simple idea of domain restriction is captured in the literature with two independent constraints: conservativity and extension. So for B and C possibly quite different predicate denotations, if it happens that the B's who are A's are the same individuals as the C's who are A's then Det A's are B's and Det A's are C's must have the same truth value. The converse holds as well. We show that D satisfies the condition in Det doctor is a vegetarian b. Det doctor is both a doctor and a vegetarian So e.
John's is CONS since John's doctor is a vegetarian is logically equivalent to John's doctor is both a doctor and a vegetarian. The apparent triviality of such equivalences suggests, wrongly, that CONS is a very weak condition. So conservativity rules out most ways we might associate properties with NP denotations. Thus it makes sense to ask whether interrogative Dets such as Which? An affirmative answer would imply that e.
Which students are vegetarians? And this seems to be the case S a true answer to either of these questions is a true answer to the other. So interrogative Which? Moreover 33 is actually just the special case of conservativity for Det1 denotations. In other words if two predicate properties B and C have the same intersection with each noun property then the Det takes the same value at the noun properties and the first predicate property as it does at the noun properties and the second predicate property.
Usually for ease of reading in what follows we just give definitions for the case of Det1 denotations, assuming the appropriate generalization to Detk's. The intuition is that Dets cannot make crucial reference to objects which fail to have the property expressed by their noun arguments. One might have thought that this was covered by Conservativity but in fact it is not.
The problem here is that once E is given a condition on ESA, the non-As in E, can always be expressed as a condition on As, and conservativity allows that we place conditions on the noun arguments A. So we now we think of a Det denotation D as a functional which chooses for each universe E a function DE from PE to the generalized quantifiers over E.
It says in effect that the head noun determines the relevant universe for purposes of the statement we are concerned with. Worth noting here is that mathematical languages such as those used in Elementary Arithmetic, Euclidean Geometry or Set Theory are special purpose in that the range of things we can talk about is fixed in advance numbers, points and line, sets.
But natural languages are general purpose S speakers use them to talk about anything they want, and common nouns in English provide the means to delimit "on line" what speakers talk about and quantify over. But Dets differ greatly among themselves with regard to how much of this information they use. Below we distinguish several subclasses together with linguistic generalizations based on these classes. Since SOME is corresponds to the existential quantifier in logic we shall sometimes refer to intersective Dets as generalized existential Dets.
We define: 38 a. Proportionality Dets like most, less than half the and over ten per cent of the are an important case in point. To decide whether most students read the Times we must be able to compare the students who do with those who don't. Just knowing which students read the Times is insufficient to decide whether most do; and just knowing which students don't is also insufficient.
Formally, 39 a. The proportionality Ds are the non-trivial boolean compounds of basic proportional ones. From 39b we see that exactly a third of the is proportional since it denotes the same function as at least a third and not more than a third. The non-triviality condition rules out ALL and NO expressible for example as more than half and less than half. These remarks lay to rest the issue about whether natural language semantics can be represented in first order logic: it can't.
We consider now the co- intersective Dets in more detail. So the properly proportional Dets do not have a monopoly on being non-first order definable.
Note also that Det2s like more To say that it is CARD just says that it depends on the cardinalities of these intersections. For further discussion of such Det2s see Keenan and Moss and Beghelli , Are there co- intersective Dets in English that are not co- cardinal? In both cases alternative analyses are certainly possible see section 3. But interpreting them as Dets they clearly fall into the co- intersective class: 40 a.
So the value of NO Here is a first, non-obvious one. This formulation eliminates the restriction to swans in favor of quantifying over all the objects in the universe, and it preserves logical equivalence with the original by replacing the original predicate property BLACK with an appropriate boolean compound of the noun property and the predicate property, SWAN 1 BLACK. Thus some does not make essential use of the domain restriction imposed by the noun argument.
The same equivalence obtains if we replace some by e. Exactly two swans are black is logically equivalent to Exactly two objects are both swans and are black. We shall say then that quantifiers like some and exactly two are sortally reducible meaning that we can eliminate the restriction on the domain of quantification compensating by building a new predicate property as some boolean compound in our examples it was boolean compounds with and of the original noun property and the original predicate property.
D is called inherently sortal if there is no such h. Query: Which English Dets are sortally reducible, and which are inherently sortal? Thus like some, all is sortally reducible. By contrast most is not. Most swans are black provably has no paraphrase of the form For most x Swan x Black x Now in fact Keenan provides a complete answer to the query: 43 A conservative D from PE to GQE is sortally reducible iff D is intersective or D is co-intersective We turn now to the role of intersectivity in providing an answer to a problem which arises in the context of generative grammar.
Consider: 44 There wasn't more than one student at the party Are there more dogs than cats in the garden? There was no one but John in the building at the time Weren't there more male than female students at the party?
Such Ss, called Existential There ET sentences, are used to affirm, deny or query the existence of objects e. NPs like more than one student which naturally occur in such Ss will be called existential NPs.
So the NPs italicized in 45 are not existential, as the Ss are either ungrammatical or assigned an unusual interpretation. In 46 we present a somewhat different solution, one that draws on theirs and on Keenan a. See ter Meulen and Reuland for extensive discussion of the empirically problematic issues here. The clause on boolean compounds correctly predicts the acceptability of Ss like There were about five dogs and more than ten cats in the kennel. Equally one proves that boolean compounds of intersective Dets are intersective, whence 46 predicts good There were neither exactly two nor exactly four cats on the mat.
Thus we have a linguistic property which correlates reasonably well with intersectivity. No corresponding property correlating with co-intersectivity is known. Namely, for each universe E, CONSE the set of conservative functions over E is exactly the functions which are buildable by boolean operations on the intersective and co- intersective functions.
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