Some Formal Properties of Higher Order Anaphors

Some Formal Properties of Higher Order Anaphors R. Zuber Laboratoire de Linguistique Formelle, CNRS and University Paris-Diderot Richard.Zuber@linguist.univ-paris-diderot.fr Abstract Formal properties of functions denoted by higher order anaphors like each other and syntactically complex expressions contain- ing each other are studied. A partial com- parison between these functions and func- tions denoted by (simple and complex) re- flexives is draw. In particular it is shown that both types of function are predicate invari- ant (in a generalised sense). These results allows us to understand the anaphoric char- acter of both reflexive and reciprocal expres- sions. 1 Introduction By higher order anaphor, I mean expressions like each other, sometimes called basic higher order anaphors, and various complex expressions syn- tactically containing each other. These complex anaphors include Boolean compounds like each other and most students, each other and them- selves and various modifications of each other like only each other or at least each other. Higher order anaphors are also expressions formed by the application of a higher order anaphoric de- terminer like each other’s or every ...except each other to a common noun (CN). All such expres- sions will be called reciprocals and sentences con- taining them (in object position) will be called re- ciprocal sentences. Higher order anaphors can be opposed to (logi- cally) simple anaphors whose basic example is the reflexive pronoun himself/herself/themselves. This simple basic anaphora can also occur in Booleanly complex anaphors like himself and most students or in modified expressions like only himself, even themselves. So the distinction between simple and higher order anaphors is of logical nature: as we will see below functions denoted by higher or- der anaphors take binary relations (or binary rela- tions and sets) as arguments and give sets of type ⟨1⟩ quantifiers as output whereas simple anaphors have arguments of the same type as higher order ones (that is their arguments are binary relations or sets and binary relations) but their output are sets (of individuals). The semantics of reciprocal sentences is a com- plex matter (as shown for instance in Dalrymple et al., 1998; Cable, forthcoming; Dotlaˇcil, forth- coming; Mari, forthcoming). In fact there does not seem to be any general agreement concerning the data and the interpretation of reciprocal con- structions (cf. Beck, 2000). In this paper I am not, strictly speaking, interested in the semantics of higher order anaphors but in the formal proper- ties of functions denoted by higher order anaphors. Two types of such properties will be discussed: those which are similar to properties of functions denoted by simple anaphors and those which make them different from functions denoted by simple anaphors. Formal properties of functions denoted by simple anaphors have been studied in Keenan (2007), Zuber (2010b) and Zuber (2011) and some formal properties of higher order anaphors are given in Sabato and Winter (2012) and Peters and Westerst˚ahl (2006). As far as I can tell, no compar- ison between the two types of function have been made. Moreover, only basic anaphors (that …

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