Towards Referential Transparent Annotations of Quantified Noun Phrases (2023)

Abstract: Using recent developments in count noun quantification, namely Referential Transparency Theory (RTT), the basic structure for annotating quantification in the nominal domain according to RTT is presented. The paper discusses core ideas of RTT, derives the abstract annotation syntax, and exemplifies annotations of quantified noun phrases partly in comparison to QuantML.

## introduction the collection of interoperable semantic annotation standards known as the semantic annotation framework (semaf) includes an annotation schema for the annotation of quantification phenomena called quantml @xcite . quantml draws on work in formal and computational semantics, in particular generalized quantifier theory @xcite , discourse representation theory @xcite , and neo-davidsonian event semantics @xcite . it aims at a considerable if not complete coverage of natural language quantification. with respect to quantified noun phrases (qnps) -that is, noun phrases which involve a quantifier word -an alternative to generalized quantifier theory (gqt) has recently been developed in terms of referential transparency theory (rtt; @xcite . 1 rtt draws its main motivation from data of natural language use as observed in dialogical interactions, where higher-order denotations postulated by gqt do not seem to be confirmed. hence, rtt pursues a witness-based approach to quantification, which arguably simplifies the representation of quantification phenomena. 1 "transparency" here -a feature of the representation of noun phrase contents -is not to be confused with transparency of quantml, where it refers to the instantiation of a metamodel @xcite . quantrtt aims at an annotation schema which makes the rtt approach available for annotation. the paper is organized as follows. section 2 introduces the key ideas of rtt that are needed to understand the annotation approach outlined in section 3. the interpretation of quantrtt annotations in the rtt framework is briefly covered in section 4. some phenomena outside the current scope of quantrtt are discussed in section 5. we conclude in section 6. ## brief primer into rtt perhaps the most consequential feature of rtt is that quantification with quantificational determiners and nouns happens entirely within the noun phrase. in other words, a qnp such as many goldfish is interpreted without reference to a so-called scope set (the property donated by the verb phrase in gqt). rtt makes crucial use of the fact that qnp contents seem to be readily structured entities, as is revealed by their anaphoric potential. consider (1): the initial sentence introduces a qnp (few environmentalists). the few environmentalists that actually came to the rally -the reference set (refset) -are picked up by the plural pronoun in (1a). however, two additional sets become accessible: the "refset environmentalists" seem to be drawn from a larger group of environmentaliststhe maximal set (max set) -, which is picked out by the plural pronoun phrase in (1b). the plural pronoun in (1c), finally, picks out those environmentalists that did not came to the rally: the complement set (compset). 2 (1) only few environmentalists came to the rally. a. but they raised their placards defiantly. b. although they all received an invitation. c. they went to a football game instead. note that compset anaphora is only licensed under certain conditions (see, e.g., @xcite . rtt offers a horror vacui-based explanation here, drawing on empty refsets @xcite .3) -see (4) below. assuming that the "qnp anatomy" (a phrase we owe to @xcite indeed hosts a set triplet, rtt develops the following qnp structure: rtt is formulated within a type theory with records @xcite . the arrow indicates a plural predicate type (ptype), that is, a predicate that expects a set-valued argument. condition c2 simply states that refset and compset add up to the maxset. obviously, the structure in (2) provides suitable antecedents for the above-given range of anaphora. the value of condition c1 is donated by the predicate type of the head noun (e.g., environmentalist, goldfish), which is distributed over all maxset members (and thereby over refset and compset). the quantificational workhorse is the quantifier condition "q-cond": it captures what can be called the descriptive meaning of a qnp. for instance, the q-cond of many states that the refset is larger than the compset (|refset| > |compset|). the quantificational condition of all has it that the compset is empty, or equivalently, that refset and maxset coincide. hence, q-cond not only expresses np-internal quantification (i.e., quantification without a scope set from the vp), it also implements quantifiers as "sieves", a metaphor due to @xcite . this is achieved since rtt is denotationally underpinned by sets of ordered set bipartitions, mathematical structures which correspond to inversely coupled pairs of the elements of the power set of the head noun's denotation. for example, let the denotation [↓] of the type bicycle be a set of three bicycles: [↓ bicycle] = {ǧ, ǧ, ǧ}. then function p returns the set of ordered set bipartitions: feature q-persp in (2) indicates whether the bipartition with the empty refset is part of a qnp's denotation; if so, its feature value is "@xmath0; otherwise "refset ̸ = ∅". nps for which q-persp is not applicable -such as proper names -have no qpersp value ("none"). thus, the q-persp value is denotationally well-founded and regiments compset anaphora: the compset is available as antecedent only if "q-persp : @xmath1. any np-internal approach to quantification needs to say something about how a qnp combines with a verb phrase (vp) into a sentence. rtt -in contrast to gqt -adopts the standard (and intuitively pleasing) notion of predication: the verb predicates of its arguments. to be more precise: vp content applies to the refsets of its arguments. that is, the meaning of a sentence like every dog barks is compositionally derived as illustrated in (5), abbreviated to the necessary degree (the pair of a situation and a situation type is an austinian proposition @xcite ; label "q-params" is abbreviated "q-p" in some paths here and in the following for reasons of space): note that (5) involves "anti-predication" of the compset. postulating multi-dimensional denotations is not uncommon in semantics, @xcite , for instance, argues for a related move. since a plural type takes a set of individuals as argument, the question arises on how exactly the predicate relates to the members of the set. the predicate bark in (5) obviously distributes to every single dog from the refset. this is distinct from collective predicates like gather, which apply to sets of individuals, and predicates like carry-a-piano, which, when asserted of a set of people, can be understood in a distributive or a collective way, and anything in between @xcite . spelling out the details of distributivity is a bit involved (see lücking, 2022, sec. 2.5), therefore quantrtt offers simple notational abbreviations, following the subtyping relation in ( 6 ): (6) ---→ ptype partial distrib./inside coll. fully distrib. the general plural type imposes no restriction onto its interpretation, whereas fully distributive and outside collective ones require what their names suggest. the types in the middle express that a plural predicate applies to individuals and to any subgroups of the refset (this is rtt's counterpart to covers, lattices of subsets under set inclusion). since these substructures can be seen from the perspective of either distributivity (in terms of partiality) or collectivity (in terms of inside collections), there are two possible ways to name these subtypes. let us briefly illustrate matters by means of a simple example: every dog chased a cat. this sentence can be used to describe situations of different kinds, namely a situation where (i) a bunch of dogs together chased a cat (outside collective), (ii) each dog from the bunch chased a different cat (fully distributive), or (iii) some dogs chased in teams (i.e., there is more than one cat but the number of cats is less than the number of dogs). how does this exposition fit to so-called narrow and wide scope readings (∀∃ vs. ∃∀)? it does not, since scope is replaced by dependent interpretations of qnps @xcite , which apply in situ and introduce a function. the relevant content parts for the every-dog-chased-acat example are shown in (7). the subject qnp introduces a refset (and a suitable q-cond), as usual. the object qnp introduces a function f which associates an individual x with a cat z. the nucleus distributively applies the predicate and the function to the subject's refset, which provides entities x (dogs in our example) as input for f , which in turn returns a cat each (i.e., an individual of type cat). the representational format of rtt -albeit presumably uncommon to most readers -is arguably more transparent than equivalent formulae of second order predicate logic. moreover, there is a systematic distinction between quantification (qparams) and predication (nucl). for this reason, the domain of markables of quantrtt is more restricted than that of quantml. this leaves a final and potentially intricate issue: definiteness. coming from a dialogical point of view, rtt employs a "referential bookkeeping mechanism", following hpsg-related work @xcite . the crucial idea is that certain nominal expressions are expected to be witnessed while others are "quantified away". this is expressed in terms of two sets of parameters, dgb-params and q-params. elements within the dialogue gameboard parameters (dgb-params; a generalization of kaplanian indices) are expected to be instantiated by an object or a set of objects known to the speaker(s), whereas quantificational parameters (q-params) need not have a specific witness. note that during dialogical clarification interaction the status of belonging to either dgb-params or q-params can switch. ## annotating with quantrtt we follow the general approach of quantml and conceive a markable m and an annotation s as an entity structure ⟨m, s⟩. markables are the strings making up noun phrases. annotations are derived from the above-introduced qnp anatomy. the relation between two or more entity structures is captured in terms of a link structure. the inventory of quantrtt looks as follows: note that we omit the annotation of q-persp since it is not involved in quantification proper but mainly regiments compset anaphora. comparing the inventories of quantml and quantrtt, we are aware of the following correspondences (∼): the attributes q-cond and involvement have some functional commonalities, but do not completely correspond to each other, as can be seen, for instance, with np negation -see ( 11 ) and (13b) below. phenomena such as inverse linking, cover interpretations or group quantification are captured in terms of dependencies (dep_on) in combination with distrib (cf. section 2). all quantrtt features have direct counterparts in the qnp anatomy. the remainder of this section presents a few examples in order to showcase quantrtt in action. a famous example for scope readings is given in (8), discussed by bunt (2020, p. 4): the reading where two languages is interpreted in the scope of everybody (i.e., the reading where there might be different pairs languages for different persons) is annotated in quantml as follows: ## interpreting annotations since annotations are derived from a qnp anatomy, annotations can be mapped onto either the basic qnp structure in (2) or the functional one in (7) of rtt, as illustrated in figures 1 and 2. note that in line with rtt's np-internal approach to nominal quantification and the distinction between quantification and verbal predication, annotations in quantrtt do not involve verb phrases (i.e., figure 3 ). figure 3 : sentence meaning of everybody speaks two languages. ## discussion an anonymous reviewer brought up the following participation example: from the perspective of rtt, (14) involves two cardinality restrictions, one on the refset (viz., "@xmath2. the latter, however, can not yet be expressed in quantrtt, simply because the annotation inventory (see section 3) lacks a corresponding annotation label. this can easily be fixed in future versions, but will still not capture recursive participant structures as in ( 15 ): we are not aware of how (or whether) such examples are to be annotated in quantml, but we imagine a nested annotation drawing on involvement and sourcedomain. the empirical phenomena that underlie the development of rtt involve count nouns. hence, currently rtt has not much to say about mass nouns and quantification with substances yet, as involved in (16) @xcite . however, given that rtt is formulated in a type theory with records, it seems to be straightforward to follow psychological work (e.g., @xcite and introduce a type subst(ance) alongside ind(ividual). given this, it seems that rtt's basic mechanisms can be adapted to substances, in which case the q-cond acts like a sieve on what can be called "refmass" and "compmass": on this view, classifiers like three cups (of milk) induce a type shift from subst to ind. furthermore, natural languages provide resources like the english adverbial modifier twice to quantify over events @xcite ): intuitively, (18) says that there have been two calling events by two children (from a certain maxset). rtt has not dealt with temporal or spatial quantification yet. however, a potential direction to account for (18) shall be indicated, drawing on the notion of string type @xcite . a string type is a concatenation of types. it can be thought of as a flip book and is used for temporally structuring an event into sub-events. accordingly, potential witnesses of string types are series of situations. event quantification on this view can be seen as a mechanism of constructing a string of copies (of a number determined by the descriptive meaning of the temporal modifier in question) from a given situation type. notating the string type ' -→ call(x) ⌢ -→ call(x)' simply by a superscript indicating the number of copies (i.e., by ' -→ call(x) 2 '), ( 18 ) is analyzed as follows (omitting details not relevant to the issue at stake): note that sit now consists of a series of two events, s 1 and s 2 . it is finally noteworthy -since it has been raised as an issue by bunt (2019b, §7) -that a type theory provides a straightforward analysis of propositional attitude verbs like believe or seek. since propositions are types @xcite , intensional verbs denote a relation between individuals and types, as shown in (20), following @xcite . vic's search will only be successful, if s/he encounters a record (a situation) that contains an individual of the type expressed by the record type p. ## conclusion we presented quantrtt, an annotation schema for quantified noun phrases based on rtt @xcite . the conceptual underpinnings have been introduced and used to derive the abstract syntax of the annotation schema. we see it as an advantage that quantrtt brings about a cleaner separation of quantification and verbal predication. furthermore, given the transparent noun phrase anatomy, quantrtt arguably lends itself to the integration with anaphora annotation projects (e.g., @xcite , contributing to the interoperability of annotations. this could involve to include q-persp in annotations to account for qnps and compset anaphora in a more systematic way. a couple of examples comparing quantml and quantrtt have been discussed. although the examples are typeset in the form of concrete xml syntax, there is no standard for quantrtt yet. to make quantrtt operable, two further, not mutually exclusive, steps are envisaged. firstly, quantrtt can be implemented as an "add-on" to quantml, for instance as a plug-in as proposed for extensions to dialogue act annotation @xcite . this move will have benefits on both sides: quantml is connected to rtt and phenomena not yet covered by rtt can be captured by appropriate quantml resources (although it remains to be seen how well both approaches interact "out of the box"). note in this context that the intersection of elements in the syntactic inventories of quantml and quantrtt is empty, meaning that they can in principle be annotated in parallel. secondly, quantrtt will be incorporated in the textannotator @xcite , an annotation suite hosting several annotation tools. this move enables to make use of annotation support from automatic natural language pre-processing tools. furthermore, due to the graphical user interface, the linking structure of dependent noun phrases can be added in a graphical display by drawing connecting edges. of course, quantrtt will develop as rtt will -a few pointers into potential research directions (e.g., mass nouns and quantificational adverbials) have been given.