Chapter 13 Abstract Objects and the Core-Periphery Distinction in the Ontological and the Conceptual Domain of Natural Language Friederike Moltmann Abstract This paper elaborates core-periphery distinctions in the ontological and the conceptual domain of natural language. The core-periphery distinction is essential for the pursuit of natural language ontology and has in fact been made implicitly by any philosopher present or past when appealing to natural language for motivating an ontological notion or view. The distinction plays a central role in the main thesis of my 2013 book Abstract Objects and the Semantics of Natural Language, that natural language permits reference to abstract objects only in its periphery, not its core. The paper explores how the core-periphery distinction relevant for ontology appears to be structurally anchored and relates to the more familiar core-periphery distinction that Chomsky drew for syntax. Keywords Abstract objects * Core-periphery distinction * Propositions * Properties * Numbers * Degrees * Functional categories 13.1 Introduction Abstract objects such as properties, numbers, and propositions have been a topic of philosophical controversy since antiquity. While there are a range of philosophical arguments for and against abstract objects, philosophers have also appealed to natural language, generally arguing that abstract objects are well-reflected in natural language, in what appear to be abstract terms, such as happiness, the number of planets or that-clauses. This has given rise to the widespread view that natural language involves a rich and philosophically controversial ontology of abstract objects. F. Moltmann () CNRS, Paris, France e-mail: fmoltmann@univ-paris1.fr © Springer Nature Switzerland AG 2020 J. L. Falguera, C. Martínez-Vidal (eds.), Abstract Objects, Synthese Library 422, https://doi.org/10.1007/978-3-030-38242-1_13 255 256 F. Moltmann A systematic linguistic investigation, based on a much greater range of data, whether and how natural language (or rather English) involves an ontology of abstract objects was the aim of my 2013 book Abstract Objects and the Semantics of Natural Language. The book rejects the common view that natural language permits pervasive reference to abstract objects and instead endorses what I will now call the Abstract Objects Hypothesis: (1) The Abstract Objects Hypothesis Natural language does not involve reference to abstract objects in its ontological core, but only in its ontological periphery. The book, for example, argues that happiness is not a term standing for an abstract property object, but rather a plural term, referring to all the particular happiness features (tropes) at once. The number of planets, the book argues, does not stand for an abstract number, but rather for a number trope (the numerical aspect plurality of the planets).1 That-clauses moreover are not considered referential terms at all standing for objects. Crucial in the formulation of the Abstract-Objects Hypothesis is the relativization to a core-periphery distinction. This is an important distinction that has always been made implicitly when philosophers appealed to natural language in support of a philosophical view, but has hardly ever been articulated and theoretically elaborated. Whenever philosophers (and linguists) appeal to natural language to motivate an ontological notion or view, they are careful to only draw on expressions from the ontologically relevant core of language ('the ontological core', for short), not its periphery. Thus, it is considered legitimate to appeal to the existence in English of expressions like happiness, the number of planets, and that it is raining when motivating properties, numbers, and propositions as abstract objects, but not expressions like the property of being happy, the number eight, and the proposition that it is raining, that is, abstract terms particular to philosophical discourse, but also legitimate for use by any competent speaker of English. Intuitively, the ontological core consists in expressions or uses of expressions not involving ontological reflection, whereas the ontological periphery consists of expressions or uses of them involving ontological reflection. This distinction between core and periphery reconciles the fact that only certain sorts of expressions or uses of expressions are considered indicative of the ontology implicit in natural language with the fact that natural language contains expressions that clearly can be used to make reference to abstract objects of various sorts and permits the introduction of new expressions (or uses of expressions) for that purpose. An important feature of the core-periphery distinction relevant for natural language ontology is that there appear to be significant constraints as to what parts of language can lead to a use in the periphery. The ontological core-periphery distinction appears to be structurally anchored, roughly going along with the lexicalfunctional distinction, with full nouns freely being able to lead to a use in the 1For the notion of a trope as a particularized property see Williams (1953). 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 257 periphery but not morphological features or functional categories. This means that the ontological core-periphery distinction is not just a somewhat elusive distinction based on mental acts or attitudes, but is also anchored in the structure of language. Natural language ontology with such a core-periphery distinction goes along with a particular conception of ontology. First, the ontology reflected in natural language should be viewed not as an ontology of what there really is but rather an ontology of appearances (Fine 2017), in the sense of consisting of objects that serve as values of referential noun phrases and bear a range of properties, but that may be derivative or merely conceived objects. Second, the ontology reflected in language should not just consist of a single domain of entities and their associated categories, but should allow for considerable flexibility and expansion of the domain, by including ontological operations introducing new entities. This sort of expandability matches that of the lexical-conceptual domain of language, which displays a similar core-periphery distinction. Thus, it has been argued that only certain concepts allow for 'conceptual engineering' (Eklund 2015, Chalmers 2011), and again, there appear to be structural constraints to what extent meanings can be modified or expanded on non-ordinary, philosophical uses of language. Core-periphery distinctions have also been made for syntax (Chomsky 1981, 1986, 1998) and for phonology (Itô and Mester 1995a, b). While the core-periphery distinction that Chomsky drew for syntax may in some ways coincide with the one relevant for ontology, the core-periphery distinction in phonology is an entirely different one. The overall aim of the paper is to elaborate the ontological core-periphery distinction in the context of the cognitive ontology reflected in natural language, focusing on apparent reference to abstract objects. The paper will start by presenting standard arguments for an against abstract objects. It will then lay out the background of natural language ontology and in particular the notion of a language-related, constructional ontology it involves. The main part of the paper consists in elaborating core-periphery distinctions for the ontological and the conceptual domains of natural language, followed by a brief discussion of core-periphery distinctions that have been made in generative syntax and phonology. Finally, the paper gives a concise presentation of the reanalyses in Moltmann (2013a) of apparent abstract terms in the core of language and proposes a way to make sense of the semantics and ontology of abstract terms in the ontological periphery. In an appendix, the paper discusses apparent problems for the Abstract Objects Hypothesis. 13.2 Abstract Objects, Natural Language, and the Core-Periphery Distinction There are a range of philosophical motivations for abstract objects of the various sorts as well as philosophical arguments against them. Properties have been considered the basis for relations of resemblance and the individuation of objects; propositions have been regarded important as the sharable contents of thought; and 258 F. Moltmann numbers, sets and other mathematical objects are obviously central in the ontology of mathematics. Yet abstract objects are also considered problematic in that they are not part of the empirical world and in particular cannot enter causal relations and act as the objects of perception, raising the issue of how we can have epistemic access to them. Moreover, it is unclear how abstract propositions can be grasped and act as the content of mental attitudes. In addition to providing metaphysical and epistemological arguments for and against abstract objects, philosophers have also drawn on natural language, generally to give support for abstract objects (but also sometimes to argue against them). In analytic philosophy, this is particularly the case in the work of Frege and Neofregeans (Wright 1986; Hale 1983), who take the syntactic category of referential noun phrases (singular or referential terms) to be indicative of objecthood ('an object is what a referential term may stand for'). Properties, propositions, and numbers then are generally considered objects as there appear to be referential noun phrases for them (the number of planets / eight, wisdom, that-clauses like that it is raining). This would extend to pure quotations (as in 'Red' means 'rouge'), which are commonly taken to involve the formation of referential terms standing for expression types. The putative involvement of objects is not limited to the role of semantic values of referential terms. Objects may also just act as implicit arguments of predicates. For example, degrees are abstract objects that are commonly regarded as implicit arguments of gradable adjectives.2 Given apparent abstract terms such as wisdom, the number of planets, and thatclauses, it has become a widespread view that natural language involves pervasive reference to abstract objects, a view which led some philosophers skeptical about abstract objects to reject language as a guide to ontology. In my own work (Moltmann 2013a), I have rejected this common view about the involvement of abstract objects in the semantics of natural language. Instead I have argued for the view that natural language does not involve reference to abstract objects in the relevant part of natural language, its ontological core, but only its periphery (the Abstract Objects Hypothesis). Relevant referential expressions that were generally considered abstract terms are now analysed differently. Instead of referring to abstract objects, they are either no longer considered referential terms or else are considered terms involving reference to something other than single abstract objects, say, pluralities of particulars, particularized properties (tropes), or variable objects with concrete entities as manifestations. Of course, there are natural language terms that cannot be regarded other than as terms standing for abstract objects, such as the property of being wise, the number eight, the proposition that it is raining, and various abstract terms that philosophers may be using. But such terms will be part of the ontological periphery, not the ontological core of language (see Sect. 13.4). The core-periphery distinction is indispensable when investigating the ontology reflected in language, and in fact it is a distinction that has been presupposed by 2Degrees can also be made explicit as in ten meters tall. 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 259 all philosophers throughout the history of philosophy that have appealed to natural language in support of a philosophical view. Thus, Frege and Neo-Fregeans knew to stay away from terms like the property of being wise, the proposition that it is raining, and the number eight, being aware that they are unsuited to make their point. Instead they drew on expressions like the number of planets and eight (Frege 1884), wisdom (Hale 1983), and that-clauses (all philosophers arguing for propositions since Frege 1918/19). Philosophers of course also stay away from expressions specific to philosophical discourse ('technical terms') and philosophical (that is, non-ordinary) uses of natural language expressions whose semantic values come from a particular philosopher's domain. Without a core-periphery distinction the ontology described by any philosophers' theory would be part of the ontology of natural language, undermining the pursuit of natural language ontology entirely. It is an undisputable fact, though, that natural language permits the introduction of expressions or uses of expressions meant to have semantic values within a particular philosopher's ontological theory (despite attempts by ordinary language philosophers to sanction such uses). Thus a philosopher can introduce a particular notion of existence, truth, or parthood and use the nouns existence, truth, and parthood to be associated with such a notion rather than whatever the ordinary meaning of the expression may be from which such nouns are derived (exist, true, part). Such uses are not illegitimate, but rather are made possible by a fundamental feature of language or rather the conceptual and ontological structure going along with it, namely what I will call expandability. Expandability consists in the possibility of conceptual modification or ontological expansion that goes along with a non-ordinary use of natural language expressions, with the introduction of new expressions, or with particular constructions already present in the language (such as constructions of the sort the property of being happy). Expandability of the conceptual and ontological domain associated with language allows for a form of creativity in more of the standard understanding of the term than the creativity of language use in the Chomskyan sense, which consists in the ability of a speaker of a language to produce and understand an indefinite number of new expressions against the background of limited experience. Creativity in the conceptual and ontological domain manifests itself in the introduction of new concepts or objects based on reflection (as opposed to implicit acceptance).3 13.3 Natural Language Ontology The following is a brief outline of the discipline whose subject matter is the ontology of natural language, natural language ontology. 3'New' in the case of objects in the ontological domain means objects that are not yet part of the domain of interpretation of the language, whether the objects are real or merely conceived. See Sect. 13.2. 260 F. Moltmann With its referential noun phrases, which take entities as semantic values as well as with its lexical predicates and constructions that involve entities in other ways, natural language reflects an ontology. This ontology is the subject matter of a particular branch of metaphysics, that of natural language ontology. More specifically, natural language ontology is part of descriptive metaphysics in Strawson's (1959) sense or what Fine (2017) calls 'naïve metaphysics'. Descriptive metaphysics has as its subject matter the ontology reflected in our ordinary judgments. Natural language ontology has as its subject matter the ontology reflected in linguistic intuitions, that is, judgments about the acceptability or grammaticality of natural language sentences and constructions. What is important about descriptive metaphysics is that it is not about the ontology of what there really is. This is instead the subject matter of a different branch of metaphysics, what Fine calls 'foundational metaphysics'.4 Descriptive metaphysics and natural language ontology in particular concerns itself with how things appear, given the data, without addressing the question of whether they are real (which is to be addressed only by foundational metaphysics). For natural language ontology; this means that no foundationalist consideration should come into play when positing objects as semantic values, such as assumptions as to whether those objects really exist (in the sense of being fundamental) or what they may be reduced to. More important is what sorts properties the semantic values of referential noun phrases may have, as is reflected (at least to an extent) in the applicability of types of natural language predicates. The domain of objects in the ontology of natural language thus may include merely conceived objects besides objects that happen to be actual ones. (This will also be important for how to make sense of the expandability of the ontological domain of the language through technical or philosophical discourse in Sect. 13.4.) The subject matter of natural language ontology is not the ontology that ordinary speakers (non-philosophers) naively accept when thinking about what there is.5 The latter is the subject matter of folk metaphysics, not natural language ontology. What speakers accept when they reflect does not matter for natural language ontology. Natural language ontology rather deals with the ontological categories, notions, and structures that are implicit in language whether or not speakers would accept them upon reflection. The ontology of natural language thus is better to be characterized as the ontology that a speaker implicitly accepts when using the language and as such is distinguished from both the reflective ontology of ordinary speakers as well as philosophers and the ontology of what there really is (Moltmann 2017, 2019). Natural language ontology is not just a new discipline (as a branch of both metaphyics and linguistics), but in a way was pursued by philosophers throughout 4For Strawson (1959), descriptive metaphysics rather contrasts with what he calls 'revisionary metaphysics'. The aim of revisionary metaphysics, for Strawson, is to conceive of a better ontology than the one we ordinarily accept. (Strawson does not specify further how 'better' is supposed to be understood.) 5Fine's (2017) term 'naïve metaphysics' thus is misleading, and 'descriptive metaphysics' a better term to use for the branch of metaphysics that comprises natural language ontology. 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 261 the history of philosophy whenever philosophers drew on natural language in support of a metaphysical notion or view. Philosophers when appealing to natural language, though, appealed to only certain types of linguistic data and not others, and thus implicitly followed a particular methodology, the same that also guides natural language ontology as an emerging contemporary discipline that is part of both descriptive metaphysics and linguistics (semantics). For example, metaphysical statements of the sort there are properties, there are numbers etc. are not considered indicative of the ontology reflected in natural language. By contrast, sortal presuppositions of lexical predicates, for example, are. An important feature of the ontology of natural language is that it involves not just a particular domain of objects and their associated ontological categories, but ontological operations that serve the introduction of new objects. In that sense it is a constructional ontology (Fine 1991). To an extent, the ontological operations go along with the compositional semantics of particular syntactic constructions. Here are some examples (though there are no unanimous views as to the semantics they involve). First, the introduction of a trope (particularized property) goes along with the construction NP's NA, where NA is the nominalization of the adjective A, as in John's happiness, Socrates' wisdom, Mary's courage (Moltmann 2013a). On a common view, formation of a kind goes along with bare (determinerless) plural and mass nouns in English (giraffes are rare, water is transparent Carlson 1977). On an equally common view, the construction of definite plurals (the students) goes along with the introduction of a sum composed of the individuals that fall under the corresponding singular noun (Link 1983).6 Finally, definite NPs with a functional noun as head or modified by an intensional relative clauses go along with the introduction of a variable object, an object that has potentially different concrete manifestations at different times and perhaps (possible) situations. Examples are the president of the US (as with is elected every four years), the water in the pool (as with decreased), and the book John needs to write (Moltmann 2013a, to appear). The linguistic ability to use those constructions goes along with the implicit acceptance of the relevant ontological operation, and an actual use goes along with the implicit acceptance of the output of the operation. In addition to syntactic constructions, there are syntactic categories and features that convey ontological categories (to give two simplified examples: verbs with events, adjectives with tropes), and syntactic knowledge of them goes along with the implicit acceptance of those ontological categories. The constructional ontology reflected in natural language involves more differentiated forms of ontological acceptance. There will not just be a single notion of acceptance, acceptance of the entities in the domain of the ontology. There will also be a notion of acceptance of ontological operations, and that with or without acceptance of the output of such operations. Implicit acceptance of an ontological operation then needs to be distinguished from the reflective acceptance that may or 6In the latter two cases, there are also arguments, however, that the NPs plurally refer, rather than leading to the composition of a new entity (Moltmann 2013a; Yi 2005, 2006). See also Sect. 13.6. 262 F. Moltmann may not apply to the output. Such reflective acceptance would then lead from the ontological core to the periphery of language, as will see. 13.4 The Core-Periphery Distinction in Ontology and in Semantics 13.4.1 The Core-Periphery Distinction in Natural Language Ontology An important distinction differentiating between linguistic data indicative of the ontology of natural language and data not indicative of it is the distinction between the ontological core and periphery of language. The distinction is reflected in the sorts of data that contemporary philosophers or linguists know to choose when they pursue natural language ontology, and it is reflected in the sorts of data that philosophers throughout history knew to choose when they in fact engaged in natural language ontology. It is a distinction that is presupposed just as much by the appeals to natural language that philosophers considered appropriate throughout history and by contemporary pursuits of natural language ontology by linguists and philosophers. It is thus a distinction that is central to natural language ontology as a historical practice and an emerging discipline. Most importantly, full nouns conveying ontological categories (sortals) are generally not taken to be indicative of the ontology of natural language. The fact that nouns like property, proposition, number, trope, kind, sum, etc. can be used, legitimately, in English to refer to entities and quantify over them is generally not taken as evidence that the ontology implicit in language includes objects of such categories. Moreover, what I call 'reifying terms', noun phrases that are formed with sortal nouns like the number eight, the proposition that it is raining, or the truth value true, have generally not been taken to be indicative of the ontology of natural language.7 Thus, Frege (1918/19) did not motivate numbers as objects appealing to the presence of the construction the number eight, and Frege (1918/19) did not motivate truth values as objects by appealing to the truth value true (but rather by considerations regarding embedded sentences and the sense-reference distinction across categories). Yet, reifying terms with a selected set of sortals are fully productive and thus lead to referential terms standing for objects, the sorts of objects that should fall under the sortal that is part of the construction. What then is the difference to the constructions discussed in the last section, which involve ontological operations yielding composite or derivative objects (tropes, kinds, pluralities, variable objects) that would fall under suitable sortals? In both 7See Moltmann (2013a, Chapter 6) for a discussion of 'reifying terms'. Reifying terms divide into close appositions such as the number eight and the truth value true and constructions of the sort the proposition that S, the fact that S, the property of being wise. 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 263 cases we have complex syntactic constructions, but only in the second case will the objects that are the output of the operation count as part of the ontology of natural language. The reason is the involvement of a sortal in the first case, which makes its output not be part of the ontology implicit in language. In the second case, the outputs are implicitly accepted by anyone making use of the construction. In the first case, the construction itself involves explicit recognition of the output as an object falling under the respective sortal, and thus an act of reflection. Even if the construction itself with the ontological object-introducing operation is part of the core of language (and thus is implicitly accepted), the use of the construction with its involvement of an act of ontological recognition (or reflection) makes the output be part of the periphery, and thus not part of the implicitly accepted ontology. The feature driving the ontological core-periphery distinction is thus ontological reflection: expressions (or uses of expressions) belong to the ontological periphery if their use involves some degree of ontological reflection, as would go along with the use of a sortal. Note that the core-periphery distinction is not a strict, but a somewhat gradual distinction. While sortals like set, sum, trope, and proposition will require not just categorial recognition, but some technical or philosophical knowledge, this is presumably not so for fact (the fact that S) or state (the state of John being happy).8 The reflection that is characteristic of the periphery goes along with the use of full sortal nouns, which involves the explicit recognition of an object as belonging to a particular ontological category. By contrast, there are linguistic categories that convey ontological categories, but do not involve reflection and do not permit conceptual modification on a non-ordinary use. An example is plural morphology. Whatever notion of plurality it may in fact involve, that notion could not possibly be altered in a context of use to accommodate a particular philosophical view of plurality. Also the object-introducing constructions mentioned in the last section are of that sort, that is, definite NPs that introduce tropes, bare plurals and mass nouns that introduce kinds, and definite NPs that introduce variable objects. The use of those constructions does not permit altering the way the so introduced objects are conceived. The ontological periphery comprises particular construction types, expressions, and uses of expressions (as well as what they stand for). As such it also includes 'technical terms', expressions introduced within a particular philosopher's theory. It includes certain sorts of uses of expressions, namely just the non-ordinary, 'philosophical' uses of expressions that have been the subject of critique of ordinary language philosophy (Wittgenstein, Ryle, Malcolm). Rather than dismissing such uses as illegitimate, from the present perspective they do have their proper place, relying on a legitimate expandability of the semantics and ontology associated with 8There are also other sorts of terms that ordinary speakers use that appear to refer to abstract object. One example are number terms of the sort the eight, which clearly are terms referring to numbers as abstract objects, but certainly are used by ordinary speakers, even children (though such number terms are rather limited, just up to the twelve in my language). (The eight arguably is also a close apposition like the number eight, but involving a silent noun NUMBER.) 264 F. Moltmann language. If expressions are used in a non-ordinary way to convey metaphysical notions, then those notions will be part of the ontological periphery of the language, not its core. One issue the ontological core-periphery distinction raises is whether the core should represent the universal part of the cognitive ontology reflected in natural language as opposed to an ontology that would be specific to particular individuals, groups of individuals, or points of views. We have seen that part of the (core) ontology of natural language goes along with the semantics of particular syntactic constructions. There is no particular reason why syntactic constructions that serve the composition or introduction of a new entity should be shared among all languages. However, it should be only in the ontological core of languages that universally shared features of a cognitive ontology can be found. 13.4.2 The Core-Periphery Distinction in Semantics and Philosophical Conceptual Theory While the core-periphery distinction is indispensable for pursuing natural language ontology, it is a distinction that may be drawn and has been drawn also for other domains associated with language, in particular lexical/conceptual meaning, syntax, and phonology. In what follows, I will briefly discuss the distinctions in those other language-related domains. Like the ontology of natural language, lexical meanings in natural language do not form a rigid domain, but rather permit a great range of flexibility and polysemy, which has given rise to theories according to which the lexicon includes operations on meanings generating other meanings (the theory of the Generative Lexicon of Pustejovsky 1995). The flexibility that lexical meanings display also consists in the possibility for a language user to precisify or otherwise modify a given lexical meaning. When a philosopher engages in such modification of conceptual meaning for philosophical purposes, this is what is called 'conceptual engineering' in recent philosophical discussions.9 Engaging in conceptual engineering is not a privilege for philosophers, of course; the very same sort of non-ordinary use of expressions and introduction of a new or modified use of an expression is available to any competent speaker of the language. Like ontological operations that lead from the ontological core to the periphery, conceptual engineering is a legitimate operation of expansion of the domain of conceptual meaning. It thereby also gives rise to a core-periphery distinction. Let me call this the semantic core-periphery distinction, a distinction roughly between what one may call 'ordinary meaning' and 'non-ordinary meaning'. 9Conceptual engineering has been advocated as a replacement of the standard approach in analytic philosophy, conceptual analysis (Cappelen 2018). 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 265 There may be another, more important core-periphery distinction in the conceptual domain, based on potential limits to conceptual engineering (Chalmers 2011; Eklund 2015; Cappelen 2018). Concepts that resist conceptual engineering have been discussed as 'bedrock concepts' by Chalmers (2011) and as 'conceptual fixed points' by Eklund (2015). These are concepts so fundamental that they permit no modification in the context of a particular philosophical view. A distinction between a core consisting of such concepts and a periphery would define a fairly wide periphery of conceptual meaning. In such a distinction, which I will call the conceptual core-periphery distinction, the core would be a universal part of the human conceptual system, which would match a potential universal part of the core ontology of a given natural language. Limits to conceptual engineering do not just concern concepts as such. Limitations to legitimate modifications of conceptual meanings can also be viewed from a more linguistic point of view. For example, it is reasonable to expect that it is not available for the meaning of light verbs (have, be, make), morphological categories (plural, singular, tense, mood), thematic roles, and syncategorematic expressions (and, or, if-then).10 Moreover, there is evidence that it depends on the syntactic category of expressions to what extent it is applicable. For example, there is a striking difference between the meaning of the verb exist and the meaning of the nominalization existence. Many philosophers, such as van Inwagen (1998), take existence to be a notion that trivially applies to everything there is, or at least every actual thing. Their use of the noun existence would be perfectly legitimate for talking about that notion. However, in natural language, the predicate exist is subject to strict conditions on the type of entity to which it can apply, it applies to material and abstract objects, but not to events (Hacker 1982; Cresswell 1986; Moltmann 2013c, 2020): (1) a. The house still exists. b. The largest prime number does not exist. (2) a. ??? The rain still exists. b. ??? The protest existed yesterday. Events do not 'exist', but 'take place', 'happen', 'occur', or 'last'. The reason why exist exhibits such restrictions, arguably, is that exist has primarily a time– relative meaning and conveys the complete presence of an entity throughout a time (Moltmann 2013c, 2020). By contrast, the verb existence can be used without imposing any restrictions on the entities it applies to (as in the existence of everything there is). What is important is that the verb exist cannot be used so as to convey the unrestricted notion of existence. The resistance of exist to events is robust: even a philosopher convinced of the univocality of the concept of existence (such as van Inwagen) will be unable to apply exist to events.11 This appears to be a 10In generative syntax, this should generally hold for what is called functional categories, as opposed to lexical categories. 11There also are various sorts of philosophers that have particular views about existence, for example that only fundamental or mereologically primitive entities exist. As long as the sortal restrictions of exist are respected, such views can attach to the use of the verb as well. 266 F. Moltmann reflection of the semantic core-periphery distinction: the meaning of the verb exist belongs to the semantic core of language, whereas at least one of the meanings of the noun existence belongs to the semantic periphery. It is also a reflection of the ontological core-periphery distinction: whenever an expression conveying a metaphysical notion undergoes legitimate conceptual modification, this involves a shift from the semantic as well as the ontological core of language to its semantic as well as its ontological periphery.12 The semantic core-periphery distinction is very similar to the ontological coreperiphery distinction. The semantic core involves concepts that are implicit in the ordinary use of language, that is, that speakers implicitly accept when they use language in the ordinary way. The semantic periphery involves doing something to ordinary conceptual meanings ('conceptual engineering') and thus adding to the given domain of lexical meaning. The ontological periphery was characterized as involving ontological reflection, and thus adding to the ontological domain that a speaker accepts when using the language. The close connection follows from the fact that metaphysical concepts also belong to the ontological domain, and conceptual engineering would send them from the ontological core to the ontological periphery.13 Another feature that the semantic and ontological core-periphery distinctions share besides a distinction between implicit acceptance and reflection is a form of true creativity that language, with its conceptual and ontological domain, permits, with choices to be made in response to particular demands or interests, quite unlike the creativity Chomsky attributed to language, which involves applying a given set of rules to produce utterances of new sentences or expressions. Creativity in the latter sense is a general feature of the morphosyntax of language, allowing for the production of an infinite number of new sentences and words. But this is not the sort of expansion that leads to a periphery. The conceptual and ontological structures associated with language allow for creativity that goes along with distinctive cognitive acts, leading to a periphery in the relevant sense. 12Somewhat similar observations can be made about the adjective true and the nominalization truth. Philosophers have various different views about truth and can use the nominalization truth to convey their respective concept of truth. Philosophers (and perhaps non-philosophers) also generally have the view that representational mental objects (with a mind-world direction of it) are truth bearers. However, true fails to apply to some of the objects that one would think are truth bearers, for example impressions and speculations (an impression cannot be said to be true or false, and neither can a speculation) (Moltmann 2018). Of course, there may be various reasons (not just conceptual fixed points) why true exhibits additional restrictions in relation to the nominalization truth. But in any case, truth conveys a reflective notion that is not the same as that conveyed by true. 13Moreover, the switch from the adjective to the noun goes along with reification of a concept (adjective) as an object (noun). So here it is not just concept modification, but also reification, and thus an ontological operation. 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 267 13.4.3 The Core-Periphery Distinction in Syntax and Phonology Chomsky (1981, 1986, 1998) introduced a core-periphery distinction for syntax. Very roughly, those parts of a natural language are considered in the periphery that are anomalous or 'added on' from influences from other languages, whereas the core of language reflects universal grammar.14 Chomsky's distinction at first sight does not seem co-extensive with the semantic and ontological core-periphery distinctions. Various sorts of complex or derived expressions that would be part of the conceptual or ontological periphery should belong to the syntactic core, including nominalizations (e.g. existence) and reifying terms. However, the syntactic coreperiphery distinction arguably goes along with the distinction between functional categories and the lexicon (Yang 2016), which means that the peripherical status of nominalizations and reifying terms could be attributed to their involvement of full nouns (sortals in the case of reifying terms). Thus there may be a sense in which the syntactic and the ontological periphery coincide after all. There are other features that the two sorts of peripheries share. One intuitive feature is that of being "additional" with respect to the core. The ontological and semantic peripheries involve additional cognitive acts besides implicit acceptance of the domain of the core: acts of philosophical reflection of some sort. In addition, the two sorts of peripheries have the purpose of separating universal features of linguistic or cognitive systems from those that are not (whether due to outside influence or explicit reflection). Only when focusing on the core of language is the pursuit of universals in the cognitive ontology reflected in language possible, such as the Abstract Objects Hypothesis. There is another core-periphery distinction that has been used in generative grammar, namely in phonology. According to that distinction, the lexicon has a core-periphery organization according to degrees of assimilation/integration of the vocabulary (Îto and Mester 1995a, b). Here the core-periphery distinction is not a binary distinction, but a gradual one, with different types of items being more or less in the periphery or core. The phonological core-periphery distinction is also based on the feature of natural language being expandable, but now allowing for the vocabulary to be expanded by importing items from other languages. The semantic and ontological peripheries, by contrast, involve expansion from within the conceptual and ontological system. 14Yang (2016) contains a more recent discussion and defense of the core-periphery distinction in syntax. 268 F. Moltmann 13.5 Abstract Objects and the Core-Periphery Distinction: A Potential Universal of Natural Language Ontology With the distinction between core and periphery in the ontological domain of natural language we can turn to the Abstract Objects Hypothesis, given again below: (3) The Abstract Objects Hypothesis Natural language does not involve reference to abstract objects in its ontological core, but only in its ontological periphery. This hypothesis is based on a range of more specific generalizations regarding various expressions in the core of English that have been taken to involve reference to abstract objects (Moltmann 2013a). The various expressions in question are reanalyzed in one of the following four ways: [1] The expression involves no reference to an abstract object, but instead to a concrete one, for example a trope (particularized property). [2] The expression involves no reference to a single abstract object, but plural reference to various actual or possible particulars. [3] The expression does not involve reference to a truly abstract object, but rather reference to an object that strictly inherits all its properties from actual or possible concrete entities. [4] The expression does not act as a referential term (in relevant environments), but rather as a nonreferential complement or subject. Analysis [3] avoids a particular notion of an abstract object, as an object that bears properties directly. When an object strictly inherits all its properties from concrete entities, then the truth conditions of statements about the object generally reduce to those of a statements just about those concrete objects. In what follows, I will just briefly indicate how various putative abstract terms are reanalyzed, referring the reader to Moltmann (2013a) for empirical and formal semantic details. First, bare (determinerless) adjective nominalizations such as wisdom have standardly been considered singular terms standing for properties (Hale 1983). Philosophers generally make use of such terms and not explicit property-referring terms such as the property of being wise when arguing for natural language involving properties as objects. That is because wisdom is regarded a term in the core of language and the property of being wise a term in the periphery. The problem for the standard view is that wisdom displays different readings with various sorts of predicates from those displayed by its explicit property-referring counterpart. This is illustrated below (Moltmann 2013a): 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 269 (4) a. Wisdom exists. b. The property of wisdom exists. (5) a. John found wisdom. b. ??? John found the property of wisdom. (6) a. Wisdom is admirable. b. ??? The property of wisdom is admirable. (7) a. True wisdom is rare. b. ??? The property of being truly wise is rare. (4a) can state only the existence of an instance of wisdom, not the existence of an abstract object as such, unlike (4b). (5a) means that John found an instance of wisdom, not an abstract property object, unlike (5b) (which could be true only in a metaphysical fantasy). (6a) means that instances of wisdom are admirable, not the abstract property as such, unlike (6b). Predicates like rare, which only care about the distribution of instances, sound natural only with bare adjective nominalizations, as in (7a), not with explicit property-referring terms as in (7b). Generally, predicates are true of what bare adjective nominalizations stand for in virtue of properties of particular instances; they do not attribute a property to an abstract object, unlike with explicit property-referring terms. In Moltmann (2004), I had adopted analysis [3] positing a type of entity (an Aristotelian 'kind') which is unable to bear properties directly but only by inheritance from its instances, namely a kind whose instances are particular tropes. In Moltmann (2013a), bare adjective nominalizations like wisdom are no longer considered singular terms, but are taken to stand for kinds conceived as 'modalized pluralities', pluralities (as many) of all the actual and possible tropes (particularized properties) (that is, they refer plurally in the sense of Yi 2005, 2006). In both cases, a predicate P when applying to a kind (an Aristotelian kind or modalized plurality) has a derivative meaning, being true of the kind just in case the original property expressed by P is true of some instances (exist and find) or instances in general (admirable). These accounts extend to bare plurals (giraffes, blue pens) and bare mass nouns (water, white rice), which again are not considered singular terms standing for kinds as abstract objects. Entities that strictly inherit all their properties from concrete ones are also posited as semantic values of definite noun phrases of the sort the water in the container (with predicates like decrease) and the book John needs to write (see also Moltmann to appear). These are variable objects, entities which at different times and in different situations have possibly different manifestations, from which they inherit their properties. In natural language semantics, it has become standard to make use of abstract objects that are degrees, namely for the analysis of positive and comparative adjectives. Tall (in John is tall) means taller than a contextually given standard degree and taller expresses a relation between two pairs each consisting of an individual and a degree. The apparent involvement of degrees in the semantics of adjectives is reanalyzed in terms of [1] and [2], replacing abstract degrees by tropes or kinds of tropes (Aristotelian kinds or modalized pluralities) (see also Moltmann 270 F. Moltmann 2009). On that view, John is taller than Mary means 'John's height (a quantitative trope) exceeds Mary's height), and John is tall 'John's height exceeds the kind of tropes that makes up the contextual standard'. Numbers as abstract objects have generally been taken to be well-reflected in natural language. Frege, in particular, appealed to natural language when arguing for numbers being objects. For Frege, terms like the number of planets as well as simple number words like eight are singular terms standing for objects. Such apparent number-referring terms are reanalyzed adopting analyses [1] and [4] (see also Moltmann 2013d). First, number words like eight are no longer considered referential terms when occurring in argument position, but rather expressions that retain the meaning they have as noun modifiers (see also Hofweber 2005). The meaning of arithmetical statements in natural language then involves what in the philosophy of mathematics is called 'the Adjectival Strategy' (Dummett 1973, Hodes 1984). Roughly, Two and two is four on that view is analysed as 'if there were two things and another two thing, then there would be four things'. Apparent number-referring terms like the number of planets are reanalyzed as terms referring to number tropes (the planets reduced to just how many they are). Propositions are abstract objects that play a central role in the semantics of attitude reports and other sentence-embedding constructions. Propositions are considered problematic not just because of their abstractness, but because of specific problems of their own, their role as truth bearers and contents of attitudes (the problems of the truth-directedness and of the unity of the proposition).15 At least since Frege, propositions have been taken to be the semantic values of that-clauses (and sentences in general) as well as the contents of attitudes. Propositions have to be abstract, so the Fregean view, since contents of attitudes are sharable among different agents. The apparent compositional semantics of attitude reports appears to require propositions, on the assumption that that-clauses act as arguments providing an argument of the relation expressed by the embedding verb. However, there is good evidence that that-causes do not in fact act as referential terms and thus that their meaning does not have the status of an object. The linguistic evidence includes the failure of that-clauses to be substitutable by the proposition that S with most attitude verbs. In Moltmann (2013a), I proposed that that-clauses instead act as plural terms standing for an ordered plurality of propositional constituents. More recently, I adopted the view that they act semantically as predicates of an object associated with the attitude verb (what I call an 'attitudinal object') (Moltmann 2014). Special quantifiers like something or everything on the traditional view have been taken to act as propositional quantifiers. But again there is strong linguistic evidence that they do not behave that way, but rather act as nominalizing quantifiers ranging over attitudinal objects or kinds (pluralities) of them, the sorts of thing nominalizations of attitude verbs stand for. The view extends to other sentenceembedding constructions such as truth predicates and modals (Moltmann 2018). Given this view, propositions no longer play a role in the semantics of natural 15See Moltmann (2013a, Chapter 5) and references therein. 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 271 language. They at best play a role as semantic values of explicit propositionreferring terms, such as the proposition that S. Expression types are also abstract objects that are generally held to be objects of reference in natural language, namely referents of pure quotations. But like that-clauses, pure quotations display features of nonreferentiality, for example by not generally permitting substitution by an explicit expression-referring term ('Red' means 'rouge' does not imply 'Red' means the expression 'red', which is unacceptable). Instead of taking pure quotations to be referential terms standing for expression types, their function is now considered that of 'presenting themselves', forming a complex predicate with the embedding verb. There are natural language terms explicitly referring to propositions, properties, numbers degrees, and expression types, of the sort the number eight, the proposition that S, the word 'rouge', the sentence S. But those terms belong to the ontological periphery of language, not its core. When it comes to the core of natural language, what appeared to be expressions referring to abstract objects are now considered either expressions referring to concrete particulars or kinds of them or expressions that do not act as referential terms in the first place (number words, that-clauses, quotations), but contribute to the meaning of the sentence in a different way. The ontology of the core of language, according to the Abstract Objects Hypothesis, thus is an Aristotelian ontology of just concrete entities, or at least objects whose involvement in a statement would guarantee truthconditional equivalence with a statement just about concrete entities. 13.6 The Ontology Reflected in Abstract Terms in the Periphery On the present view, expressions and uses of expressions in the ontological periphery are legitimate parts of language (or its use). The question then arises for the theoretician how to handle their semantics and ontology. Of course, the semanticist cannot but accept the meanings that expressions are meant to obtain on a non-ordinary use, and so for newly introduced 'technical' expressions. But there is a question of how to regard the ontology that goes along with referential noun phrases on a non-ordinary or technical use. Here it is important to keep in mind that the ontology of natural language in general should be understood as an ontology of appearances. A referential noun phrase that is part of the ontological periphery thus should have a semantic value that is an object, but which may be a merely conceived object (though it may also turn out to be an actual one). There are also referential noun phrases whose semantic values are part of the periphery due to their compositional semantics. In particular, these would be reifying terms of the sort the number eight, the proposition that S, and the property of being wise. While this is not the place to discuss the syntax and semantics of 272 F. Moltmann reifying terms in detail, one general condition that should certainly obtain is that their semantics needs to be sufficiently general to allow for them to have semantic values on any given philosophers' theory of abstract objects. Yet the compositional semantics of reifying terms is also indicative of the role and nature of abstract objects in the core of natural language. The structure of reifying terms arguably involves a nonreferential expression or use of an expression following the sortal: eight in the number eight is still an adjective, and as such just mentioned rather than used (Moltmann 2013a). This then suggests a context-dependent semantics of reifying terms along the following lines. In a context of use c, the reifying term the N X stands the object o that is obtained on the basis of statements in which X occurs (referentially or nonreferentially), where it depends on the background assumptions in c what S consists in. If the object o is conceived as an object whose nature is exhausted by the attribution of predicates obtained from true statements in which X occurs, this would amount to a pleonastic account of abstract objects (Schiffer 1996, 2003). For example, given the Adjectival Strategy, divisible by two in eight is divisible by two roughly means 'any possible plurality of eight things consists in two equal-membered subpluralities'. Then the predicate is divisible by two can be attributed to o in virtue of the pleonastic equivalence below: (8) The number eight is divisible by two iff eight is divisible by two. Similarly, explicit property-referring terms would introduce objects whose nature is exhausted by pleonastic equivalences such as the one below (Schiffer 2003): (9) John has the property of being happy iff John is happy. However, the semantics of reifying terms should not commit itself to a pleonastic account of abstract objects, that is, an account on which there is nothing more to a number or a property than what can be attributed to it in virtue of an equivalence such as (8) or (9). Rather it should be compatible with various other philosophical views of abstract objects and be able to accommodate the use of sortals (in reifying terms) based on philosophical definition. Thus, (8) is compatible with a view on which the number eight is a set-theoretical construct or else a type of collection or just a light object whose properties are to be read of equivalences as in (8). Similarly, (9) is compatible with a view on which properties are platonic objects, collections of similar tropes, or else light objects, 'mere shadows of predicates'. The pleonastic account of abstract objects as semantic values of sortal nouns is no longer appropriate to pursue once the ontological periphery of natural language is acknowledged with its space for multiple ontologies that serve the semantic values of sortal nouns. What natural language tells us about abstract objects is not that they could not be objects of reference (and of the sort that a particular philosophical theory takes them to be). It only tells us that they are not part of the ontological core of natural language. They can be part of the ontological periphery, and (at least as merely conceived objects) fall under various philosophical views, something that semantic theory needs to be able to accommodate. 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 273 13.7 Conclusion The core-periphery distinction is crucial for natural language ontology, and it is particularly important when approaching abstract objects from the point of view of natural language. It is the basis for the Abstract Objects Hypothesis, a putative universal of natural language ontology. The ontological core-periphery distinction is due to the legitimate expandability of language, the same feature of language that is grounds for a distinction between core and periphery for lexical/conceptual meaning and that permits 'conceptual engineering'. In both cases, expansion goes along with an additional cognitive effort of reflection. In the case of the ontological periphery, such an act of reflection consists in the recognition of an object as belonging to a particular ontological category, by the application of a sortal noun. Recognizing both core and periphery allows taking language with its ontology to be a cognitive system that comprises both implicit acceptance and reflection and gives justice both to the ontology implicit in natural language and the reflective ontology of philosophers and nonphilosophers. Acknowledgments Part of the material of this paper has been presented in courses on natural language ontology in Duesseldorf, ESSLLI in Sofia (Bulgaria), and Munich in 2018 as well as at NYU (May 2019) and the IHPST, Paris (July 2019). The paper has benefitted greatly from discussions with the audiences as well as from numerous conversations with Kit Fine, exchanges with Noam Chomsky and Donca Steriade, as well as comments by Matti Eklund, Jonathan Schaffer, and an anonymous referee on a previous version of this paper. Appendix: Some Potential Issues for the Abstract Object Hypothesis In this appendix, I will briefly discuss some issues that appear problematic for the Abstract Objects Hypothesis. First, there is a notion of an abstract state that, it has been argued, plays a role not just for the semantic values of reifying terms of the sort the state of being happy, but also as implicit arguments of stative verbs. Thus, Maienborn (2007) introduced a distinction between abstract states ('Kimian states' as she calls them Kim 1976) and concrete states ('Davidsonian states' as she calls them Davidson 1967). Both sorts of states, she argues, act as arguments of stative verbs. Abstract states are implicit arguments of of abstract state verbs such as owe, own, know, be, have, and resemble (see also Moltmann 2011). Unlike concrete state verbs like sleep and sit, abstract state verbs do not accept a range of adverbial modifiers, such as causal and locational adverbials, and that is, so Maienborn, because of the ontology of abstract states, which lack causal roles and a spatial location. Abstract state verbs certainly are part of the core of language, which is in conflict with the Abstract Objects Hypothesis. Maienborn's view, though, is not uncontroversial. There are 274 F. Moltmann alternative accounts of the 'stative adverb gap', as it is called. One of them takes stative verbs to lack an event argument position to be filled in by states (Katz 2003). Another account takes adverbial modifiers to apply to truth makers rather than event arguments of verbs, or at least when modifying stative verbs (Moltmann 2007). Another apparent issue for the Abstract Objects Hypothesis is kinds and the applicability of existence predicates (Moltmann 2020). Existence predicates impose particular conditions on what sorts of objects they can apply to, with or without temporal or spatial modifier. Roughly, the generalization is that existence predicates can apply to an entity o relative to a location l just in case o is completely present throughout l (Moltmann 2020). The complete presence condition throughout a location has the consequence that existence predicates with temporal modifiers apply only to enduring objects and not events (Sect. 13.4.2). With spatial modifiers existence predicates can apply to only few sorts of entities (illnesses, languages, and kinds), and that is because those sorts of objects have abstract part structures permitting them to be present at multiple spatial locations at once. As entities with an abstract part structure, kinds then will have to be abstract objects themselves (rather than, say, pluralities of possible and actual particulars). However, other predicates than existence predicates just do not treat kinds as abstract objects, as we have seen in Sect. 13.5. There is a solution to this puzzle, and that is to take the constitutive features of kinds not to be abstract properties, but rather kinds of tropes (particularized properties). Kinds of tropes are present throughout a location just in case they have instances throughout the location. Another issue with the core-periphery distinction is that philosophers drawing on natural in support of an ontological view sometimes make use of expressions from the ontological periphery of language. For example, Vendler (1967) made use of the reifying term the fact that S to argue for a distinction between facts and events. I myself in Moltmann (2004, 2013a) made of use of reifying terms like the property of being wise to argue for a distinction between properties and kinds of tropes (the semantic values of terms like wisdom). Here the use of reifying terms is legitimate as it serves to show an ontological difference between a type of entity from the core and a type of entity already in the periphery. The term from the periphery is associated with intuitions ordinary speakers have – the periphery is not reserved for philosophers only. Of course, using technical terms that a particular philosopher may have introduced in the context of a philosophical theory would be a different matter. References Cappelen, H. (2018). Fixing language: An essay on conceptual engineering. Oxford: Oxford University Press. Carlson, G. (1977). A unified analysis of the English bare plural. Linguistics and Philosophy, 413– 457. Chalmers, D. (2011). Verbal disputes. Philosophical Review, 120(4), 515–566. 13 Abstract Objects and the Core-Periphery Distinction in the Ontological. . . 275 Chomsky, N. (1981). Lectures on government and binding. The Pisa lectures (Studies in Generative Grammar, no. 9). Dordrecht: Foris Publications. Chomsky, N. (1986). Knowledge of language. Its nature, origin, and use. Westport/London: Praeger. Chomsky, N. (1998). New horizons in the study of language and mind. Cambridge: Cambridge University Press. Cresswell, M. J. (1986). Why object exists, but events occur. Studia Logica, 45, 371–5. Davidson, D. (1967). The logical form of action sentences. In N. Rescher (ed.). The Logic of Decision and Action (pp. 81–95). Pittsburgh: Pittsburgh University Press. Dummett, M. (1973). Frege: Philosophy of language. Cambridge, MA: Harvard University Press. Eklund, M. (2015). Intuitions, conceptual engineering, and conceptual fixed points. In C. Daly (Ed.), Palgrave handbook of philosophical methods. New York: Palgrave Macmillan. Fine, K. (1991). The study of ontology. Nous, 25(3), 262–294. Fine, K. (2017). Naïve metaphysics (J. Schaffer, Ed.). Philosophical Issues, 27. Frege, G. (1884). Die Grundlagen der Arithmetik (Foundations of Arithmetic). Frege, G. (1918/9). Thoughts. In B. McGuinness (ed.). Collected Papers on Mathematics, Logic, and Philosophy (pp. 351–372). Blackwell, Oxford, 1984. Hacker, P. M. S. (1982). Events, Ontology, and Grammar. Philosophy 57, 477–486. Hale, B. (1987). Abstract Objects. Blackwell, Oxford. Hodes, H. (1984). The ontological commitment of arithmetics. Journal of Philosophy, 81, 123– 149. Hofweber, T. (2005). Number determiners, numbers, and arithmetics. Philosophical Review, 114(2), 179–225. Itô, J., & Mester, A. (1995a). Japanese phonology. In J. Goldsmith (ed.). The handbook of phonological theory (pp. 816–838). Malden: Wiley Blackwell. Itô, J., & Mester, A. (1995b). The core-periphery structure of the lexicon and constraints on Reranking. In J. Beckman, S. Urbanczyk, & L. Walsh (Eds.), University of Massachusetts Occasional Papers in Linguistics [UMOP] (Vol. 18, pp. 181–209). Amherst: University of Massachusetts/GLSA. Katz, G. (2003). Event arguments, adverb selection, and the stative adverb gap. In E. Lang (Eds.). Modifying Adjuncts, de Gruyter. Kim, J. (1976). Events as Property Exemplifications. In M. Brand & D. Walton (Eds.). Action Theory. Reidel: Dordrecht, pp. 310–326. Maienborn, C. (2007). On Davidsonian and Kimian states. In I. Comorovski / K. von Heusinger (eds.): Existence: Semantics and syntax. Springer, Dordrecht, 107–130. Moltmann, F. (2004). Properties and kinds of tropes. New linguistic facts and old philosophical insights. Mind, 123(1), 1–41. Moltmann, F. (2007). Events, tropes, and truthmaking. Philosophical Studies, 134, 363–403. Moltmann, F. (2009). Degree structure as trope structure A trope-based analysis of comparative and positive adjectives. Linguistics and Philosophy, 32, 51–94. Moltmann, F. (2013a). Abstract objects and the semantics of natural language. Oxford: Oxford University Press. Moltmann, F. (2013b). On the distinction between abstract states, concrete states, and tropes. In A. Mari, C. Beyssade, & F. Del Prete (Eds.), Genericity (pp. 292–311). Oxford: Oxford University Press. Moltmann, F. (2013c). The semantics of existence. Linguistics and Philosophy, 36(1), 31–63. Moltmann, F. (2013d). Reference to numbers in natural language. Philosophical Studies, 162(3), 499–534. Moltmann, F. (2014). Propositions, attitudinal objects, and the distinction between actions and products. Canadian Journal of Philosophy, 43(5–6), 679–701. Moltmann, F. (2017). Natural language ontology. Oxford Encyclopedia of Linguistics, online. Moltmann, F. (2018). Truth Predicates, Truth Bearers, and their Variants. In J. Wyatt (ed.). Special issue Truth: Concept meets Property, Synthese, online first. Moltmann, F. (2020). Existence Predicates. Synthese, 197(1) 311–335 276 F. Moltmann Moltmann, F. (2019). Natural language and its ontology. In A. Goldman & B. McLaughlin (Eds.), Metaphysics and cognitive science (pp. 206–232). Oxford: Oxford University Press. Moltmann, F. (to appear). Variable objects and Truthmaking. In M. Dumitru (Ed.), The philosophy of Kit Fine. New York: Oxford University Press. Pustejovsky, J. (1995). The generative lexicon. Cambridge, MA: MIT Press. Schiffer, S. (1996). Language-created and language-independent entities. Philosophical Topics, 24(1), 149–167. Schiffer, S. (2003). The things we mean. Oxford: Clarendon Press. Strawson, P. (1959). Individuals. An essay in descriptive metaphysics. London: Methuen. van Inwagen, P. (1998). Meta-ontology. Erkenntnis 48, 233–250. Vendler, Z. (1967). Linguistics in philosophy. Ithaca: Cornell University Press. Williams, D. (1953). On the elements of being. Review of Metaphysics, 7, 3–18. Wright, C. (1983). Frege's Conception of Numbers as Objects. Aberdeen UP: Aberdeen. Yang, C. (2016). The price of linguistic productivity. How children lean to break the rules of language. Cambridge, MA: MIT Press. Yi, B.-Y. (2005). The logic and meaning of plurals. Part I. Journal of Philosophical Logic, 34, 459–506. Yi, B.-Y. (2006). The logic and meaning of plurals. Part II. Journal of Philosophical Logic, 35, 239.