Anaphoric Conservativity

Abstract

The notion of anaphoric conservativity, that is a property of specific functions taking sets and binary relations as arguments is studied. Such functions are denotations of anaphoric (reflexive or reciprocal) determiners forming nominal anaphors. It is shown that anaphoric conservativity is strictly stronger that ordinary conservativity of this type of functions. In consequence some novel semantic descriptions of reflexive and reciprocal pronouns are provided and a semantic universal stating that reflexive and reciprocal non-possessive determiners denote anaphorically conservative functions is proposed.

1 Introduction

The notion of the classical conservativity, that is a specific property of type $\langle 1,1\rangle$ quantifiers denoted by unary determiners, has been extended and generalised in various ways. In particular it appeared necessary for theoretical reasons to extend it in order to apply to quantifiers of other types than just type $\langle 1,1\rangle$. Thus Keenan and Moss (1985) extensively study binary determiners (that is determiners taking two common nouns to form an NP and then taking a VP to form a sentence) and show that the corresponding type $\langle \langle 1,1\rangle 1\rangle$ quantifiers they denote are conservative (in a naturally generalised way). They justify the definition of conservativity for type $\langle \langle 1,1\rangle 1\rangle$ quantifiers (denotations of binary determiners) given in (1):

• (1) A type $\langle \langle 1,1\rangle 1\rangle$ function F is conservative iff $F(X_{1}, X_{2})(Y)=F(X_{1},X_{2})(Y\cap (X_{1}\cup X_{2}))$. The function given in (3) which is denoted by the binary determiner more... than... as used in (2) is conservative in the sense of definition given in (1):

• (2) More students than teachers danced.

• (3) $MORE(X_{1},X_{2})(Y)=1$ iff $|X_{1}\cap Y|>|X_{2}\cap Y|$. Similarly, Beghelli (1994) indicates that it is necessary to introduce the notion of conservativity for determiners taking even more nominal arguments. Finally, extensions of the notion of conservativity have been proposed for quantifiers having not only sets but additionally relations as arguments (Keenan and Westerståhl 1997), Keenan (2018). In particular Keenan (2018) notes that quantifiers taking two sets and one relation as arguments and having truth-value as result, are also conservative. Conservativity of such quantifiers is defined in (4a) and it concerns complex determiners occurring in sentence like the one given in (4b):

• (4a) A type $\langle 1,1,2\rangle$ function F is conservative iff $F(X,Y,R)=F(X,Y, ((X\times Y)\cap R)$

• (4b) Different students answered different questions. Conservativity of the quantifier denoted by the complex determiner different... different in (4b) means that to check whether (4b) holds we do not need to check every pair belonging to ANSWER relation but only those whose first co-ordinates are students and whose second co-ordinates are questions. The important point is that even if the notion of conservativity given in (4a) has been "devised" for specific constructions like the one in (4b), it holds for simple "ordinary" constructions as well. For instance a function involved in a simple transitive sentence with ordinary DPs in subject and object positions is also conservative in the sense of (4a). Thus the equivalence in (5) holds for simple sentences where $D_{1}$ and $D_{2}$ are functions denoted by unary determiners and R is a binary relation denoted by a transitive verb:

• (5) $D_{!}(X)$ R $D_{2}(Y)$ iff $D_{1}(X)$ $(R\cap (X\times Y))$ $D_{2}(Y)$. The second reason for which a generalisations of the notion of conservativity became necessary is empirical. To account for the existence of natural language determiners which are not (classically) conservative, such as only or mostly and in order to preserve the conservativity universal it became necessary to weaken the classical notion of conservativity (Zuber 2019, Zuber and Keenan 2019). As we will see the conservativity proper to anaphors studied in this paper can be non-trivially related to specific properties of determiners like only and mostly. In this article I study some properties of anaphoric conservativity, that is conservativity of some functions taking a binary relation and a set as arguments. As we will see, such functions are denotations of specific anaphoric determiners found in the following examples:

• (6a) Dan criticised every philosopher except himself.

• (6b) Dan shaved seven philosophers, including himself.

• (7a) Bo and Dan admire every philosopher except each other.

• (7b) Bo and Dan admire most philosophers including themselves. One observes that sentences with such anaphoric determiners give rise to specific entailments concerning NP subjects. For instance (6a) and (6b), entail (8a) and (7a) and (7b) entail (8b) but (9) below, in which no anaphor occurs, does not entail (8a):

• (8a) Dan is a philosopher.

• (8b) Bo and Dan are philosophers.

• (9) Bo and Dan admire no philosopher/most/seven philosophers. In addition, (10), but not (11), entails that some monks are philosophers:

• (10) Most monks admire no philosophers except each other and themselves.

• (11) Most monks admire three/most/all philosophers. Finally we observe that (12a) entails (12b) and cannot mean (13):

• (12a) Dan admires seven teachers including two monks.

• (12b) Two monks are teachers.

• (13) Dan admires seven teachers and two monks which are not teachers.

In this paper I study various properties of functions denoted by anaphoric expressions. I provide in part novel semantic description of anaphoric constructions. It will be shown that entailments indicated in (6) - (10) are related to a specific property of functions denoted by anaphoric expressions. This property is a generalisation of the classical conservativity and as such allows us to propose a universal constraint concerning anaphoric determiners. The fact that unary determiners in natural languages denote conservative functions is considered as a semantic universal. In this paper the universal that I propose concerns anaphoric (non-possessive) determiners: it will be suggested that all such determiners denote anaphorically conservative functions.

2 Formal Preliminaries

We will consider binary relations and functions over a universe E, assumed to be finite and containing at least two elements throughout this paper. If R is a binary relation Dn(R) denotes the domain of R.R. The relation I is the identity relation: $I=\{\langle x,y\rangle : x=y\}$. If R is a binary relation and X a set, then $R/X=R\cap (X\times X)$. The binary relation $R^{S}$ is the greatest symmetric relation included in R, that is $R^{S}=R\cap R^{-1}$ and $R^{S^{i}}=R^{S}\cap I'$. If R is an irreflexive symmetric relation (i.e. $R\cap R^{-1}\cap I=\emptyset$) then $\Pi (R)$ is the coarsest partition of R (treated as a set of pairs) such that every of its blocks is of the form $(A\times A)\cap I'$. A partition is trivial iff all its blocks are singletons. For any relation R and any $a\in E$, the set aR is defined as $aR=\{x: x\in E\wedge \langle a,x\rangle \in R\}$.

Usually functions taking relations (including sets) as arguments and giving truth-values as output are called quantifiers. Since functions to be discussed here have relations as arguments but do not give truth-values as outputs I will not call them quantifiers and I will use a notation more transparent for them, I believe, that the usual one. If a function takes only a binary relation as argument, its type is noted $\langle 2:\tau \rangle$, where $\tau$ is the type of the output; if a function takes a set and a binary relation as arguments, its type is noted $\langle 1,2:\tau \rangle$. The variable $\tau$ can take two values: if $\tau =1$ then the output of the function is a set of individuals and if $\tau = \langle 1\rangle$ then the output of the function is a set of sets (of individuals). We will thus consider two types of functions: functions of type $\langle 2:1\rangle$ and functions of type $\langle 1,2:1\rangle$. For instance, the function SELF, denoted by the reflexive himself and defined as $SELF(R)=\{x:\langle x, x\rangle \in R\}$, is of type $\langle 2:1\rangle$ and the function denoted by the anaphoric determiner every...but himself is of type $\langle 1,2:1\rangle$: it takes a set and a binary relation as arguments and gives a set as a result. Similarly, when $\tau$ corresponds to the set of type $\langle 1 \rangle$ quantifiers and thus $\tau$ equals, in Montagovian notation, $\langle \langle \langle e,t\rangle t\rangle t\rangle$ the corresponding functions are of type $\langle 1,2:\tau \rangle$: they take a set and a binary relation as arguments and give a set of type $\langle 1\rangle$ quantifiers as result. For instance, as we will see, the function denoted by the anaphoric determiner no...except each other is of type $\langle 1,2: \langle 1\rangle \rangle$.

Basic type $\langle 1 \rangle$ quantifiers are functions from sets to truth-values, which means that they are sets of sets. As such they have Boolean complements. In addition it is possible to define the post-negation of a type $\langle 1\rangle$ quantifier: if Q is a type $\langle 1\rangle$ quantifier, then $Q\lnot$, the post-negation of Q is defined as: $Q\lnot =\{X: X'\in Q\}$, where $X'$ is the Boolean complement of X. Atomic quantifiers are sets which contain just one set as element. A quantifier which has the set A as its only element will be noted $Q_{A}$.

A set of sets is the "basic" denotation of type $\langle 1\rangle$ quantifiers. In this case they are denotations of subject NPs. However, NPs can also occur in oblique positions and in this case their denotations do not take sets (denotations of VPs) as arguments but rather denotations of transitive VPs (relations) as arguments. To account for this eventuality one extends the domain of application of basic type $\langle 1 \rangle$ quantifiers so that they apply to n-ary relations and have as output an (n–1)-ary relation (Keenan 2016). Since we are basically interested in binary relations, the domain of application of basic type $\langle 1 \rangle$ quantifiers will be extended by adding to their domain the set of binary relations. In this case the quantifier Q can act as a "subject" quantifier or a "direct object" quantifier. When the NP denoting the quantifier Q acts as the direct object it denotes an accusative extension $Q_{acc}$ of Q, which is a type $\langle 2:1\rangle$ function defined as follows (Keenan and Westerståhl 1997):

D1: For each type $\langle 1 \rangle$ quantifier Q, $Q_{acc}R=\{a: Q(aR)=1\}$.

Various motivations for the accusative case extension and some of its application are given in Keenan (2016).

Type $\langle 1\rangle$ quantifiers have the property of living on a set (cf. Barwise and Cooper Barwise and Cooper (1981)). Thus quantifier Q lives on a set A iff for all $X\subseteq E$, $Q(X)=Q(X\cap A)$. The fact that quantifier Q lives on A will be noted: Li(Q, A). Observe that if Li(Q, A) and $A\subseteq B$ then Li(Q, B). Similarly, if Q is monotone increasing, non-trivial and Li(Q, A) then $A\in Q$.

Unary determiners are expressions which take one common name as argument and give a determiner phrase (or an NP) as result. They denote type $\langle 1,1\rangle$ quantifiers (functions), that is functions from sets to type $\langle 1\rangle$ quantifiers. For instance the determiner two denotes a type $\langle 1,1\rangle$ function TWO such that $TWO(X)(Y)=1$ iff $|X\cap Y|\ge 2$. A type $\langle 1 \rangle$ quantifier Q is called plural, $Q\in PL$, iff neither the empty set nor any singleton belongs to Q. The notion of a plural quantifier is needed since sentences with reciprocals necessitate plural subject NPs.

It is generally considered that type $\langle 1,1\rangle$ functions denoted by unary natural language determiners are conservative in the following sense:

D2: A type $\langle 1,1\rangle$ function D is conservative iff $D(X)(Y)=D(X)(X\cap Y)$, for any $X,Y\subseteq E$.

Conservative type $\langle 1,1\rangle$ functions form a Boolean algebra. This algebra has as a sub-algebra the set of intersective functions INT (cf. Keenan 1993) defined in D3:

D3: A type $\langle 1,1\rangle$ function F is intersective iff $F(X)(Y)=F(X\cap Y)(X\cap Y)$.

For instance determiners some and no denote an intersective function.

Conservativity defined in D2 will be occasionally called classical conservativity. It is related to the property of living on: if Q is a type $\langle 1\rangle$ quantifier formed from the type $\langle 1,1\rangle$ function D applied to the set A, that is if $Q=D(A)$ and D is conservative then Q lives on A.

Accusative extensions of type $\langle 1 \rangle$ quantifiers are specific type $\langle 2:1\rangle$ functions. This is because, informally, they form VPs (which denote sets), by applying to transitive verbs (which denote binary relations). They are just those functions that satisfy the invariance called accusative extension condition AE (Keenan and Westerståhl (1997)):

D4: A type $\langle 2:1\rangle$ function F satisfies AE iff for R and S binary relations, and $a,b\in E$, if $aR=bS$ then $a\in F(R)$ iff $b\in F(S)$.

Observe that if F satisfies AE then for all $X\subseteq E$ either $F(E\times X)=\emptyset$ or $F(E\times X)= E$. Given that $SELF(E\times A)=A$ the function SELF does not satisfy AE. The function SELF satisfies the following weaker predicate invariance condition PI:

D5: A type $\langle 2:1\rangle$ function F is predicate invariant (PI) iff for R and S binary relations, and $a\in E$, if $aR=aS$ then $a\in F(R)$ iff $a\in F(S)$.

This condition is also satisfied by the function ONLY-SELF defined as: ONLY-$SELF(R)= \{x: xR=\{x\}\}$. Given that ONLY-$SELF(E\times \{a\})=\{a\}$, the function ONLY-SELF does not satisfy AE. Following Keenan (2007) predicate invariant functions which do not satisfy AE will be called anaphoric functions. They are denoted by anaphors,

In this paper we are mainly interested in type $\langle 1,2:1\rangle$ functions. When a NP (a determiner phrase) occurring on the direct object position denotes the accusative extension of the type $\langle 1\rangle$ quantifier which is of the form D(A) (where D is a type $\langle 1,1 \rangle$ quantifier), then D can be considered as a type $\langle 1,2:1\rangle$ function. For instance, the denotation of the VP see most students, which is supposed to denote a set, can be seen as obtained by the application of the denotation of most to the CN student and the transitive verb meet. This means that with any type $\langle 1,1\rangle$ function D we can associate a type $\langle 1,2:1\rangle$ function, called the accusative determiner extension of D. This can be done in the following way:

D6: Let D be a type $\langle 1,1\rangle$ function. Then $D_{acc}$ is a type $\langle 1,2:1\rangle$ function defined as follows: $D_{acc}(X,R)=\{y: D(X)(yR)=1\}$

Type $\langle 1,2:1\rangle$ functions which are accusative extensions of conservative determiners satisfy a generalisation of AE, called accusative extension for one place determiners or D1AE, for short (cf. Zuber (2011)

D7: A type $\langle 1,2:1\rangle$ function satisfies D1AE iff for any $a,b\in E$, any $X\subseteq E$ and any binary relations R, S, iff $aR\cap X=bS\cap X$ then $a\in F(X,R)$ iff $b\in F(X,S)$.

For functions satisfying D1AE the following is true:

Fact 1: If a type $\langle 1,2:1\rangle$ function satisfies D1AE then $F(X, (E\times (A\cap X)))=\emptyset$ or $F(X, (E\times (A\cap X)))=E$, for any $X\subseteq E$.

The existence of type $\langle 1,2:1\rangle$ functions and definition D6 raise various questions. The first question is whether there is a property similar to the classical conservativity that type $\langle 1,2:1\rangle$ functions denoted in natural languages have, and, in addition, whether this property for functions obtained by the determiner accusative extension is related to the conservativity of the determiner. Zuber (2010a) proposes the following natural generalisation of the notion of conservativity which applies to type $\langle 1,2:1\rangle$ functions:

D8: A type $\langle 1,2:1\rangle$ function F is conservative iff $F(X,R)= F(X, (E\times X)\cap R)$.

As in the case of the classical conservativity the conservativity of type $\langle 1,2:\tau \rangle$ functions can be defined equivalently as in Fact 2:

Fact 2: A type $\langle 1,2:\tau \rangle$ function is conservative iff $F(X,R_{1})=F(X,R_{2})$ whenever $(E\times X)\cap R_{1}=(E\times X)\cap R_{2}$.

Observe that for $\tau =1$ definition D8 applies to type $\langle 1,2:1\rangle$ functions and for $\tau =\langle 1\rangle$ to type $\langle 1,2:\langle 1\rangle \rangle$ functions.

The following proposition, proved in Zuber (2010a), can be considered as a justification of definition D8:

Proposition 1

Let D be a type $\langle 1,1\rangle$ quantifier and F a type $\langle 1,2:1\rangle$ function defined as $F(X,R)=D(X)_{acc}(R)$. Then F is conservative iff D is conservative.

Consequently the generalised definition of conservativity for type $\langle 1,2:1\rangle$ functions given in D8 is strictly related to the classical conservativity of type $\langle 1,1\rangle$ functions.

Proposition 1 shows how to construct non-conservative type $\langle 1,2:1\rangle$ functions; it is enough to take the accusative determiner extension of a non-conservative type $\langle 1,1\rangle$ function. For instance it is easy to show that the type $\langle 1,1\rangle$ function $D_{=}$ defined as $D_{=}(X)(Y)=1$ iff $X=Y$ is not conservative. Given this the type $\langle 1,2:1\rangle$ function F defined as $F(X,R)=\{y: X=yR\}$, which is the accusative determiner extension of $D_{=}$, is not conservative.

As is well known only can be considered as a determiner and it is not classically conservative. In example (14) the determiner only occurs in the object position and denotes the type $\langle 1,2:1\rangle$ function $ONLY_{o}$ given in (15) and which can be considered as the accusative determiner extension of ONLY:

(14) Leo reads only short stories.

(15) $ONLY_{o}(X,R)=\{y: yR\subseteq X\}$

Function $ONLY_{o}$ is not conservative because $ONLY_{o}(X, (E\times X)\cap R)=E$ for any X and R whereas $ONLY_{o}(X,R)$ is not constant.

What is more interesting is the fact that many type $\langle 1,2:1\rangle$ functions denoted by various "natural" expressions of natural languages are conservative. This is in particular the case with various "comparative determiners" which cannot occur in subject NPs (cf. Keenan 2016). Consider for instance the expressions more... than Dan or the same... as Bo. These expressions form a VP taking a CN and a transitive verb as "arguments": the VP to read the same books as Bo is obtained by the "application" of the determiner the same as...Bo to CN books and to the transitive verb read. Thus semantically the determiner the same...as Bo denotes the function given in (16):

(16) $SAME_{b}(X,R)=\{y: yR\cap X=bR\cap X\}$.

Function in (16) is conservative (cf. Zuber 2011).

For our purposes more important is the fact that anaphoric determiners denote type $\langle 1,2:1\rangle$ functions which are also conservative. In the next section the anaphoric functions of this type denoted by reflexive determiners are studied.

3 Reflexive Determiners

In this section anaphoric conservativity of functions denoted by reflexive determiners is studied. I distinguish, as in Zuber (2017), anaphoric reflexive determiners (RefDets) and anaphoric reciprocal determiners (RecDets). In this section I study conservativity of reflexive determiners, that is, roughly speaking determiners related to himself/herself/themselves such as every... except himself, no... but himself, most..., including herself and Bo, etc.

As we have seen anaphoricity of functions denoted by reflexive pronouns is characterised by the property called predicate invariance of type $\langle 2:1\rangle$ functions denoted by such pronouns. Type $\langle 1,2:1\rangle$ functions denoted by RefDets satisfy a similar invariant condition (Zuber 2010b):

D9: A type $\langle 1, 2:1\rangle$ function F satisfies D1PI (predicate invariance for unary determiners) iff for R and S binary relations $X\subseteq E$, and $x\in E$, if $xR\cap X=xS\cap X$ then $x\in F(X,R)$ iff $x\in F(X,S)$.

The following proposition indicates an equivalent way to define D1PI :

Proposition 2

A type $\langle 1, 2:1\rangle$ function F satisfies D1PI iff for any $x\in E$, $X\subseteq E$, any binary relation R one has $x\in F(X,R)$ iff $x\in F(X,(\{x\}\times X)\cap R)$

Condition D1PI is a weakening of the condition D1AE. There are functions which satisfy D1PI and do not satisfy D1AE. As an example consider the RefDet no... but himself and Dan. This anaphoric determiner denotes a type $\langle 1, 2:1\rangle$ function given in (17):

(17) NO-$SELF_{d}(X,R)=\{y: yR\cap X=\{y,d\}\}$.

To show that the function in (17) satisfies D1PI suppose that for some $a\in E$ and some binary relations R, S we have $aR\cap X=aS\cap X$ and that $a\in NO$-$SELF_{d}(X,R)$. This means, given (7), that $aR\cap X=\{a,d\}$ and thus that $aS\cap X=\{a,d\}$ and consequently that $a\in NO$-$SELF_{d}(X,S)$ and thus the function in (17) satisfies D1PI.

To show that the function in (17) does not satisfy D1AE we can use Fact 1. If $A=\{d\}$ and X is such that $d\in X$ then NO-$SELF_{d}(X, (E\times \{d\})=\{d\}$ and thus NO-$SELF_{d}$ does not satisfy D1AE.

Observe that if F is type $\langle 2,1:1\rangle$ function satisfying D1PI and $G_{A}$ is a type $\langle 2:1\rangle$ function such that $G_{A}(R)=F(A,R)$ then $G_{A}$ is an anaphoric function (for a non-trivial A). For this reason functions which satisfy D1PI and do not satisfy D1AE will be called anaphoric reflexive dets. They are denoted by RefDets.

It is also easy to show that the function in (17) as well as many other functions denoted by RefDets are conservative. In fact this is not surprising because we have the following proposition:

Proposition 3

If a type $\langle 1, 2:1\rangle$ function satisfies D1PI then it is conservative.

To prove Proposition 3 suppose that F satisfies D1PI and, a contrario, is not conservative. This means that $F(X,R)\ne F(X,(E\times X)\cap R)$. Thus for some $a\in E$ we have $a\in F(X,R)$ and $a\notin F(X, (E\times X)\cap R)$ (or $a\notin F(X,R)$ and $a\in F(X,(E\times X)\cap T)$). Since F satisfies D1PI, if $a\in F(X,R)$ then, given Proposition 2, $a\in F(X,(\{a\}\times X)\cap R$ and $a\notin (F(X, (E\times X)\cap R)$. But this is impossible because $a(\{a\} \times X)\cap R)\cap X=a((E\times X)\cap R)\cap X$ and F satisfies D1PI.

What Proposition 3 says is that the notion of conservativity (as defined in D8) is in some sense useless for reflexive anaphoric functions since such functions are necessarily conservative. So the question is whether there is a property similar to conservativity which is proper to anaphoric functions. Such a property, called anaphoric conservativity, is partially studied in Zuber (2010a):

D10: A type $\langle 1,2:\tau \rangle$ function F is anaphorically conservative (a-conservative for short) iff $F(X,R)=F(X, (X\times X)\cap R)$.

As in the case of the classical conservativity a-conservativity can be defined in an equivalent way. The following proposition makes probably clearer what a-conservativity is:

Proposition 4

A type $\langle 1,2:\tau \rangle$ function F is a-conservative iff for any $X\subseteq E$ and any binary relations $R_{1}$ and $R_{2}$ if $(X\times X)\cap R_{1}=(X\times X)\cap R_{2}$ then $F(X,R_{1})=F(X,R_{2})$.

We show first the if-part. Let the equality (1) holds: (1) $(X\times X)\cap R_{1}=(X\times X)\cap R_{2}$ and suppose that F is a-conservative. Then, given a-conservativity of F, we have $F(X,R_{1})=F(X,(X\times X)\cap R_{1})$. Hence, given (1), $F(X,R_{1})=F(X, (X\times X)\cap R_{2})$. But since F is a-conservative we get the conclusion that $F(X, (X\times X)\cap R_{2})=F(X,R_{2})$.

To show the only if--part assume that the implication (i) holds for any $X\in E$, and any binary relations $R_{1}$ and $R_{2}$: (i): if $(X\times X)\cap R_{1}=(X\times X)\cap R_{2}$ then $F(X,R_{1})=F(X,R_{2})$. Suppose, a contrario, that $F(X,R)\ne F(X, (X\times X)\cap R$. Take $R=R_{1}$ and $R_{2}=(X\times X)\cap R$. Then we get the equality (2): $(X\times X)\cap R_{1}=(X\times X)\cap R_{2}$ and thus, given (i), $F(X,R)= F(X, (X\times X)\cap R$. Contradiction.

Finally, using simple set theoretic calculation one can show that a-conservativity can also be expressed as in Fact 3:

Fact 3: F is a-conservative iff $F(X,R)=F(X, (X\times X)'\cup R)$, where $(X\times X)'$ is the Boolean complement of the relation $X\times X$.

Thus, informally, definition D10, Proposition 4 and Fact 3 indicate that in order to check whether an element is in the domain of an a-conservative function having X and R as arguments we do not have to check every pair in the relation R but only those whose co-ordinates are both in X.

As an example consider the RefDet no.. except himself which denotes the function given in (18):

(18) $NO_{SELF} (X,R)=\{y: yR\cap X= \{y\}\}$.

Observe that $NO_{SELF} (X,R)=NO_{SELF} (X,(X\times X)\cap R)$ because $aR\cap X=\{a\}$ iff $a(R\cap (X\times X)\cap X=\{a\}$. Thus the function in (18) is a-conservative. We get the same result if we recall that the function in (18) satisfies D1PI.

Here are two examples of conservative type $\langle 1, 2:1\rangle$ functions which are not a-conservative. As the first example consider the accusative extension of the determiner EVERY (cf. Zuber 2017). Let $R=E\times A$, for some $A\subseteq E$ and let $F(X,R)=EVERY(X)_{acc}(E\times A)=E$ if $X\subseteq A$ but in this case $F(X, (X\times X)\cap R)= EVERY(X)_{acc}((X\times X)\cap (E\times A)=X$. Thus $F(X,R)\ne F(X,(X\times X)\cap R)$ which means that $F(X,R)= EVERY(X)_{acc}(R)$ is not a-conservative (though it is conservative).

As the second example consider the function in (16) denoted by the comparative determiner the same... as Bo. If we compare the set $\{y: yR\cap X=bR\cap X\}$ with the set $\{y: y((X\times X)\cap R)=b((X\times X)\cap R))\}$ we see that these sets are different for $R=E\times X$ and X such that $b\notin X$. In this case $SAME_{b}(X,R)=E$ and $SAME_{b}(X,(X\times X)\cap R)=X'$. This means that the function in (16) is not a-conservative.

There are thus conservative functions which are not a-conservative. However, the following proposition is true:

Proposition 5

Any type $\langle 1,2:1\rangle$ a-conservative function is conservative.

Suppose F is a-conservative. Then, on the one hand, $F(X,R)=F(X, (X\times X)\cap R$ and, on the other hand, $F(X, (E\times X)\cap R)=F(X, (X\times X)\cap (E\times X)\cap R)=F(X, (X\times X)\cap R)$. Thus $F(X,R)= F(X, (E\times X)\cap R$, which means that F is conservative.

Predicate invariance is not sufficient for a-conservativity since there are functions satisfying D1PI which are not a-conservative. In (19) we have an example of such a function:

(19) $F_{A}(X,R)=\{y: yR\cap X=A\}$.

The question I want now to answer is what are additional properties that anaphoric reflexive functions have to have in order to be a-conservative.

When we look at the definition conservativity of type $\langle 1,2:1\rangle$ functions given in D8, we can say, informally that it involves the range of the relational argument R since in computing the values of such functions only some elements of the range of R are involved, only those which also belong to the argument X. We can restrict in the same way the elements of the domain of the relational argument R. In this case we get a new, "symmetric" notion of conservativity of type $\langle 1,2:1\rangle$ functions. This new property of type $\langle 1,2:1\rangle$ functions will be called cons1:

D11: A type $\langle 1,2:1\rangle$ function F is cons1 iff $F(X,R)=F(X, (X\times E)\cap R)$.

A sufficient condition for anaphoric type $\langle 1,2:1\rangle$ functions to be cons1 is given in:

Proposition 6

If F is a type $\langle 1,2:1\rangle$ function satisfying D1PI and the condition $F(X,R)\subseteq X$, for any $X\subseteq E$ and any R binary relation, then F is cons1.

Indeed, if $F(X,R)\subseteq X$ then for any $a\in E$ we have $aR\cap X=a((X\times E)\cap R)\cap X$ and thus, since F satisfies D1PI, $F(X,R)=F(X,(X\times E)\cap R)$.

The sufficient condition indicated in Proposition 6 will be called inclusion condition.

Let us see more formally how we have obtained the property cons1. Classical conservativity can be said to be "conservativity with respect to the second argument" or "conservativity on the right" of a type $\langle 1,1\rangle$ function. This is because conservativity thus defined permits a restriction of the second argument (the argument "on the right") of the function to its intersection with the first argument. Formally it is possible to define "conservativity with respect to the first argument" or "conservativity on the left" (cons1, for short). This is done in D12:

D12: A type $\langle 1,1\rangle$ function F is cons1 iff $F(X)(Y)=F(X\cap Y)(Y)$.

It has been observed that unary determiners which do not denote classically conservative functions denote functions which are conservative "on the right" or conservative in the sense of D9 (cf. Zuber 2019, Zuber and Keenan 2019). This is in particular the case with the determiners only and mostly. For this reason, as we have seen, the accusative extension of only given in (15) is not conservative type $\langle 1,2:1\rangle$ function. However, this function is cons1 since $yR\cap X=$ $y((X\times E)\cap R)\cap X$.

Thus formally cons1 for type $\langle 1,2:1 \rangle$ functions is an extension or a generalisation of cons1 for type $\langle 1,1\rangle$ in the same way as conservativity for type $\langle 1, 2:1\rangle$ is a generalisation of the classical conservativity.

There are of course type $\langle 1,2:1 \rangle$ functions which are neither conservative nor cons1. The function $F_{=}(X,R)=\{y: yR=X\}$ is such a function.

It is easy now to prove Proposition 7:

Proposition 7

A type $\langle 1,2:1\rangle$ function F is a-conservative iff F is conservative and cons1.

To show the implication from left to right we have to show that a-conservative functions are cons1 since, given Proposition 5 any a-conservative function is conservative. Let F be a-conservative and consider $F(X. (X\times E)\cap R$.By a-conservativity $F(X. (X\times E)\cap R=F(X,(X\times X)\cap (X\times E)\cap R))= F(X,R)$ which means that F is cons1.

To show the implication from right to left suppose F if cons1 and conservative. Then $F(X,R)=F(X,(X\times E)\cap R)$=$F((E\times X)\cap (X\times E)\cap R))$ and thus $F(X, (X\times X)\cap R))$.

Functions which are clearly a-conservative because they are conservative and cons are accusative extensions of intersective functions. The reason is that intersective functions are conservative on the left and on the right (cf. D3). However, such functions are not anaphoric.

We have thus two sources of a-conservativity, so to speak. A type $\langle 1,2: 1\rangle$ function is a-conservative because it, informally, conservative on the left and conservative on the right, or because it satisfies predicate invariance and inclusion condition. All natural language RefDets seem to satisfy the inclusion condition. This is in particular the case with examples (6a), (6b), (7b), (17) and (18). Consequently functions denoted by determiners in these examples are a-conservative.

4 Reciprocal Determiners

Reciprocal anaphoric determiners, or RecDets, for short are expressions which apply to a CN and give an expression playing the role of a direct object, which I will call a generalised NP (or GNP). Such GNP apply to transitive verbs and form VPs. Morphologically RecDets are related to the reciprocal each other. It has been pointed out, at least since Langendoen (1978), that simple reciprocal each other is often used to express a "similar" relation between members of a given group even if this relation does not hold between all members of the group. Accordingly various weaker readings of the simple reciprocal have been distinguished (Langendoen 1978, Dalrymple et al. (Dalrymple et al. (yyy)). For simplicity I will discuss only full (logical) reciprocity. Furthermore, in most cases the properties we discuss are independent of this distinction of readings of reciprocals.

An additional warning concerning some empirical aspects of this paper may be necessary. To illustrate various properties of anaphoric functions I will use examples which in some cases are probably of limited acceptability. This limitation has a pragmatic basis since often discussed examples have peculiar truth conditions. I will ignore this problem given that in general provided examples are easily interpretable.

As we have seen many RefDets are formed from the reflexive himself and a "part" of inclusive or exclusive determiner like including NP or except NP. Not surprisingly, RecDets can also be formed from inclusive and exclusive "ordinary" determiners. Thus in (20a) and (20b) we have RecDets based on inclusive "ordinary" determiners and in (21a) and (21b) - RecDets based on exclusive determiners:

• (20a) Leo and Lea hate most vegetarians, including each other.

• (20b) Most teachers admire some Japanese, including each other and themselves.

• (21a) Leo and Lea admire no philosopher except each other and Plato.

• (21b) Three linguists admire every linguist except each other.

This way of constructing RefDets and RecDets from ordinary inclusive and exclusive determiners is productive in many languages.

As indicated above I assume that expressions forming direct objects in the above sentences denote type $\langle 2:\langle 1\rangle \rangle$ (type $\langle \langle \langle e,t\rangle t\rangle t\rangle$ in Montague notation) functions. This means in particular that they give as their output a set of type $\langle 1\rangle$ quantifiers. Let me make some remarks which justify this assumption.

Observe first that any type $\langle 2:1\rangle$ function whose output is denoted by a VP can be lifted to the type $\langle 2:\langle 1\rangle \rangle$ function. This is in particular the case with the accusative extension of a type $\langle 1\rangle$ quantifier. For instance the accusative extension of a type $\langle 1 \rangle$ quantifier can be lifted to type $\langle 2:\langle 1\rangle \rangle$ function in the way indicated in (22). Such functions will be called accusative lifts. More generally, if F is a type $\langle 2:1\rangle$ function, its lift $F^{L}$, a type $\langle 2:\langle 1\rangle \rangle$ function, is defined in (23):

• (22) $Q^{L}_{acc}(R)=\{Z: Z(Q_{acc}(R))=1\}$.

• (23) $F^{L}(R)=\{Z: Z(F(R))=1\}$.

The variable Z above runs over the set of type $\langle 1 \rangle$ quantifiers.

For type $\langle 2: \langle 1\rangle \rangle$ functions which are lifts of type $\langle 2:1\rangle$ functions we have:

Proposition 7

If a type $\langle 2:\langle 1\rangle \rangle$ function F is a lift of a type $\langle 2:1\rangle$ function then for any type $\langle 1\rangle$ quantifiers $Q_{1}$ and $Q_{2}$ and any binary relation R, if $Q_{1}\in F(R)$ and $Q_{2}\in F(R)$ then $(Q_{1}\wedge Q_{2})\in F(R)$

For type $\langle 2:\langle 1\rangle \rangle$ functions which are accusative lifts we have:

Proposition 8

Let F be a type $\langle 2:\langle 1\rangle \rangle$ function which is an accusative lift. Then for any $A,B \subseteq E$, any binary relation R, $Ft(A)\in F(R)$ and $Ft(B)\in F(R)$ iff $Ft(A\cup B)\in F(R)$.

Consider now the following examples;

• (24a) Leo and Lea hug each other.

• (24b) Bill and Sue hug each other.

• (25) Leo, Lea, Bill and Sue hug each other.

Clearly (24a) in conjunction with (24b) does not entail (25). Thus, given proposition 3, functions denoted by reciprocal GNPs are are not lifts of type $\langle 2:1\rangle$ functions. Hence, to avoid the type mismatch and get the right interpretations we will consider that the GNPs each other denotes a type $\langle 2:\langle 1\rangle \rangle$ function and RecDets denote type $\langle 1,2:\langle 1\rangle \rangle$ functions.

Given the above facts we need to specify for type type $\langle 2:\langle 1\rangle \rangle$ function and type $\langle 1,2:\langle 1\rangle \rangle$ functions "higher order" constraints and invariance principles similar to those specified above for type $\langle 2:1\rangle$ and type $\langle 1,2:1\rangle$ functions. Such principles are discussed in some details in Zuber (2014). Here I recall only some of them directly related to reciprocal anaphors and reciprocal determiners.

A generalisation of the PI which applies to type $\langle 2:\langle 1\rangle \rangle$ functions is the higher order predicate invariance HPI defined in D13. (cf. Zuber 2014. An equivalent way to define HPI is given in Proposition 9:

D13: A type $\langle 2: \langle 1\rangle \rangle$ function F satisfies HPI (higher order predicate invariance) iff for any type $\langle 1 \rangle$ quantifier Q, any $A\subseteq E$, any binary relations R, S, if Li(Q, A) and $\forall _{a\in A}(aR=aS)$ then $Q\in F(R)$ iff $Q\in F(S)$.

Proposition 9

function F satisfies HPI iff $\forall (A\in E)$ if Li(Q, A) then $Q\in F(R)$ iff $Q\in F((A\times E)\cap R)$

The relation between PI and HPI is indicated in the following proposition proved in Zuber (2014):

Proposition 10

If F is a type $\langle 2:1\rangle$ function which satisfies PI then $F^{L}$ satisfies HPI.

The above definitions of HPI can be extended to type $\langle 1, 2:\langle 1\rangle \rangle$ functions in the following way:

D14: A type $\langle 1, 2:\langle 1\rangle \rangle$ function F satisfies D1HPI (higher order predicate invariance for unary dets) iff for any type $\langle 1 \rangle$ quantifier Q, any $A\subseteq E$, any binary relations R, S, if Li(Q, A) and $\forall _{a\in A} (aR\cap X=aS\cap X)$ then $Q\in F(X,R)$ iff $Q\in F(X,S)$.

The condition D1HPI can also be characterised as in:

Proposition 11

A type $\langle 1, 2:\langle 1\rangle \rangle$ function F satisfies D1HPI iff $\forall (A\in E)$ if Q lives on A then $Q\in F(X,R)$ iff $Q\in F(X, (A\times X)\cap R)$

Higher order predicate invariance entails conservativity:

Proposition 12

If a type function F satisfies D1HPI then F is conservative.

To prove Proposition 12 suppose F satisfies D1HPI and let $Q\in F(X,R)$. Suppose Li(q, A). Then, on the one hand, given Proposition 7, $Q\in F(X,R)$ iff $Q\in F(, (A\times X)\cap R)$ and, on the other hand, $Q\in F(X (E\times X)\cap R)$ iff $Q\in F(X, (A\times X)\cap R)$. Then $Q\in F(X,R)$ iff $Q\in F(X,(E\times X)\cap R)$, which means that F is conservative.

Thus higher order anaphoric functions are conservative. To give a sufficient condition for such functions to be a-conservative we define the notion of higher order inclusion condition (HIC:

D15: A type $\langle 1,2:\langle 1\rangle \rangle$ function F satisfies higher order inclusion condition, $F\in HIC$, iff $Q\in F(X,.R)$ then Li(Q, X), any Q, X, R.

Proposition 13

Any type $\langle 1,2: \langle 1\rangle \rangle$ function satisfying D1HPI and HIC is a-conservative.

Proof

Suppose $F\in D1HPI\cap HIC$. Then, by conservativity of F, $Q\in F(X,R)$ iff $Q\in F(X,(E\times X)\cap R)$. Hence, given Proposition 11 and condition HIC we have $Q\in F(X,R$ iff $Q\in F(X,(E\times X)\cap R)$ iff $Q\in F(X, (X\times X)\cap R)$. $\square$

The negation of the relational argument preserves predicate invariance and higher order invariance. Given that $aR\cap X= aS\cap X$ iff $aR'\cap X=aS'\cap X$ we have in particular :

Fact 4: Let $G(X,R)=F(X,R')$. Then $F\in D1HPI$ iff $G\in D1HPI$.

In order to check whether functions denoting RecDets have above properties we need to provide a semantic description of them. I must say that in general this is a somewhat cumbersome and often difficult task I present below first a type $\langle 1, 2:\langle 1\rangle \rangle$ function corresponding to the reciprocal each other and then the function corresponding to the RecDet no... but each other.

To define the type $\langle 2:\langle 1\rangle \rangle$ function EA denoted by the reciprocal each other we use the partition $\Pi (R^{S^{i}})$. The function EA is given in (26):

(26) (i) $EA(R)=\emptyset$ if $R^{S^{i}}=\emptyset$ or $\Pi (R^{S^{i}})$ is trivial (ii) $EA(R)=\{Q: Q \in PL\wedge \exists _{B\in \Pi (R^{^i})} \exists _{C\subseteq Dn(B)} |C| \ge 2 \wedge Q(C)=1\}\cup \{Q: Q\in PL\wedge \exists _{K\subseteq E} \forall _{B\in \Pi (R^{S^{i}})} |(Dn(B)\cap |K|\ge 1\wedge |K|\ge 2\wedge Q=\lnot TWO(K)\}$ if $R^{S^{i}}\ne \emptyset$ and $\Pi (R^{S^{i}})$ is not trivial

Some comments about the above description are in order. First, quantifiers denoted by subject NPs of sentences with each other as direct object, have to be plural. Second, since $R^{S^{i}}=R\cap R^{-1}\cap I'$, if $R^{S^{i}}=\emptyset$ or $\Pi (R^{S^{i}})$ is trivial then no two individuals are mutually in the relation R and thus no two objects with whatever property are in such a relation. This is accounted by clause (i). Observe in addition that in this case the NP no two objects could be a subject of the corresponding sentence but we exclude it since the quantifier $\lnot TWO(E)$ denoted by this NP is not plural. For the same reason we exclude any consequence of $\lnot TWO(E)$.

According to clause (ii) if $R^{S^{i}})\ne \emptyset$ and $|Pi(R^{S^{i}})$ is not trivial two sets compose the output of the function EA: (1) the set of these type $\langle 1\rangle$ quantifiers which are true of a domain or a sub-domain of a relation forming a block of the partition and (2) the set of quantifiers which are false of the set of individuals which are not in mutual relation because their intersection with sets being in a mutual relation (domains of blocks of the partition) contains at most one element and thus they are not in the the mutual relation.

Function EA satisfies HPI. To show this we use Proposition 11 and he fact that for any block there exists one-to-one correspondence between blocks B of $\Pi (R^{R^{i}})$ and blocks $B_{1}$ of $\Pi ((A\times E)\cap R)^{S^{i}}$ since we have: $B_{1}=(A\times E)\cap B$. Suppose that $Q\in EA(R)$ and Li(Q, A). If $Q\in F(R)$ because is a block B of the partition $\Pi (R^{S^{i}})$ and $C\subseteq Dn(B)$ such that $Q(C)=1$ then $|A\cap C|\ge 1$ because $Q\in PL$. Hence $(A\times A)\cap B$ is a block of $\Pi ((A\times E)\cap R)^{S^{i}})$ and $Q(C\cap A)=1$. Thus $Q\in F((A\times E)\cap R)$. Finally, if $Q=\lnot TWO(H)$ for H such that for any $B\in \Pi (R^{S^{i}})$ we have $|H\cap Dn(B)|\ge 2$ then $|A\cap H|\ge 2$ and the set $A\cap H\cap Dn(B)$ is a subset of some block of the partition $\Pi ((A\times E)\cap R)^{S^{i}}$.

Suppose now that $Q\in EA((A\times E)\cap R)$. If this is the case because there is a block B1 of $\Pi ((A\times E)\cap R)^{S^{i}}$ and $C\subseteq Dn(B_{1})$ such that $Q(C)=1$, then $Dn(B_{1}=A\cap Dn(B)$ for some block B of $\Pi (R^{S^{I}})$. But then $(A\cap C)\subseteq Dn(B)$ and $Q(A\cap C)=1$. Hence $Q\in EA(R)$. Suppose now that $Q\in EA((A\times E)\cap R$ because $Q=\lnot TWO(K)$ for some K such that $|K|\ge 2$ and $|K\cap Bn(B_{1}|\le 1$ for any block $B_{1}$ of $\Pi ((A\times E)\cap R)^{S^{i}}$. Thus $|K\cap A\cap Dn(B)|\le 1$ for any $B\in \Pi (R^{S^{i}})$. Since $\lnot TWO (K)\in PL$, we have $|A\cap K|\ge 2$. Thus $|K\cap Dn(B)|\le 1$ for any block B of $\Pi (R^{S^{i}})$. Hence $Q\in EA(R)$.

Consider now the RecDet No... except each other. It denotes a type $\langle 1,2:\langle 1\rangle \rangle$ function $NO_{EEA}$ in (27):

• (27) $NO_{EEA}(X,R)= \{Q: Q\in PL\wedge \exists _{B\in \Pi (R/X)^{S^{i}}} Q(\{y: y\in X\wedge |X|\ge 2 \wedge yR\cap X=Dn(B)\}=1\}$ To obtain the function $EVERY_{EEA}$ denoted by every... except each other one can use the fact that, roughly, EVERY is related to NO by the negation of the second argument R. Thus we have:

• (28) $EVERY_{EEA}(X,R)=NO_{EEA}(X,R')$.

Functions $NO_{EEA}$ and $EVERY_{EEA}$ satisfy D1HPI. The reason is that if Li(Q, A) and $Q\in PL$ then $|A\cap Y|\ge 2$ for any $Y\in Q$. Thus B is a block of $\Pi (R/X)^{S^{i}}$ iff $(A\times A)\cap B$ is a block of $\Pi ((A\times X)\cap R)^{S^{i}}$. Hence $Q\in NO_{EEA}(X,R)$ iff $Q\in NO_{EEA}(X, (A\times X)\cap R)$.

Finally we consider the semantics of a class of RecDet which are of the form Det..., including each other, where Det is a unary determiner. In (29) are some examples of such a determiner:

• (29a) Leo and Lea draw every student, including each other.

• (29b) Most mathematicians admire some logicians, including each other.

• (29c) Two monks washed five priests, including each other. In order for sentences with the RecDet of the form Det..., including each other the determiner Det has to denote a type $\langle 1,1\rangle$ function which is monotone on the second argument, since sentences like (30) are impossible. In (31) is indicated the semantics of such determiners:

• (30) *Dan and Bo washed few students/no students, including each other.

• (31) (i) $F_{DIA}=\emptyset$ if $\Pi (R_{S^{i}}/X)=\emptyset$.

• (ii) $F_{DIA}=\{Q: (\{y:y\in X\wedge |X|\ge 2\wedge \exists _{B\in \Pi (R^{S^{i}})} Dn(B)\subseteq yR\wedge D(X)(yR)=1\}\}$ if $\Pi (R_{S^{i}}/X)\ne \emptyset$.

Observe now that if $y\in X$ then $yR\cap X=y(R\cap (X\times X)$ and $\Pi ((R^{S^{i}}/X)=\Pi (X\times X)\cap R)^{S^{i}}/X)$. This entails that functions in (27), (28) and (31) are a-conservative. We notice also that in (7a) the monotone increasing quantifier denoted by Bo and Dan lives on the set PHILOSOPHER and for this reason (7a) entails (8a). For similar reasons (29a) entails that all mathematicians are logicians and (29b) entails that all monks are priests.

5 Conclusive Remarks

Various empirical observations related to nominal anaphors and to reflexive and reciprocal determiners forming them and different formal results made in this paper allow me to propose a semantic universal concerning the property that anaphoric determiners have. We have seen that reflexive determiners denote type $\langle 1,2: 1\rangle$ functions which are anaphorically conservative and reciprocal determiners denote type $\langle 1,2:\langle 1 \rangle \rangle$ functions which are anaphorically conservative. The definition of anaphoric conservativity is the same for both types of functions. Furthermore, even if there are other determiner-like expressions which denote anaphoric conservative functions, these expressions cannot be considered as anaphoric determiners because the functions they denote do not satisfy predicate invariance, a property specific to anaphoric functions. The proposed language universal based on the above observations is given in (32):

• (32) All anaphoric determiners denote a-conservative functions. The universal constraint proposed in (32) should be refined in various ways. In particular more detailed information is needed concerning its scope. For instance some languages have possessive reflexive anaphoric determiners corresponding, roughly, to his/her own. In Zuber (2009) I discuss such possessive determiners in Polish and show that functions they denote are anaphoric but are not a-conservative. The reason is that they do not satisfy the inclusion condition. Furthermore, the underlying definition of the anaphor used in this paper is semantic and related to properties of functions taking binary relations as arguments represented by transitive verbs. Anaphors can occur in many other positions. For instance the pronoun himself/herself can appear not only as argument of verbs taking three arguments but also can appear twice in constructions involving such verbs. Anaphors formed from anaphoric determiners can also occur on other than direct object positions though probably not twice: (33), (34) and (35) are easily interpretable but (36) is not:

• (33) Bo protected herself from herself.

• (34) Leo protected Bo from every student except from Dan.

• (35) Dan protected Leo and Lea from every students except from each other.

• (36) ?Leo and Lea protected each other from each other.

We observe also that (34) entails that Dan is a student but not that Leo is a student and (35) entails that Leo and Lea are students and does not entail that Dan is a student. Consequently anaphoric determiners have a use in which they denote functions taking a set, a type $\langle 1\rangle$ quantifier and a ternary relation as arguments. In this case they can also be said to be a-conservative, in a slightly generalised sense. All these and similar cases need additional elaboration, which may be not straightforward.