Richard Zuber
email: richard.zuber@linguist.jussieu.fr

Factives and Intensionality

Notion of normal intensionality and of intensionality preserving negation are introduced. Both these notions apply to intensional sentential operators. When applied to factive operators they allows us to understand the way these operators give rise to presuppositions and their duals, assertions. They allow us also to distinguish semantically emotive factives from non-emotive ones.

Introduction

Factives are verbs of propositional attitudes. As is well-known verbs of propositional attitudes and operators formed from them are traditionally considered as posing special problems in formal semantics of natural languages on the one hand and as giving rise to particular questions in the philosophy of language and of knowledge. The difficulties that formal semantician encounters when approaching these verbs and constructions in which they occur are usually related to the fact that these verbs form intensional contexts and thus violate in various ways some general principles accepted in formal semantics [START_REF] Partee | Semantics-Mathematics or Psychology[END_REF]). In particular their strong intensionality, roughly speaking is not easily compatible with a compositional treatment of their semantics [START_REF] Pelletier | The Principle of Semantic Compositionality[END_REF]. In this paper I want to discuss something which, as far as I can tell, not has been explicitly discussed in the context of verbs of propositional attitudes : these are, roughly speaking, specific entailment like relations which hold between sentences (expressions) formed from verbs of propositional attitudes or their arguments and which are caused precisely by the intensionality of these verbs. Thus I will discuss the well-known relation of presupposition of factive verbs and the less discussed, but related relation of assertion. As we will see both these relations are easily obtainable in the context of some sub-set of verbs of propositional attitudes and a specific but natural negation of them.

Two warnings are needed at the outset. First, though I will discuss some semantic properties of sentential operators formed from verbs of propositional attitudes, I will not, strictly speaking discuss the semantics of these verbs themselves. Their semantics has many sides and probably is not separable from various pragmatic aspects. I will discuss those semantic aspects of sentential operators formed from verbs of propositional attitudes which are essential for presuppositions and assertions which can be associated with them. this means in particular that I will treat such operators as syntactically and semantically indecomposable wholes. Thus I will mostly ignore the fact that such verbs take grammatical subjects usually referring to human beings. I will also ignore the fact that such verbs can take quantificational noun phrases as grammatical subjects. This does not seem to be harmful since usually the subject position is considered as not giving rise to intensionality. Consequently I will only analyse sentential operators of the form A V that where A is a proper name and V a verb of propositional attitudes.

The second warning concerns the notion of presupposition. I want to stress that this paper does not offer any general theory of presuppositions. There is a huge literature concerning presuppositions both from theoretical and empirical points of view. An important topic in these considerations concerns the problem of presupposition projection that is the way presuppositions of more complex expressions are obtained from presuppositions of their simpler components. Again, this will not be considered here. Presuppositions that we will discuss are presuppositions of factives only. As will be seen, the way the existence of presuppositions of factives is explained does not easily generalise to other presupposing constructions. The reason is that the notion of normal intensional negation which plays essential role in the understanding of presuppositions of factives does not apply to cases in which intensionality is not involved.

As indicated above, I will discuss various entailments between sentences formed from sentential operators (which are formed from the verbs of propositional attitudes). The notion of entailment is taken in rather an intuitive sense. It takes for granted the notion of a possible world. Thus simply sentence S is said to entail a sentence T iff T is true in every possible world in which S is true. No more sophisticated notion of a logical model will be used.

There are at least three cases of entailments, or more generally of implications, under intensional operators interesting from the semantic point of view. The first is the case of factives predicates presupposing their sentential complements, the second is the entailment between two similar sentences with the same factive predicate in both sentences and differing only by the complementizer (that vs whether ) of the factive. Finally there is a case of "neg-transportation" entailments (implications) to which some intensional predicates give rise. I will only mention this third type of entailment since it does not involve factive verbs. In the next section I recall more specifically these three cases with particular intention paid to the first two types involving factive verbs. Then in the next section I propose a specific notion of negation applicable to verbs of propositional attitudes which allows us to understand both types of entailments to which factives give rise.

Factives

Since the publication of [START_REF] Kiparsky | Progress in Linguistics[END_REF] the term factive refers in linguistic semantics to verbs and predicates like to know, to forget, to regret, be sad, be strange, etc. when they take sentential complements. These predicates have an interesting semantic property : when completed by the grammatical subject referring to a human (which is usually the most natural case for them) they form sentential operators which presuppose the truth of the sentential complement that they take as argument. More specifically (positive forms of) sentences formed from factive predicates and their (natural) negations both entail the complement sentence. Thus (1a) and (1b) (1a) Leo knows/regrets that P (1b) Leo does not know/not regret that P both seem to entail, on the most natural and generally accepted reading, that the sentence P is true.

Although before Kiparsky's article some philosophers also have tried to analyse certain verbs taking sentential complements with the help of the notion of presupposition (cf. Filmore 1963[START_REF] Odegard | On defining 'S knows that p[END_REF], their work was much more complete, empirically rich and gave rise to the renewal of the research on presuppositions. Indeed, sentences with factives represented a new set of presupposing constructions, different from those traditionally studied in the context of definite descriptions for instance. These new constructions showed that, as in the case of existential presuppositions, the notion of negation needed in the definition of presupposition must be different from the classical negation. This is because, as often has been pointed out, the sentence Leo does not know that P can be considered as true in the case when P is false (cf. Leo does not know that P because not-P ).

Kiparsky's distinguish in part by syntactic means, two type of factive predicates: non-emotive and emotive ones. The first class includes verbs like know, remember, realise and to emotive factives belong verbs like regret, resent or predicates like be sad, be strange, be interesting etc, when they take impersonal subjects.

It is not clear to what extent the syntax alone is sufficient to distinguish clearly emotive from non-emotive factive predicates. Some of criteria for such a distinction seem to be language dependent. Let me mention here the differences which are important for the purposes of this article. One difference between those two classes is that, informally, emotive factives seem to be sensitive to more, contingent and non-contingent, sentences than non-emotive ones. Consequently emotive factives presuppose "more" than non-emotive ones. Consider the following examples:

(2a) Leo knows that the bottle is half empty. (2b) Leo knows that the bottle is half full. (3a) Leo regrets that the bottle is half empty. (3b) Leo regrets that the bottle is half full.

Very likely sentences in (2) have both the same truth value whereas those in (3) have different truth values. We express this by saying that Leo regrets that is intensionally stronger than Leo knows that. The notion of the "strength of intensionality" will be made more precise in what follows.

The second difference between emotive and non-emotive factives is that emotive, but not non-emotive, seem to presuppose in addition "knowledge" of the truth of the complement sentence [START_REF] Zuber | Decomposition of factives[END_REF]. For instance (4a) presupposes (4b) and(5a) presupposes (5b) : (4a) Leo regrets that P. (4b) Leo knows that P. (5a) It is strange that P. (5b) It is known that P.

Obviously sentences with non-emotive factives do not have such a presupposition. For instance (4b) does not presuppose itself nor (5b). Neither do these sentences presuppose a belief of any kind by the human beings denoted by the subject noun phrases occurring in them.

Finally, non-emotive factives differ from emotive ones by the fact that only non-emotive can take in addition the complementizer whether. Thus we have know whether and remember whether but not *regret whether or *it is sad whether.

The fact that intensional verbs forming non-emotive factives can take two complementizers (that vs whether ) can be related to another semantic relation to which non-emotive factives give rise. This relation, which can be called assertion (in opposition to presupposition). It can be illustrated by the following examples:

(6a) Leo knows that P. (6b) Leo knows whether P. (7a) Leo does not know that P. (7b) Leo does not know whether P.

As already has been indicated, it is generally accepted, also by logicians, that (6a) entails (6b). Although it has not been often claimed so, it is also true that (7a), taken as presupposing P, entails (7b). Indeed, suppose a contrario that (7a) is true and (7b) is false. The falsity of (7b) means that either (1) Leo knows that P or (2) Leo knows that not-P. This is, however, impossible: (1) cannot be true because (7a) is true and (2) cannot be true because (7a) presupposes the truth of P. Notice that the negation in (7a) and (7b) cannot be interpreted by the classical propositional negation (obviously syntactically it is not) and in particular it must be a presupposition preserving negation in (7a). Thus (8a) can be true and (8b) false in the situation in which Leo knows that not-P is true: (8a) It is not true that Leo knows that P. (8b) Leo knows whether P.

The last case of intensional sentence-embedding predicates giving rise to specific entailments concerns the so-called Neg-raising predicates. As I said I will not discuss them in detail but I only indicate their main property. These predicates have the following property: when negated sentences with such predicates imply a corresponding sentence in which the negation takes scope in the embedded clause. Thus, intuitively (9a) implies (9b):

(9a) Leo does not think that life is sad. (9b) Leo thinks that life is not sad.

Notice that the negation in (8a) cannot be interpreted as just the metalinguistic or classical propositional negation. Indeed it is easy to see that (10) does not imply (9b):

(10) It is not true that Leo thinks that life is sad.

For instance in most cases when Leo is sleeping ( 10) is true and (9b) is false.

Concerning the relation between ( 9a) and (9b) one could also ask whether these two sentences are equivalent: should one consider that (9b) also implies (9a). We will not discuss this possibility since it does not concern factive operators.

We can thus see that in the all three cases of entailments (or implications) presented above we need a special negation. The purpose of this paper is to show that we have the same negation in the case of presupposition of factives and in the case of assertion of factives..

Intensionality preserving negation

Obviously entailments and implications concerning presuppositions, assertions and neg-transportation essentially involve a negation. Moreover the cases we are considering involve intensional functional expressions (intensional sentential operators). This means that defining the negation we have to take into account precisely the fact that it has to apply to intensional functional expressions. So first we have to precise the notion of intensionality of such expressions and then define the appropriate negation of them.

Intensionality of functional expressions that we will use is characterised by the fact that the Leibniz' law does not hold in their context. Since we are basically interested in intensional operators taking sentences as arguments, the failure of Leibniz' law just means that the substitution of sentential arguments by logically equivalent sentences does not preserve truth-value. Since we are interested in sentential operators formed from verbs of propositional attitudes which take as grammatical subjects NPs referring to human beings it is preferable to relativise he notion of intensionality to a possible world. More precisely we will adopt the following definition: Definition 1. A sentential operator O (an expression of the category S/S) is normally intensional iff for some sentence P O(P ) is true in some world w and for every sentence P and every possible world w, if O(P ) is true in w, then there exists a sentence P ′ such that P and P ′ have the same truth-value in w and O(P ′ ) is false in w.

Few remarks about the definition. First, as already indicated, we will treat syntactically complex expressions consisting of a grammatical subject and a verb of propositional attitude as operators having syntactically non complex form. This maybe be a simplification. Second, as we will see, definition D1 does not make directly the difference between the intensionality of classical modal operators and the intensionality of operators formed from verbs of propositional attitudes. Indeed we know that modal operators display weaker intensionality than the propositional attitude verbs (Creswell 1975). In addition they do not give rise to semantic relations discussed above. In particular the necessity operator does not presuppose, but only entails, the truth of its argument. One could make the indicated difference by imposing the condition of contingency on sentences used in substitution; none of the sentences which substitute the other can be a necessary sentence. We will see that it is also possible to make the difference by the appropriate notion of negation of normally intensional operators.

Finally, it follows from definition, 1 that for a sentential operator to be normally intensional it must be possible to show that the appropriate substitution never preserves truth-value (and not only in some cases as it is the case for "classical" intensionality). This feature can be related to the fact that intensional operators we consider here are strictly speaking syntactically complex being composed of a subject NP and a verb phrase. The subject NP usually refers, directly or indirectly, to human beings, which are non omniscient and sensitive to different ways of presentation of the same semantic content. Different logically equivalent sentences can be considered as different ways of presentation of the same (logical) content.

One can check that the sentential operators formed with verbs of propositional attitudes (not only factive predicates) are normally intensional. This is true not only of predicates whose subject NPs refers to human beings but also of predicates whose subject NP is the impersonal pronoun it. The classical modal operator of necessity is also normally intensional. the possibility operator is not normally intensional.

Although we will not be interested in (Booleanly) complex intensional operators nor in the problem of calculating presuppositions and assertions of complex sentences notice the following simple facts concerning some compositions of normally intensional operators: We want now to define a natural negation of intensional operators. We observe that some of such intensional operators can be considered as being related to other intensional operators by a specific lexical negation making them contrary to each other. This is the case for instance with believe that vs doubt that or remember that vs forget that. Notice that both members of these pairs are normally intensional. So we can suppose that there not only lexical but also a syntactic (and semantic) way to negate normally intensional operators. This can be done by the normal negation n whose semantics is defined as follows:

Definition 2. Let O be a normally intensional operator. Then n-O is the normal negation of O iff for any sentence P , n-O(P ) has the same truth value as ¬(O(P )) and n-O is a normally intensional operator (where ¬ is the classical negation, and, since O is of the category S/S, n is of the category (S/S)/(S/S)).

Thus n does the job of ordinary negation, but in addition it preserves the intensionality of the operator to which is applies.

Obviously normal intensional negation n entails the "ordinary" extensional negation but not the other way around. In addition the ordinary negation ¬ does not preserve the normal intensionality. We can show this with the following example. Let O be a factive operator Leo knows that (which is normally intensional) and consider the complex propositional operator ¬O: It is not true that Leo knows that. Clearly, if P is false ¬O(P ) is true. However since P is false there is no sentence P 1 equivalent to P such that ¬O(P 1 ) would be false. Thus ¬O is not normally intensional even though O is.

Notice also that definition 2 does not guarantee that all normally intensional operators have a normal negation. For instance the modal operator of necessity does not have a normal negation.

The importance of the normal negation is shown in the following proposition which indicates that normal negation is precisely the negation which is used when factives give rise to presuppositions: Proposition 2. If O is a normally intensional operator such that for any P , O(P ) entails P , then n-O(P ) also entails P .

Proof. [START_REF] Zuber | Note on why factives cannot assert what their complement sentences express[END_REF]: Suppose a contrario that there exists a world w such that n-O(P ) is true in w and P is false. This means, given definition 2, that there exists P ′ with the same truth value as P in w and such that O(P ′ ) is true in w. But this is impossible since O(P ′ ) entails P ′ .

⊓ ⊔

Thus factives are verbs of propositional attitudes which, roughly speaking, etail the sentential complement to which they apply. Consequently we are justified in calling factives verbs of propositional attitudes which entail their sentential argument because their normal negation also entails it.

As indicated there are two types of factives, emotive and non-emotive ones. Examples like (2) and (3) above, and like the following ones, suggest that emotive factives distinguish a "stronger semantic equivalence" than the equivalence distinguished by non-emotive factives:

(11a) Leo knows that Bill bought a car from Sam. (11b) Leo knows that Sam sold a car to Bill.

(12a) Leo regrets that Bill bought a car from Sam. (12b) Leo regrets that Sam sold a car to Bill.

For our purposes we can assume, at least for some possible world w, that (11a) and (11b) have the same truth value (in w) whereas (12a) and (12b) differ in their truth value (in w). This supposition means that the emotive factive operators (Leo regrets that) are in some sense intensionally stronger than the non-emotive ones.

It follows from the above examples that A regrets that is intensionally stronger than A knows that in the sense that is is sensitive to more arguments. Similarly it has been noticed (Creswell 1975) that verbs of propositional attitude are more strongly intensional than for instance the classical modal operator of necessity. On way to show this difference is to point out that intensionality of verbs of propositional attitudes can be tested by a pair of materially equivalent sentences such that both members of the pair are contingent sentences. This is not the case with the operator of necessity: we can show that this operator is intensional only using a pair of equivalent sentences such that one of the sentences is contingent and the other one non-contingent (necessarily true). So our definition of the strength of intensionality has to take into account these facts. Furthermore we want also that if O is normally intensional then O and n-O should have the same intensional strenght. These observations lead to a series of a new notions. First we define a detector of intensionality: Definition 3. The set of two sentences P, P ′ is a detector of intensionality of the operator O (in the world w), iff P and P ′ have the same truth value in w, but O(P ) and O(P ′ ) have different truth-values in w.

Obviously extensional propositional operators do not have detectors of intensionality (in no possible world). Furthermore, any detector of intensionality of the modal operator of necessity contains a necessarily true sentence as a member. Normally intensional operators formed from verbs of propositional attitudes can have detectors of intensionality which contain only sentences.

With the help of the detector of intensionality we define the notion of the degree of intensionality of a given sentential operator, which needs not to be normally intensional: For instance the degree of intensionality of an extensional sentential operator (in any possible world) is the empty set.

We can now precise how to compare the intensional strength of two sentential operators: Notice that since the set-inclusion constitutes a partial order there are normally intensional operators which cannot be compared from the point of view of their strength of intensionality.

Concerning normally intensional operators and their normal negation the following fact is true: Proposition 3. For any possible world w and any normally intensional operator O the following equality holds: DI(w, O) = DI(w, n-O).

Thus normal intensional negation preserves the degree of intensionality of a given intensional operator. This means in particular that normal intensional negation is not intensional operator. In fact in some sense it is even "stronger" than the clasical extensional negation. This observation will be used when discussing some properties of assertions.

Definitions 2 and 5 allow us to prove the following proposition which accounts for the presuppositions of emotive factives [START_REF] Zuber | Some universal constraints on the semantic content of complex sentences[END_REF]): Proof. Suppose a contrario that there exists a world w such that n-O(P ) is true in w and O(P ) is false (in w). Since O is intensionally stronger than O ′ this means, given definition 5 and proposition 3, that there exist sentences P 1 and P 2 which are detectors of intensionality of O but are not detectors of intensionality of O ′ (in w). Consider now the following two sets of sentences {P, P 1 } and }P, P 2 } as possible detectors of intensionality of O and O ′ (in w). Suppose first that {P, P 1 } ∈ DI(O, w). It follows from this supposition that {P, P 1 } / ∈ DI(O ′ , w) and thus that O ′ (P 1 ) is false in w. This is impossible, however, because O(P 1 ) is true in w (since n-O(P ) is true in w). Suppose now that {P, P 1 } / ∈ DI(O, w). It is easy to show that in this case {P, P 2 } ∈ DI(O, w) and {P, P 2 } / ∈ DI(O ′ , w). But this is again impossible because in this case O(P 2 ) would be true and O ′ (P 2 ) false in w. This a contrario supposition leads to a contradiction.

⊓ ⊔

Notice that proposition 4 is a generalisation of proposition 2, if we suppose that any sentence P is equivalent to T (P ), where T is the classical truth operator.

The reason is that T can be considered, according to definition 5 as intensionally weaker than any normally intensional operator. Concerning the relation of assertion one can show that it is related to the identical intensionality strength of sentential operators. More precisely the following proposition is true [START_REF] Zuber | Some universal constraints on the semantic content of complex sentences[END_REF] Proof. Suppose a contrario that there exists a world w such that n-O(P ) is true in w and n-O ′ (P ) is false (in w). Since n-O is normally intensional there is a sentence P ′ which forms with P a detector of intensionality of O and n-O in w. Given the hypothesis of the proposition the set {P, P 1 } is also the detector of intensionality of O ′ and n-O ′ . But then O(P 1 ) would be true and O ′ (P 1 ) false in w which is impossible.

⊓ ⊔

Proposition 5 can be illustrated by examples in ( 6) and ( 7): since (6a) entails (6b) and Leo knows that and Leo knows whether have the same intensional strength, we have to conclude, given proposition 5, that (7a), the intensional negation of (6a) entails (7b), the intensional negation of (6b). Thus we can say, roughly, that know that asserts know whether. And this is true of many other non-emotive factives.

Conclusions

A negation preserving intensionality applying to verbs of propositional attitudes have been introduced in order to explain the mechanism producing in particular presuppositions of factive prdicates. Though the idea of such a negation is intuitivcely clear its precise rendering, I must say, is not obvious. It is quite possible that another non-equivalent definition of intensional negation, in which partial functions are used, is necessary. The machinery proposed for the explication of factive presuppositions have also been used to justify semantically the distinction between emotive and nonemotive factives. It is well-know that there is not a general agfreement concerning the data to which give rise emotive factive. I considered, following Zuber 1977, that they presuppose knowledge. Some researches consdider that they presuppose only belief. Still some others conside that they are not even veridical (for the review of diferent stands see [START_REF] Egré | Question-Embedding and Factivity[END_REF]. It seems to me that one can reach the final conclusion in these matters after considering other syntactic and semantic factors (cf. [START_REF] Egré | Question-Embedding and Factivity[END_REF][START_REF] Zuber | Some universal constraints on the semantic content of complex sentences[END_REF].

The above analyses cannot be generalised to implicative verbs [START_REF] Karttunen | Implicative verbs[END_REF]. The reason is that they are not normally intensional (at the predicate level). More specifically it is not true that given an implicative verb (to manage, to forget, etc.) and a predicate to which it applies, one can always find another co-extensional predicate such that the substitution of the one predicate by the other one will (systematically) lead to the change of the extension of the whole complex (modified by the implicative verb) predicate. In fact it is often very difficult to decide whether different VPs have the same denotation. The reason is that that VPs with implicative modifiers probably denote events. [START_REF] Sharvit | Trying to be Progressive: the Extensionality of Try[END_REF] shows that the verb try lexically related to many implicative modifiers is not intensional (though not exactly in the sense used here). And indeed implicatives give rise to different implications than factives. This is in particular true with negated implicatives.

Proposition 1 .

 1 Let O 1 and O 2 be two normally intensional operators such that O 1 = ¬O 2 (where ¬ is the classical negation), T the classical truth operator and N the classical propositional negation. Then: (i) O 1 ∧ O 2 and T ∧ O 1 are normally intensional. (ii) O 1 ∨ O 2 and N ∨ O 1 are not normally intensional.

Definition 4 .

 4 Degree of intensionality of the sentential operator O in the possible world w, noted DI(w, O), is the set of the detectors of intensionality of O in w.

Definition 5 .

 5 The sentential operator O is intensionally stronger than the intensional operator O ′ iff for any possible world w the set DI(w, O ′ ) is strictly included in the set DI(w, O). If these sets are equal then O and O ′ have the same intensional strength.

Proposition 4 .

 4 If O is a sentential operator which is intensionally stronger than O ′ and for any P , O(P ) entails O ′ (P ) then n-O(P ) entails O ′ (P ).

  ): Proposition 5. If O and O ′ have the same intensional strength and O(P ) entails O(P ′ ) then n-O(P ) entails n-O ′ (P ).


R Zuber
email: richard.zuber@linguist.jussieu.fr

Some generalised comparative determiners

Functions denoted by specific comparative expressions called generalised comparative determiners are analysed. These expressions form verb arguments when applied to common nouns. They denote functions which take sets and a binary relation as argument and give a set as result. These functions are thus different from denotations of "ordinary" determiners. However, they do obey some similar constraints, properly generalised. It is shown that verbal arguments obtained from such generalised determiners extend the expressive power of NLs since functions that they denote are not just case extensions of type 1 quantifiers used to interpret "ordinary" determiner phrases.

Introduction

The basic meaning of the notion of determiner as established in modern formal linguistics is that of an expression which when applied to one (or more) common nouns (CNs) forms a noun phrase (NP) or rather a determiner phrase (DP). NPs are (syntactic) arguments of verb phrases and thus they can occur in various argumental positions, for instance as grammatical subjects, grammatical objects or various oblique objects. When a DP occurs in the direct object position (of a simple sentence of the form NP TVP DP) one can consider that the determiner which forms it denotes a function which takes a set (the denotation of the CN) and a binary relation (denotation of the TVP) and gives a set (the denotation of the TVP+DP) as result. In this paper I will discuss some functions of this type in cases where they are denoted not by "ordinary" determiners but by what can be called generalised determiners, that is expressions which take one or more common nouns as arguments and which give a verbal argument as result.

There are verbal arguments which cannot occur in every argumental position and thus are not strictly speaking NPs. Consider (1), ( 2) and (3):

(1) Leo washed himself.

(2) Leo washed himself and Lea.

(3) a. *Himself washed Leo.

b. *Himself and Lea washed them/themselves/Leo

In (1) himself occurs in direct object position and is thus similar to a an ordinary noun phrase. In addition it can be conjoined with an ordinary NP to make a Boolean complex NP which can occur in the direct object position, as shown in (2). However, neither himself alone nor its Boolean composition with an ordinary NP can occur in the subject position, as seen in ( 3a) and (3b).

The case of himself and of its Boolean compound indicated above is well known: it illustrates (nominal) anaphora. What is less well-known is the fact that there are expressions forming nominal anaphors in a way similar to the one in which "ordinary" DPs are formed: they can be formed by an application of an anaphoric determiner to a CN. To see this consider the following examples:

(4) Leo shaved every philosopher except himself.

(5) Lea admires most philosophers, including herself .

In (4) the DP-like expression every philosopher except himself and in (5) the DP-like expression most philosophers, including herself are (complex) nominal anaphors. What is important for us is that they are formed by the application of (complex) determiner-like expressions every..., except himself and most..., including herself, respectively, to the common noun philosophers. These expressions are called anaphoric determiners and, obviously, anaphoric determiners are generalised determiners.

It is possible to mention many differences between functions denoted by ordinary determiners and functions denoted by anaphoric determiners (Zuber 2010a). In particular it is useful to consider that anaphoric determiners denote functions from sets (denotations of common nouns) to functions from binary relations (denotations of transitive verbs) to sets (denotations of verb phrases). Or, equivalently, anaphoric determiners denote functions which take a set (or sets) and a binary relation as argument and give a set as result.

Some languages, for instance Scandinavian, Slavic or Latin, have possessive anaphoric determiners which are morphologically simple, in contraposition to those given in (4) and (5). Slavic type anaphoric determiners and various properties of functions denoted by them are studied in Zuber 2010b and Zuber 2010c.

In his paper I am going to study another sub-class of generalised determiners, which I will call comparative determiners. I will in particular show that functions denoted by comparative determiners, though of the same type as functions denoted by anaphoric determiners, have different formal properties.

As an example of a comparative (generalised) determiner consider (6):

(6) Leo knows more philosophers than Lea.

The expression more...than Lea in (6) takes the common noun philosophers as argument and makes an expression which can be considered as "syntactic" argument of the verb know. In addition this argument cannot occur in the subject position. Consequently more...than Lea is a generalised determiner.

The following technical and notational preliminary will be useful. Since we are interested basically in comparatives and proportional quantifiers, both of which are naturally interpreted only in finite universes, we will assume that our universe of discourse E is finite. Thus all sets (of individuals) considered are sub-sets of E. For any set A, |A| is the cardinality of A and for any binary relation R the set aR is defined as follows: aR = {x : a, x ∈ R}.

Since the functions I will discuss are ofen related to quantifiers let me recall some basic notions concerning generalised quantifiers. Functions from sets (subsets of E) to truth-values are type 1 quantifiers. Functions from pairs of sets to truth-values or binary relations between sets are type 1, 1 quantifiers. They are denotations of unary determiners. Type 1, 1, 1 quantifiers are ternary relations between sets. They are denotations of binary determiners. In fact for some syntactic reasons, one can distinguish two sub-classes of type 1, 1, 1 quantifiers: (1) quantifiers whose type is noted 1, 1 1 and (2) quantifiers whose type is noted 1 1, 1 . The first class corresponds to denotations of binary determiners which take two nominal arguments (as in More students than teachers danced ) and the second class corresponds to denotations of binary determiners which take two predicative arguments (as in More students danced than sang).

We will also use the following notation for functions from sets or relations to sets. A type 1, 2 : 1 function is a function having a set and a binary relation as argument and giving a set as result. A type 1, 1, 2 : 1 function is a function which takes two sets and one binary relation as arguments and gives a set as a result. Such functions are denoted by unary and binary generalised determiners.

We are interested in the interpretation of sentences of the form N P T V P GDP where T V P is a transitive verb phrase and GDP is a generalised determiner phrase (an expression obtained by the application of a generalised determiner to a common noun). In such sentences N P is interpreted by a type 1 quantifier, which is a set of sets, and T V P s is interpreted by a binary relation. Concerning GDP there are two possibilities: if it is an "ordinary" DP , it is interpreted by a function which is an accusative extension of a type 1 quantifier or, if it is not an DP it is interpreted by a function (from binary relations to sets) which is not an accusative extension of a type 1 quantifier. An accusative extension Q acc of a type 1 quantifier Q is defined as follows [START_REF] Keenan | On Semantics and the Binding Theory[END_REF]:

Definition 1. For each type 1 quantifier Q, Q acc (R) = {a : Q(aR) = 1}.
Thus the accusative extension of a quantifier is a function from binary relations to sets induced by the quantifier in the way indicated in Definition 1. Accusative extensions of quantifiers permit one to compute directly denotations of verb phrases formed from transitive verb phrases and a noun phrase in the position of the direct object.

Accusative extensions of type 1 quantifiers are specific type 2 : 1 functions. They are specific because they satisfy the following accusative extension condition AE (Keenan and Westerstahl 1997):

Definition 2 (AE). A type 2 : 1 function F satisfies AE iff for R and S binary relations, and a, b ∈ E, if aR = bS then a ∈ F (R) iff b ∈ F (S). Not all type 2 : 1 functions satisfy AE. For instance the function SELF defined as SELF (R) = {x : x, x ∈ R} which interprets the reflexive pronoun (as it occurs in (1) for instance) does not satisfy AEC. Similarly the type 2 : 1 function F (R) = M ORE l,P h (R) = {x : |xR ∩ P h| > |lR ∩ P h|} (where l is the individual referred to by Lea and P h is the set of philosophers), which is denoted by the verbal argument more philosophers than Lea in (6), does not satisfy AE.

The two functions above satisfy conditions weaker than AE. The function SELF satisfies the following anaphor condition AC: Definition 3 (AC). A type 2 : 1 function F satisfies the anaphor condition iff for R and S binary relations, and a ∈ E, if aR = aS then a ∈ F (R) iff a ∈ F (S).

The AC condition, sometimes called predicate invariance (Keenan and Westerstahl 1997), is obviously weaker than AE. The function M ORE l,P h above satisfies another weakening of AE, the so-called argument invariance condition AI:

Definition 4 (AI). A type 2 : 1 function F is argument invariant iff for any binary relation R and a, b ∈ E, if aR = bR then a ∈ F (R) iff b ∈ F (R).
The conditions AE, AC and AI concern type 2 : 1 functions, considered here as being denoted by "full" verbal arguments. We are interested in denotations of generalised determiners, that is expressions forming verbal arguments when applied to common nouns. When such determiners are unary, that is when they apply to one common noun, they denote type 1, 2 : 1 functions. The accusative extension condition for such functions is as follows:

Definition 5 (D1AE). A type 1, 2 : 1 function F satisfies D1AE iff for R and S binary relations, X ⊆ E and a, b

∈ E, if aR ∩ X = bS ∩ X then a ∈ F (X, R) iff b ∈ F (X, S).
Denotations of ordinary determiners occurring in DPs which take direct object position satisfy D1AE. More precisely if D is a type 1, 1 quantifier, then the function F (X, R) = D(X) acc (R) satisfies D1AE. Denotations of anaphoric determiners do not satisfy D1AE. For instance the function F (X, R) = {y : X ∩ yR = {y}} denoted by the anaphoric determiner no... except himself/herself does not satisfy D1AE.

Anaphoric functions satisfy the following condition (Zuber 2010c):

Definition 6 (D1AC). A type 1, 2 : 1 function F satisfies D1AC ( anaphor condition for unary determiners) iff for R and S binary relations X ⊆ E, and

a ∈ E, if aR ∩ X = aS ∩ X then a ∈ F (X, R) iff a ∈ F (X, S).
In this article we are interested in functions denoted by comparative generalised determiners. As we will see they do not satisfy D1AC. The condition which characterises such functions is as follows:

Definition 7 (D1AI). A type 1, 2 : 1 function F satisfies D1AI ( argument invariance for unary determiners) iff for any binary relation R,

X ⊆ E and a, b ∈ E, if aR ∩ X = bR ∩ X then a ∈ F (R) iff b ∈ F (R).
The following property gives a justification of condition D1AI:

Proposition 1. If the function F of type 1, 2 : 1 satisfies D1AI then the function G A of type 2 : 1 defined as G A (R) = F (A, R) satisfies AI.
What Proposition 1 informally says is that functions satisfying D1AI are those from which we get functions satisfying AI when fixing their set argument.

Functions from binary relations and sets to sets which satisfy D1AI have the following obvious property:

Proposition 2. If a function F from binary relations to sets satisfies D1AI then for any X, Y ⊆ E one has F (X, E × Y ) = ∅ or F (X, E × Y ) = E.
The conditions specified above are related to the fact that "proper" anaphors, anaphoric determiners or comparative generalised determiners cannot occur in subject position. We have seen that such expressions are close, in some sense, to expressions denoting various quantifiers. As we will see functions denoted by generalised comparative determiners, in addition to property of argument invariance satisfy a natural generalisation of the conservativity property characteristic of "ordinary" determiners.

Conservativity and related properties (intersectivity, etc) are properties of quantifiers. It is possible, however to naturally generalise this notion to functions of the type studied here. We have the following definition (Zuber 2010a):

Definition 8. A function F of type 1, 2 : 1 is conservative iff F (X, R) = F (X, (E × X) ∩ R)
The following property gives plausibility to the above definition of generalised conservativity [START_REF] Zuber | Semantics of Slavic anaphoric possessive determiners[END_REF]):

Proposition 3. Let D be a type 1, 1 quantifier and F a type 1, 2 : 1 function defined as:

F (X, R) = D(X) acc (R). Then F is conservative iff D is conservative.
It is easy to check that the anaphoric function denoted by the anaphoric determiner no... except himself and the comparative function denoted by the generalised determiner more...than Lea (both mentioned above) are conservative. As is well established conservative quantifiers have various important subclasses, such as classes of intersective, cardinal, etc. quantifiers (cf. Keenan and Westerstahl 1997). It is also possible to generalise the notions of intersective, co-intersective and cardinal quantifiers in such a way that they apply to type 1, 2 : 1 functions. Thus we have the following definitions (Zuber 2010a):

Definition 9. A type 1, 2 : 1 function is intersective (resp. co-intersective) iff F (X 1 , R 1 ) = F (X 2 , R 2 ) whenever (E × X 1 ) ∩ R 1 = (E × X 2 ) ∩ R 2 (resp. (E × X 1 ) ∩ R ′ 1 = (E × X 2 ) ∩ R ′ 2 ).
The following proposition, similar to Proposition 1, can be considered as justifying the above definition: Proposition 4. Let D be a type 1, 1 quantifier and F a type 1, 2 : 1 function defined as:

F (X, R) = D(X) acc (R). Then F is intersective (resp. co-intersective) iff D is intersective (resp. co-intersective).
It is easy to see that both functions mentioned above are intersective. Concerning co-intersective functions it is easy to show that the function EV ERY (X)-BU T -SELF (R) defined in ( 7), and denoted by the anaphoric determiner every... but himself, is co-intersective:

EVERY(X)-BUT-SELF(R) = {x : X ∩ xR ′ = {x}} (7)
It is also possible to generalise other sub-properties of conservativity. Consider so-called cardinal quantifiers. A type 1, 1 quantifier

F is cardinal iff F (X 1 )(Y 1 ) = F (X 2 )(Y 2 ) whenever |X 1 ∩ Y 1 | = |X 2 ∩ Y 2 |; numerals denote car- dinal quantifiers.
Type 1, 2 : 1 cardinal functions are defined as follows:

Definition 10. A type 1, 2 : 1 function is cardinal iff F (X 1 , R 1 ) = F (X 2 , R 2 ) whenever ∀y(|X 1 ∩ yR 1 | = |X 2 ∩ yR 2 |)
Obviously, cardinal, intersective and cardinal functions are conservative. In the next section we will discuss various examples of comparative generalised determiners which denote functions having one of the properties defined in definitions 2, 3 and 4.

Some unary comparative determiners

In this section I will basically discuss numerical comparative (and superlative) generalised determiners, that is, semantically, quantifiers and more generally functions applying to sets and relations involving (numerical) comparisons of cardinalities of various sets. In that way the results will be clear and at the same time directly extensible to the case of adjectival comparison. However, I will discuss adjectival comparative (superlative) constructions only briefly since they give rise to generalised determiners only in a special case.

Let me first clarify the distinction between what have been sometimes called comparative quantifiers and comparative functions in which we are interested here and which are not comparative quantifiers (or their case extensions).

Numerical comparisons make it clear that various entities can be compared in comparative constructions. To show this, let me present two numerical comparative constructions in semantics of which binary (comparative) quantifiers but not comparative type 1, 2 : 1 functions are involved. In (8) we have comparative constructions involving what are now known as binary determiners (Keenan and Mos 1985[START_REF] Beghelli | Structured Quantifiers[END_REF][START_REF] Zuber | A semantic constraint on binary determiners[END_REF]):

(8) a. More students than priests dance b. More students dance than talk

In (8a) we have a binary determiner denoting a type 1, 1 1 quantifier and in (8b) a binary determiner denoting a type 1 1, 1 quantifier. Their semantics is given in ( 9) and ( 10) respectively:

MORE(S)THAN(P )(D) = |S ∩ D| > |P ∩ D| (9) MORE(S)(D)THAN(T ) = |S ∩ D| > |S ∩ T | (10)
We know that DPs formed with binary determiners occur in object position:

(11) Leo met more students than priests.

One can say that in (11) we have a binary generalised determiner which denotes a type 1, 1, 2 : 1 function. But clearly this is not the case of a genuine generalised determiner: the DP it forms can occur in subject position (as in (8a), the function it denotes is just the accusative extension of the function given in (9a). Later on we will see that there are genuine generalised determiners which denote type 1, 1, 2 : 1 functions.

Consider example (12), similar to the one in ( 6) used above to illustrate the case of genuine generalised determiners:

(12) Leo met more students than Lea.

When comparing ( 12) with (11) we see that different things are compared in these sentences: in (11) we compare properties that one individual has whereas in (12) we compare individuals which share one (non trivial) property.

In (12) we have a genuine generalised determiner, more...than Lea. It is genuine because the function it denotes is not an accusative extension of a type 1, 1 quantifier. Informally this can be seen by the following reasoning in which condition D1AE is used. Suppose the set of Xs that Bill met is the same as the set of Xs that Leo met. It does not follow from this that the sentence Bill met more Xs than Lea has the same truth value as the sentence Leo met more Xs than Lea.

The function which is denoted by the generalised determiner more...than Lea is given in ( 13):

(13) F (X, R) = M ORE l (X, R) = {y : |yR ∩ X| > |lR ∩ X|}, where l is the individual referred to by Lea
It is obvious that the function in (13) satisfies the argument invariance for unary determiners. Indeed, suppose that (14a) holds. Then clearly (14b) holds as well:

(14)a. aR ∩ X = bR ∩ X b. a ∈ M ORE l (X, R) iff b ∈ M ORE l (X, R)
It is also easy to show that function M ORE l is conservative. We have to show that M ORE l (X, R) = M ORE l (X, (E × X) ∩ R). But this is obvious given the semantics in ( 12) and the fact that a((E × X) ∩ R) = X ∩ aR.

In fact, the function M ORE l has a stronger property than conservativity: it is cardinal (in the sense of definition 4 above). To show this we have to prove that (15) holds if (16) holds:

(15) M ORE l (X 1 , R 1 ) = M ORE l (X 2 , R 2 ) (16) ∀y(|X 1 ∩ yR 1 | = |X 2 ∩ yR 2 |). .
We have the following chain of equivalent statements:

x ∈ M ORE l (X 1 , R 1 ) iff (given (12)) |xR 1 ∩ X 1 | > |lR 1 ∩ X 1 | iff (given (16)) |xR 2 ∩ X 2 | > |lR 2 ∩ X 2 | iff x ∈ M ORE l (X 2 , R 2 ). Thus M ORE l is a cardinal type 1, 2 : 1 function.
The above example suggests various questions concerning the number, the variety of patterns and the specificity of properties of generalised comparative determiners and of the functions they denote. For instance one would like to know, roughly speaking, whether there are many other patterns of generalised comparative determiners which denote cardinal functions, whether there are generalised determiners which denote intersective non cardinal functions, or which denote conservative non intersective functions, etc. of course such questions cannot be fully answered here and, in addition some of them should be made more precise. In what follows I will try to give partial answers to some of these questions.

Observe first that in the above example we can replace the expression more by less or the same number (of ). Obviously the generalised determiners thus obtained will still denote cardinal functions.

We can still make generalised determiners more complex by adding explicitly numerals to the determiners discussed above. Consider the example in (17). It is probably ambiguous with the readings indicated in (18a) and (18b):

(17) Leo read 5 more books than Lea. (18)a. Leo read exactly 5 more books than Lea.

b. Leo read at least 5 more books than Lea.

In the above examples we have the generalised comparative determiners 5 more... than Lea and exactly 5 more... than Lea. They denote type 1, 2 : 1 functions which are instances of functions given in (19a) and (19b) respectively:

(19)a. EXACT LY -M ORE n,i (X, R) = {x : |xR ∩ X| = |iR ∩ X| + n} b. AT -LEAST -M ORE n,i (X, R) = {x : |xR ∩ X| > |iR ∩ X| + n}
It is easy to show that functions in (19a) and (19b) are cardinal and satisfy condition D1AI. Moreover, the above examples show that there is an infinite number of generalised determiners denoting cardinal functions.

It is still possible to make comparative generalised determiners more complex by making Boolean compounds of them. Such Boolean complex determiners are given in the following examples:

(20) Leo read more books than Lea but less than 17. (21) Leo read more books than Lea and more than Bill. ( 22) Leo read less books than Lea but more than Bill.

Again, functions involved in the semantics of above examples are cardinal functions. This follows from the observation that functions discussed here form Boolean algebras, the fact that I will not comment in more detail.

Recall that cardinal functions are intersective ones. One may wonder whether there are generalised comparative determiners denoting intersective non cardinal functions. I think that a good example of such a determiner is the same...as Lea as it occurs in (23). The function denoted by this determiner is an instance of the function given in ( 24):

(23) Leo read the same books as Lea.

(24

) T HE-SAM E i (X, R) = {y : yR ∩ X = iR ∩ X}.
The function in ( 24) is intersective. We have to show that (26) follows from (25). This is true because (25) entails the two equalities given in ( 27):

(25) (E × X 1 ) ∩ R 1 = (E × X 2 ) ∩ R 2 (26) {y : yR 1 ∩ X 1 = iR 1 ∩ X 1 } iff {y : yR 2 ∩ X 2 = iR 2 ∩ X 2 } (27) (i) yR 1 ∩ X 1 = yR 2 ∩ X 2 , (ii) iR 1 ∩ X 1 = iR 2 ∩ X 2
We get the needed result by replacing in (26) the equal parts indicated in (27).

It is also easy to show that function T HE-SAM E i given in (24) does not satisfy D1AE and that it satisfies D1AI.

In quite the same way one can show that the generalised determiner different... than Lea, as it occurs in (28), denotes an intersective comparative function:

(28) Leo knows different languages than Lea.

One can consider that ( 28) is ambiguous with possible readings in which the set of languages known by Leo and by Lea are either "just" different or they are disjoint. These two readings can be distinguished by two different functions DIF F EREN T i indicated in (29a) and (29b) respectively:

(29)a. DIF F EREN T i (X, R) = {y : yR ∩ X = iR ∩ X} b. DIF F EREN T i (X, R) = {y : yR ∩ X ∩ iR∩ = ∅}
Both functions, the one in (29a) and the one in (29b), are intersective.

All the examples of generalised determiners presented above contain a complementizer than or as. We discussed basically the cases when these complementizers are followed by a proper name. In fact this is not necessary: they can be followed by virtually any NP. Thus we can have generalised determiners like more... than most students, the same... as some philosophers, different ... than the ten teachers, etc. The presentation of semantics of such determiners necessitates the generalisation of the notation aR to the notation Q nom (R), where Q is a type 1 quantifier, a denotation of an NP. Q nom (R), the nominative extension of Q is defined as follows:

(30) Q nom (R) = {x : Q(Rx) = 1}, where Rx = {y : y, x ∈ R}
The nominal extension of a type 1 quantifier is different from its accusative extension. The difference can be illustrated by the following example. Let Q = M OST (S). Then M OST (S) nom (R) is the set of objects to which most Ss are in the relation R whereas M OST (S) acc (R) is the set of objects which are in the relation R to most Ss.

To see the usefulness of the notation Q nom (R) consider the following example in (31). The generalised determiner the same... as most students occurring in (31) denotes the function T HE-SAM E M OST (S) given in ( 32) :

(31) Leo knows the same languages as most students.

(32

) T HE-SAM E M OST (S) (X, R) = {y : yR ∩ X = M OST (S) nom (R) ∩ X}
The extension of the complements of than (or of as) enriches, though somewhat trivially, all the classes of generalised comparative determiners we have considered. There is, however one class of comparative functions formally defined in the preceding section, which do not seem to be denoted in natural languages: these are co-intersective functions.

Some superlative constructions also give rise to generalised comparative determiners. It is enough for our purpose to consider that superlatives correspond to specific conjunctions of comparatives: roughly the oldest man is the man who is older than m 1 , and older than m 2 ... and older than m n . So roughly speaking, to get a superlative we form a conjunction in which conjuncts are the complements of then. We will indicate implicitly such a conjunction by the expression than anybody else or anything else. Consider now (33a), its semantics in (33b) and its superlative counterpart in ( 34 By applying the above idea of conjunction of comparatives we get ( 35):

(35) M OST (T )(D) = 1 iff ∀X(X ∩ D = ∅) → |T ∩ D| > |T ∩ X|
The equivalences in ( 36) and ( 37) are easy to be proved. From them and (35) follow the two equivalences in ( 38):

(36) For all sets X and Y

, |X∩Y | > |X ′ ∩Y | iff ∀Z(X∩Z = ∅) → |X∩Y | > |Z∩Y | (37) For any set X, Y , |X| = |X ∩ Y | + |X ∩ Y ′ | (38) M OST (T )(D) = 1 iff |T ∩ D| > |T ∩ D ′ | iff 2 × |T ∩ D| > |T |
Thus the superlative associated with the comparative construction in which a binary determiner occurs is the "classical" determiner most (of ) denoting M OST . It is not a genuine generalised determiner. However, a "generalised superlative determiner" can be associated with the comparative construction in (39). Its superlative counterpart is given in (40a) and its denotation is given in (40b):

(39) Leo read more books than Lea. (40)a. Leo read more books than anybody else.

b.

N SU P (X, R) = {x : ∀y(y = x → |xR ∩ X| > |yR ∩ X|}
The function N SU P is denoted by the following generalised determiners, supposedly equivalent, the most, more than anybody else or the greatest number of. Interestingly, this function is not anaphoric: using fact 2, one shows that it does not satisfy D1AC. Informally, suppose that Leo studies precisely those languages that he knows. It does not follow from this that Leo studies the greatest number of languages is equivalent to Leo knows the greatest number of languages.

To conclude this section I give an example of a non-numerical comparative and superlative constructions which contain a generalised determiner. Consider (41) which contains the determiner an older... than Lea: Sentence (41) has two readings: absolute, in (42a) and relative, in (42b):

(41) Leo hugged an older woman than Lea. (42)a. Leo hugged a woman older than Lea.

b. Leo hugged a woman that was older than a woman hugged by Lea.

When the determiner is used in a DP in subject position, as in ( 43), the relative reading disappears:

(43) An older woman than Lea was dancing.

Any gradable adjective introduces a (total) relation; in the above case we have the relation O corresponding to be older than. Consequently in the semantics of (41) two type 1, 2 : 1 functions are involved:

(44)a. F O,l (X, R) = {x : ∃y(y ∈ xR ∩ X ∧ y, l ∈ O} b. F O,l (X, R) = {x : ∃y, z(y ∈ xR ∩ X ∧ z ∈ lR ∩ X ∧ y, z ∈ O}
Only the function in (44b) is a genuine comparative type 1, 2 : 1 function.

There may be some problems with the above example since it involves some unicity conditions. The absolute and relative readings are better seen in superlative constructions (the classical paper concerning this problem is Szabolcsi 1986: (45) is ambigous with the two meanings given respectively in (46a), absolute reading, and (46b), relative reading:

(45) Leo hugged the oldest woman (46)a. Leo hugged a woman who was older than any other woman b. Leo hugged a woman who was older than any other woman hugged by anybody else.

Clearly the different readings of the superlatives in (46a) and ( 46b) are related to the different comparatives from which they originate. Thus an explanation of these different readings is easy to conceive along the lines here proposed. In particular one observes that when the superlative occurs in subject position it can have only the absolute reading as in ( 47):

(47) The oldest woman lives in Japan.

The above observation suggests that superlatives with relative readings are not related to an accusative extension of (the denotation of) any noun phrase, and thus in particular the superlative itself cannot be considered as an "ordinary" noun phrase denoting a type 1 quantifier. This is indeed the case since it does not satisfy the AE. We show this informally, just using English examples, by showing that the corresponding "comparative-anaphoric" form of superlative, the one given in (46b) does not satisfy AE. Suppose that (48) holds:

(48) The persons that Leo hugged are the same as those that Bill kissed.

In (48) we have an instance of the conditional part of the AE condition. One observes now that, given (48), the sentence in (49a) needs not hold the same truth value as the one in (49b):

(49)a. Leo hugged a woman who is older than any other woman hugged by anybody else b. Bill kissed a woman who is older than any other woman kissed by anybody else.

On the other hand the function interpreting the absolute reading of the superlative does satisfy the AE condition : from (48) follows the identity of truth values between (50a) and (50b) :

(50)a. Leo hugged a woman who is older than any other woman.

b. Bill kissed a woman who is older than any other woman.

The generalised determiner we have in (45) corresponds to the expression the oldest when it forms the superlative DP which occurs in the object position gives rise to the relative reading. It denotes the function in ( 51):

(51) F O,a (X, R) = {x : ∃y(y ∈ xR ∩ X ∧ ∀z(z = y ∧ z ∈ aR ∩ X) → y, z O}
In the next section I discuss some binary comparative determiners.

Some binary generalised determiners

Recall that natural languages have binary or even n-ary determiners, that is expressions which take two or n common nouns to form a DP. As Keenan and Moss 1985 noted, n-ary determiners can be easily obtained by the conjunctions of common nouns in the "syntactic" scope of an unary determiner:

(52) Most students, teachers and priests were sleeping. This sentence probably means that most students and most teachers and most priests were sleeping and not that most individuals which are students, teachers and priests ("at the same time") were sleeping. Under this reading the determiner most..and...and... is a ternary determiner.

Deterniners taking many common nouns as arguments as illustrated by the above example have an obvious property: their denotations are Booleanly reducibly to a conjunction of denotations of unary determiners (see [START_REF] Keenan | Generalized quantifiers and the expressive power of natural language[END_REF]. It has been observed, however that natural languages have also binary determiners whose denotations are not reducible in that sense (Keenan andMoss 1985, Beghelli 1994). For instance, the quantifier denoted by the binary determiner in (10) above is not Booleanly reducible (cf. [START_REF] Beghelli | Structured Quantifiers[END_REF].

Of course binary determiners occurring in DPs in object position can be considered as generalised binary determiners: if D 2 is a type 1, 1 1 quantifier then the function F ((X 1 , X 2 , R) = D 2 (X 1 , X 2 ) acc (R) is a well-defined type 1, 1, 2 : 1 function. In what follows I indicate, however, that natural languages have genuine generalised binary, or even n-ary, comparative determiners. Though the notion of Boolean reducibility of n-ary determiners will not be made more precise, it will be intuitively clear that such determiners can be either Boolean reducible or Boolean irreducible. Moreover, functions denoted by these determiners have similar properties to the functions denoted by unary determiners: they are "at least" conservative and satisfy the condition of argument invariance.

We need first to define various properties of functions denoted by binary determiners, similar to those which have functions denoted by unary generalised determiners. Such properties are well-defined for "ordinary" binary or n-ary determiners (cf. [START_REF] Keenan | Generalized quantifiers and the expressive power of natural language[END_REF][START_REF] Beghelli | Structured Quantifiers[END_REF][START_REF] Zuber | More Algebras for Determiners[END_REF][START_REF] Zuber | A semantic constraint on binary determiners[END_REF]). I give here some such definitions for type 1, 1, 2 : 1 functions. For conservativity we have the definition11 and the proposition 5 (Zuber 2010a):

Definition 11. A type 1, 1, 2 : 1 function F is conservative iff for any X 1 , X 2 ⊆ E and any binary relations R 1 and R 2 , if E × X 1 ∩ R 1 = E × X 1 ∩ R 2 and E × X 2 ∩ R 1 = E × X 2 ∩ R 2 then F (X 1 , X 2 , R 1 ) = F (X 1 , X 2 , R 2 ). Proposition 5. A type 1, 1, 2 : 1 function F is conservative iff for any X 1 , X 2 ⊆ E and binary relation R one has F (X 1 , X 2 , R) = F (X 1 , X 2 , (E ×(X 1 ∪X 2 ))∩R).
Since many of the determiners we will present denote cardinal functions, here is the corresponding definition:

Definition 12. A type 1, 1, 2 : 1 function is cardinal iff F (X 1 , Y 1 , R 1 ) = F (X 2 Y 2 , R 2 ) whenever ∀y(|X 1 ∩ yR 1 | = |X 2 ∩ yR 2 |) and ∀x(|Y 1 ∩ xR 1 | = |Y 2 ∩ xR 2 |)
Finally, the condition of argument invariance for type 1, 1, 2 : 1 functions is formulated as follows:

Definition 13 (D2AI). A function F of type 1, 1, 2 : 1 satisfies argument invariance condition for binary determiners (D2AI) iff for any a

, b ∈ E, X, Y ⊆ E and R a binary relation, if a((E×X)∩R) = b((E×X)∩R) and a((E×Y )∩R) = b((E × Y ) ∩ R) then a ∈ F (X, Y, R) iff b ∈ F (X, Y, R).
Let us see now some examples of binary generalised determiners which denote conservative argument invariant functions. Consider sentence (53): one of its readings is given in (54a). In (53) we have a binary generalised determiner which denotes an instance of the type 1, 12 : 1 function given in (54b):

(53) Leo read more books and articles than Lea. (54)a. Leo read more books than Lea and more articles than Lea.

b.

M ORE i,j (X, Y, R) = {y : |yR ∩ X > |iR ∩ X| ∧ |yR ∩ Y > |jR ∩ Y |}
It is easy to show that function M ORE i,j is cardinal and thus conservative. It is also argument invariant. The fact that (34a) is equivalent to (34b) qand that we have a conjunction in (35) shows that this function is Booleanly reducible. Furthermore, since the number of CNs which can occur as conjuncts in (34a) is not limited, example (34a) shows how to construct n-ary comparative determiners.

A binary generalised determiner denoting the non-reducible type 1, 1, 2 : 1 function is given in (55a); the corresponding function is given in (55b):

(55)a. Leo read more books than Lea articles.

b. M ORE 2,i (X, Y, R) = {y : |yR ∩ X| > |iR ∩ Y |}
Using the method similar to the one used in connection with the example (24) above one shows that the function in (55b) is cardinal and argument invariant.

We get similar examples be replacing in the above examples more by less or the same number of. Similarly we can make some Boolean compounds as in ( 56), where we also have a generalised binary determiner denoting a cardinal function:

(56) Leo read more books than Lea articles but less than 17 altogether.

Consider finally (57a) and the function in (57b) which is denoted by the generalised determiner three times more... than Lea...: The type 1, 1, 2 : 1 function in (57b) is cardinal.

Conclusive remarks

By analogy with anaphoric determiners I distinguished a subclass of generalised determiners called comparative determiners. A generalised determiner is an expression which when applied to one or more common nouns forms a generalised DP, that is an expression which can serve as argument of a verb phrase. A genuine generalised DP is an expression which cannot serve as the grammatical subject of a sentence. Anaphoric determiners (studied in Zuber 2010a, 2010b) are genuine generalised determiners because when applied to a common noun they form nominal anaphors which are verbal arguments which cannot occur in subject position. Since I was basically interested in the logical properties of comparative determiners, no attempt has been made to justify their category syntactically. From the logical point of view it appears that their denotations have many striking similarities with denotations of "ordinary" determiners: for instance, they are conservative in a naturally generalised sense. Furthermore, as with ordinary determiners, there are not only unary generalised determiners but also n-ary ones. Semantically, comparative DPs, in the same way as nominal anaphor, cannot be interpreted by functions corresponding to generalised (type 1 ) quantifiers denoted by ordinary NPs. Formally this amounts to saying that denotations comparative DPs do not satisfy the specific invariant condition for type 1 quantifiers given in called accusative extension condition. They satisfy, however the strictly weaker condition given of argument invariance. Consequently one can say that comparative (generalised) DPs essentially augment the expressive power of, say, English, since the expressive power of English would be less than it is if the only noun phrases we need were ones interpretable as subjects of main clause intransitive verbs. The reason is that such DPs must be interpreted by functions from relations to sets which lie outside the class of generalised quantifiers as classically defined, that is type 2 : 1 functions which are not extensions of type 1 quantifiers.

It might be interesting to notice the analogy with nominal anaphors. [START_REF] Keenan | Semantic Case Theory[END_REF][START_REF] Keenan | On Semantics and the Binding Theory[END_REF][START_REF] Keenan | On the denotations of anaphors[END_REF] shows that something similar is true because of the existence of nominal anaphors. More specifically, anaphors like himself, herself (considered as the second nominal argument of transitive verbs) also must be interpreted by functions which do not satisfy the AE, and thus the generalised type 1 quantifiers are not enough for their interpretation. Anaphoric functions interpreting nominal anaphors satisfy another weakening of the AE, the condition AC.

  ): (33)a. More teachers danced than sang. b. M ORE(T )(D)T HAN (S) = |T ∩ D| > |T ∩ S| (34) More teachers danced than did anything else.

(

  57)a. Leo read at least three times more books than Lea articles b. M ORE 3×,i (X, Y, R) = {y : |yR ∩ X| ≥ 3 × |iR ∩ Y |}