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Bare Plurals in Spanish are Interpreted as Properties

McNally, Louise

Abstract

In this paper, I argue that bare plural noun phrases in Spanish unambiguously denote properties of individuals. I begin by using evidence from their incompatibility with kind-level predicates to show that Spanish bare plurals do not denote kinds. I then point to crucial ways in which their interaction with quantifiers is unlike that of other indefinite NPs (specifically, they have obligatory narrowest scope and cannot contribute the main restriction on a quantifier), and I conclude that bare plurals must therefore have a different semantics from other indefinites. I present a formal semantic analysis which allows for any verbal predicate in Spanish to combine with a propertytype noun phrase, showing how this analysis can also account for certain facts involving discourse anaphora to bare plurals. Finally, I discuss the advantages of the proposed analysis over those which try to maintain a uniform semantics for bare plurals and indefinites and indicate some of the general implications of the proposal.

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Abstract In this paper, I argue that bare plural noun phrases in Spanish unambiguously denote properties of individuals. I begin by using evidence from their incompatibility with kind-level predicates to show that Spanish bare plurals do not denote kinds. I then point to crucial ways in which their interaction with quantifiers is unlike that of other indefinite NPs (specifically, they have obligatory narrowest scope and cannot contribute the main restriction on a quantifier), and I conclude that bare plurals must therefore have a different semantics from other indefinites. I present a formal semantic analysis which allows for any verbal predicate in Spanish to combine with a propertytype noun phrase, showing how this analysis can also account for certain facts involving discourse anaphora to bare plurals. Finally, I discuss the advantages of the proposed analysis over those which try to maintain a uniform semantics for bare plurals and indefinites and indicate some of the general implications of the proposal. Key words: semantics, properties, bare nominals, incorporation *I am grateful to audiences at SUNY Buffalo, UC Santa Cruz, UC San Diego, and Stanford University for the opportunity to present this material and for useful comments. This work has also benefitted from discussion with Donka Farkas, Bill Ladusaw, Kari Swingle, and especially Josep M. Fontana, as well as from comments by two reviewers for the Catalan Journal of Linguistics. Catalan Journal of Linguistics 3, 2004 115-133 Bare Plurals in Spanish are Interpreted as Properties* Louise McNally Universitat Pompeu Fabra. Departament de Traducció i Filologia La Rambla, 30-32. E-08002 Barcelona [email protected] Table of Contents Preface 1. The problem 2. Distinguishing properties of Spanish BPs 3. Analysis 4. Why not treat BPs as ‘special’ indefinites? 5. Conclusion: some final support for the property analysis of BPs Appendix References Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 115 Preface This paper was originally presented at the Formal Grammar Conference at the European Summer School in Logic, Language and Information (ESSLLI) in Barcelona in 1995; a written version appeared in the proceedings of the conference, edited by Richard Oehrle and Glyn Morrill and distributed during ESSLLI as a spiral-bound set of photocopies. Under normal circumstances I would have subsequently revised the paper for publication; however, the circumstances were not normal – I was in the middle of moving to Barcelona, with all of the personal and professional changes and tràmits that that entailed – and the paper ended up abandoned, though it has circulated actively as an unpublished manuscript. I have chosen to leave the paper essentially in its original form for two reasons. First, it has been cited often enough that it seemed reasonable to make it available in a properly published form. But second and more importantly, given the amount of research that has been done on property-type arguments and on bare nominals in the years since the paper was written, revising it would have meant starting from scratch and writing an entirely new paper or, more likely, set of papers. The main theoretical goal of the paper was to present a formalization – the first, that I know of, together with that presented in Van Geenhoven (1995) – of the intuition that property-type nominals can systematically appear in argument positions normally occupied by referential or quantificational NPs. This formalization has since become familiar in natural language semantics largely through Van Geenhoven’s work and some joint work of ours (e.g. McNally and Van Geenhoven 1997, Van Geenhoven and McNally, in press) under the term semantic incorporation. However, there are now two important alternative compositional semantic treatments of property-type arguments, namely those in Farkas and de Swart (2003) and Chung and Ladusaw (2004). Carrying out a comprehensive comparison of these three approaches to handling property-type arguments is a pending task, but one that would have gone well beyond the scope of the work presented here. It would also have made little sense to revise substantially what is said here about Spanish bare plurals, given the large body of research on the semantics of bare nominals in the Romance languages has been carried out over the last decade. Perhaps the most comprehensive discussion of Spanish specifically appears in Bosque (1996) and the references cited there. An exhaustive study of the descriptive and theoretical issues, not only for Spanish but for the Romance languages more generally, has been conducted over a number of years by Dobrovie-Sorin and Laca (e.g. Laca 1996, Dobrovie-Sorin 1997, Dobrovie-Sorin and Laca 1998, and most notably Dobrovie-Sorin and Laca 2003, the last of these with a comprehensive bibliography). There is little that I could have added to what has already been said in these works. The proposal that bare nominals in the Romance languages denote properties when they appear in argument positions is not uncontroversial. Chierchia (1998) offers several arguments that such bare nominals denote kinds, and not properties. Some of these arguments have been questioned or refuted in e.g. Van Geenhoven (2000), Chung (2000), and Dobrovie-Sorin and Laca (2003); Krifka (2003) sug116 CatJL 3, 2004 Louise McNally Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 116 gests a compromise position between those adovcated by Chierchia and these latter researchers. The complexity of the data discussed in these works clearly suggests that this is an area in need of additional, very detailed empricial investigation, and not only on the Romance languages. Though the present paper has not been changed substantially, I have made some minor modifications. I have corrected various errors, changed examples or bits of text to improve readability, added a couple of footnotes (indicated as new in the text), suppressed some others, and updated the references, a number of which were originally cited as manuscripts and have since been published, in some cases in substantially revised form. 1. The problem In English, Spanish, and a number of other languages with article systems, both bare nominals and nominals marked with indefinite determiners1have similar distributions, and both have what appear to be existential interpretations in certain linguistic contexts. For example, all of the English examples in (1), and their Spanish counterparts in (2), have a reading truth conditionally equivalent to (3):2 (1) a. Martha bought an orange. b. Martha bought some oranges. c. Martha bought oranges. (2) a. Marta compró una naranja. Marta bought a.sg orange b. Marta compró unas naranjas. Marta bought a.pl oranges c. Marta compró naranjas. Marta bought oranges (3) (∃x: orange(x))[bought(m,x)] Although Carlson’s (1977) classic analysis of bare plurals (BPs) treated them as unambiguously kind-denoting (in contrast to indefinite singular noun phrases, Bare Plurals in Spanish are Interpreted as Properties CatJL 3, 2004 117 1. Hereafter, the term ‘indefinites’ will be used to refer to nominals which carry some kind of determiner, and will exclude determinerless nominals. Throughout this paper the term ‘noun phrase’ (NP) will be used purely descriptively – I will not address the issue of whether nominals carrying a determiner are better analyzed as NPs or Determiner Phrases. 2. Abbreviations in the interlinear translations are: sg = singular, pl = plural, acc = accusative, dat = dative, inf = infinitive, imp = imperfect, subj = subjunctive, cond = conditional. Unless otherwise marked, all verbs are indicative and any past tense is preterite (as opposed to imperfect). All verbs agree with their subjects in person and number, and in general determiners and adjectives agree with the nouns they modify in number and gender. Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 117 which were given a more complicated analysis), in at least one major subpart of the recent formal syntax and semantics literature it has been widely assumed (following e.g. Kratzer 1995, Diesing 1992, Wilkinson 1991) that BPs and indefinites can both be interpreted as ‘variables’ (see Kamp 1984, Heim 1982 for slightly different implementations). Though this analysis improves in several ways on Carlson’s original proposal, assigning BPs and indefinites to the same semantic type predicts that, all things being equal, either the two sorts of NPs should have identical semantic properties, or else there should be a principled explanation for any differences between them. In the case of Spanish, this prediction is incorrect. The goal of this paper is therefore to argue that the differences between Spanish BPs and indefinites follow naturally if we posit a simple difference in semantic type between the two sorts of NPs: specifically, I argue that Spanish BPs are unambiguously interpreted as properties, while this is not so for indefinites. This analysis is much in the spirit of Carlson’s proposal and reflects a long-held, if never fully formalized, intuition about the semantics of Spanish BPs (see Alonso 1954, Masullo 1992, Laca 1996, Bosque 1996 and references cited there; contrast Chierchia 1998 for a different position on Italian). In contrast, I will make no particular claims concerning indefinites in this paper, other than that, unlike BPs, they are not unambiguously interpreted as properties. For present purposes I will assume that they can be given the same treatment proposed for English indefinites in Heim (1982). The paper proceeds as follows. In section 2, I describe two essential differences between BPs and indefinites in Spanish. An analysis of these facts is presented in section 3. In section 4, I show why the proposed analysis is more successful than one which does not posit a type difference between BPs and indefinites. Section 5 contains some concluding comments on the implications of the analysis. 2. Distinguishing properties of Spanish BPs 2.1. Preface: Differences between Spanish and English BPs In order to situate the reader already familiar with the English facts, let me point out two differences between Spanish and English BPs. First, while English BPs can denote kinds (see (4)), BPs in Spanish and other Romance languages cannot (as seen in (5); (5b) is from Laca 1996: 262): (4) Tigers are becoming extinct in India. (5) a. En la India se están extinguiendo los tigres. in the India se are becoming-extinct the tigers ‘Tigers are becoming extinct in India.’ b. *En la India se están extinguiendo tigres. in the India se are becoming-extinct tigers 118 CatJL 3, 2004 Louise McNally Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 118 Thus, in claiming that Spanish BPs are interpreted as properties, I am committed to the position that a property is distinct from a kind. While it will be clear below how I distinguish the two formally, I will have nothing to say about the substantive nature of the distinction here. Spanish BPs also lack the ‘functional’ reading discussed at length for English in Condoravdi (1994), and illustrated in (6a). This reading is intuitively universal but not generic: the BP in (6a) refers to all of the union representatives at the factory (as opposed to just some of them)–thus, is universal as opposed to existential, but it is not generic because it does not refer to all union representatives in general. To express this reading in Spanish, a definite plural NP must be used in place of the BP3(compare the translations of (6a) in (6b,c), where the BP, if acceptable at all,4gives rise only to a ‘weak’ existential interpretation): (6) a. Last week Nissan announced that it would lay off one-third of its employees at its Barcelona factory. Union representatives have protested vehemently. b. La semana pasada Nissan anunció que despediría un tercio de the week passed Nissan announced that fire.cond one third of sus empleados en Barcelona. Los representantes sindicales han its employees in Barcelona. the representatives syndical have protestado enérgicamente. protested energetically. c. ...??Representantes sindicales han protestado enérgicamente. representatives syndical have protested energetically. A third difference between Spanish and English will be evident in the data below. 2.2. Spanish BP/indefinite contrasts For now, I will focus on two differences between the behavior of BPs and indefinites in Spanish. First, BPs must take narrowest possible scope with respect to any other operators in the same clause, including negation, intensional predicates, and modals: All of the (b) examples below are unambiguously assigned the narrow scope (d) readings. In contrast, indefinites may take variable scope; the (a) examples are all interpretable with either the wide scope (c) or narrow scope (d) readings Bare Plurals in Spanish are Interpreted as Properties CatJL 3, 2004 119 3. For this reason it is tempting to try to relate the absence of the functional reading in Spanish to the absence of the kind reading, though Condoravdi (personal communication) has expressed pessimism on this point. I refer the reader to her work for additional discussion. 4. There is a rather strong preference in Spanish for putting BPs postverbally except in certain discourse contexts (see e.g. Laca 1994 for cases where they can appear preverbally), which makes sentences such as (6c) sound unnatural. Nonetheless, according to the speakers I consulted, this particular sentence sounds even worse with the BP in postverbal position. The fact that it is difficult to construct felicitous, natural-sounding examples with BPs which could even be candidates for the functional reading underscores the fact that this reading is not expressed in Spanish by BPs at all. Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 119 ((7a,b) and (8a,b) are from Laca 1996:253,254; see Carlson 1977 and Wilkinson 1991 for similar contrasts in English): (7) a. A la reunión no asistieron algunos profesores. (c or d) at the meeting not attended some.pl professors ‘The meeting was not attended by some professors.’ b. A la reunión no asistieron profesores. (d only) c. (∃x: professor(x)) [¬attend(x,m)] d. ¬(∃x: professor(x))[attend(x,m)] (8) a. María cree que Sara ha robado unos libros. (c or d) Maria believes that Sara has stolen a.pl books ‘Maria believes that Sara has stolen some books.’ b. María cree que Sara ha robado libros. (d only) c. (∃x : book(x))[believe(m, ^stole(s,x))] d. believe(m,^(∃x: book(x))[stole(s,x)]) (9) a. Hoy Juana tiene que leer unos artículos. (c or d) Today Juana has that read.inf a.pl articles ‘Today Juana has to read some articles.’ b. Hoy Juana tiene que leer artículos. (d only) c. (∃x: article(x))[[read(j,x)]] d. (∃x: article(x))[read(j,x)] A second difference between bare plurals and indefinites in Spanish is that bare plurals cannot supply the primary restriction on adverbial quantification over individuals,5while indefinites can. For instance, the indefinite in (10a) naturally expresses a generalization about members of parliament, as in (10b). (10) a. Un miembro del parlamento trabaja normalmente ocho horas a.sg member of-the parliament work normally ocho hours al día. to-the day. ‘A member of parliament normally works eight hours per day.’ b. (normally x: mp(x))[8hrs/day(work(x))] 120 CatJL 3, 2004 Louise McNally 5. If one takes the position that adverbs necessarily quantify over eventualities or cases, then this quantification could perhaps alternatively be described as being over equivalence classes of cases which are distinguished by the identity of some participant. However, for simplicity, I will talk as if adverbs can quantify over individuals. See e.g. de Swart (1993) on adverbial quantification. Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 120 In contrast, (11a) with the BP can only express quantification over some set of contextually-determined situations, and not over members of parliament (see (11b), where CR stands for a contextual restriction on the domain of situations). In other words, (11a) would be false in a situation in which 80 out of 100 members of parliament have eaten at La Panadella restaurant, but only one time each, while the vast majority of the time, the meals eaten by members of parliament are eaten at the bar in the train station: (11) a. Miembros del parlamento comen a menudo en La Panadella. members of-the parliament eat often in La Panadella ‘Members of parliament often eat at La Panadella.’ b. (often e: CR(e))[(∃x: member-parliament(x))[eat(x,p,e)]] In this respect, Spanish BPs appear to contrast with English BPs; both sentences in (12) have a reading that appears to express a generalization about welltrained dogs: (12) a. A well-trained dog always sits quietly. b. Well-trained dogs always sit quietly. To summarize, we have seen so far that Spanish BPs contrast with indefinites in two ways: (1) bare plurals must take narrow scope with respect to any operators in the sentence, while indefinites are not limited in their scope possibilities; and (2) bare plurals cannot contribute the primary restriction on adverbial quantification over individuals, while indefinites can. 3. Analysis I will embed the analysis of these facts in a slightly modified version of Heim’s (1982) File Change Semantics (FCS). In FCS, like other theories of dynamic interpretation (e.g. DRT, Kamp 1984; Dynamic Montague Grammar, Groenendijk and Stockhof 1990), the semantic value of a sentence – its context change potential – is characterized as its ability to affect the shared assumptions of the participants within a conversational context C. Here, the context change potential for a sentence will be a function from ‘files’ (i.e. discourse models) to files, where a file for a context Cis a pair <Dom(C),Sat(C)>. Dom(C) is treated here as a set of indexed variables (corresponding to referents in the domain of discourse); Sat(C), as the set of pairs of possible worlds and variable assignment functions <w,g> such that <w,g> satisfies all of the sentences in C. Truth can in turn be defined in terms of satisfaction: A sentence Sis true in a model Mand world w with respect to the file for a context Ciff there is some gsuch that <w,g> is an element of Sat(C+S). Specific interpretation rules will be provided below; see Heim (1982) for additional details. The assumptions underlying the analysis can be summarized as follows. First, as stated above, Spanish BPs will be interpreted unambiguously as properties. How Bare Plurals in Spanish are Interpreted as Properties CatJL 3, 2004 121 Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 121 exactly properties are characterized is not crucial for my purposes, other than that they be kept distinct from kinds. In order to avoid having to introduce a lot of formal machinery, I will treat them in a familiar way as functions from indices to sets of individuals (i.e. type <s,<e,t>>).6Indefinites, in contrast, will be interpreted as formulas (i.e., type t) containing a variable which must be assigned a value. Correlated with this type difference is the fact that the indefinite introduces a discourse referent (though perhaps a quantificationally or modally subordinated one, see Heim 1982 for further discussion), while the BP does not.7 The second assumption is that it must be possible for a predicate like comprar (‘to buy’) to combine with a property-type expression while retaining fundamentally the same content as when it combines with an ordinary entity-denoting NP. A simple way to do this is to associate Spanish verbal predicates with ‘floating’ types subject to a special interpretive convention. Specifically:8 (13) For all n-ary predicates P(n>1), and for all i, 1≤i≤n, if P’s i-th argument is of type e, then P’s i-th argument can also be of type <s,<e,t>>. The interpretation rule for the combination of a predicate Pwith a propertytype argument Q9will be as in (14): (14) For all Q of type <s,<e,t>>, and for all n-ary predicates Pas in (13) with ordered arguments <al... an>, <w,g> ∈Sat(C+P(al,...Qi, ...an)), iff there is some xand some g ′ (where g ′ agrees with g on all elements in the Dom(C)) such that g′(x) ∈[[Q]] and <w,g′> ∈Sat(C+ P(al, ...xi,... an)). 122 CatJL 3, 2004 Louise McNally 6. And I assume, following Carlson (1977), that kinds are entities, though I will not discuss kinds further here. 7. Casting the analysis in terms of unification-based grammar formalism would perhaps sharpen this relationship: for example, BP and indefinite counterparts could be specified with identical values for a CONTENT feature, while only indefinites would receive a value for a REFERENTIAL INDEX feature. Such a formalism could perhaps also allow for a simpler and more general account of the way predicates combine with indefinites and BPs. For example, combining a transitive verb with either an indefinite or BP would have the syntactic effect of saturating the argument corresponding to the position filled by the NP; the semantics could simply unify the content of the NP with that of the verb. In the case of BPs, what would get unified would simply be a condition on the description of whatever individual ultimately satisfies the participant role associated with the BP; in the case of indefinites, there would be an added condition that this individual be a possible value for the referent xiintroduced by the indefinite (and which will therefore be novel with respect to the discourse). 8. Since some predicates in Spanish do not allow BP arguments, or allow such arguments only under certain circumstances, perhaps the option of accepting a property-type argument ultimately should not be available to all verbs in the language, contrary to what is proposed in (13). How best to handle the idiosyncratic limitations on the distribution of BPs – whether the conditions are syntactic, semantic, pragmatic or some combination of these is a matter of ongoing research. 9. I assume the syntactic rules of the language will ensure that Qcorresponds to an NP as opposed to e.g. an AP or VP. Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 122 This rule is very much in the spirit of Carlson’s (1977) interpretation rule for bare plural arguments of stage-level predicates, insofar as any existential force associated with the bare plural comes from the lexical entailments of the predicate, and not from either inherent quantificational force associated with the bare plural or the equivalent of Heim’s (1982:Chap. 2) ‘text-level existential closure.’10 Let us now see how these assumptions account for the facts discussed in the previous section. Since indefinites will be treated essentially as in Heim (1982), I will discuss only the BP facts. Consider first the obligatory narrow scope of BPs in sentences such as (7b), repeated in (15): (15) A la reunión no asistieron profesores. at the meeting not attended professors ‘The meeting was not attended by (any) professors.’ A Heim-style Logical Form (LF) for this sentence under the analysis proposed here would be translated as in (16), where xicorresponds to a familiar discourse referent and ^P is the constant corresponding to the property of being a professor: (16) ¬[meeting(xi) ∧attend(^P, xi)] Following an adapted version of the rules in Heim (1982:363ff.), provided in the Appendix, the context change potential for (15) is as in (17). Since the sentence is quantificationally closed and therefore does not introduce any new, persistent discourse referents, the domain of the file will not change; the resulting satisfaction set will be the complement of the set of world-assignment function pairs that would have satisfied the scope of the negation in (16), most crucially, the second conjunct: (17) Context change potential for (15): Dom(C+(15)) = Dom(C); Sat(C+(15)) = {<w,g> ∈Sat(C) | there is no pair <w,g′> (where g′agrees with g on all elements in Dom(C)) such that <w,g′> ∈Sat((C+meeting(xi)) + attended(^P, xi))}. A world-assignment function pair <w,g> will be in Sat(C+ attended(^P,xi)) iff there is some xsuch that g(x) is a professor and attended the meeting in w. The complement of this set of world-assignment function pairs will be all those in which there is no such x. The effect of narrow scope with respect to negation has thus been achieved. The analysis will work analogously no matter what operator is involved. The obligatory narrow scope of Spanish BPs therefore follows directly, in essentially the same way as it followed on Carlson’s analysis of English BPs. Now consider the second fact discussed above, namely that BPs cannot constitute the primary restriction on adverbial quantification over individuals. We need Bare Plurals in Spanish are Interpreted as Properties CatJL 3, 2004 123 10. See also Van Geenhoven (1995) and Glasbey (1998) for very similar proposals. Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 123 Second, the analysis suggests why BPs resemble incorporated nominals (see Masullo 1992, Van Geenhoven 1995) without requiring that they be incorporated in any interesting syntactic or morphological sense – a desirable result since Spanish BPs are syntactically quite independent. On the view taken here, incorporated nominals (which, like BPs, are generally determinerless; see e.g. Mithun 1984) and BPs would resemble each other in their semantic type – both interpreted as properties, as e.g. van Geenhoven (1993, 1995) proposes – and therefore likely to combine with predicate denotations via similar semantic operations. Although the special pragmatic functions of incorporation are still poorly understood, given the fact that it is so widespread cross-linguistically, it seems quite plausible that speakers of languages without morphological or syntactic incorporation constructions would find use for other linguistic devices (such as BPs with their special semantics) that would allow them to approximate the pragmatic functions of incorporation. Third, the analysis raises questions about our understanding of the semantics of English BPs. In particular, we should reconsider in light of the results presented here the consequences of the fact that English BPs can denote kinds, in an effort to see whether the availability of this denotation accounts for the various differences between English and Spanish BPs. Such a reevaluation of the literature on English BPs could have important implications for our understanding of the contribution of kind-denoting expressions to characterizing sentences. Linguists have uncovered increasing evidence over the past several years that natural languages make considerable use of property-type nominals. This evidence emphasizes the need for a formal semantics that can make use of the interpretations of such expressions without paradox or loss of generalization. The use of properties leads to new ways of thinking about the existential force we intuitively associate with NPs, about the ways in which discourse anaphora is licensed, about what it means for an NP to be ‘weak’ (see Ladusaw 1994, McNally 1998, and even about intensionality (see Zimmermann 1993). I hope that this preliminary analysis of the Spanish facts and the theoretical and descriptive issues it raises will encourage further development of semantic theories that can make insightful use of property-type argument expressions. Appendix General interpretation rules, relative to a Model M= <E,[[·]]>, Ea set of entities, [[·]] an interpretation function (adapted from Heim (1982:363ff.): 1. Let ϕ be an atomic formula of the form β ( α 1,... α n). Then: Dom(C+ ϕ ) = Dom(C) ∪{ α l,... α n}; Sat(C+ ϕ ) = {<w,g> ∈Sat(C) | <g( α l),...g( α n)> ∈[[ β ]] in w}. 2. Let ϕ be a formula with immediate constituent formulas ϕ 1... ϕ nin that order. Then: Dom(C+ ϕ ) = Dom((C+ ϕ 1) ... + ϕ n); Sat(C+ ϕ ) = Sat((C+ ϕ 1) ... + ϕ n). 130 CatJL 3, 2004 Louise McNally Cat.Jour.Ling. 3 001-182 10/9/04 13:26 Página 130 3. Let ϕ be a quantified formula consisting of a binary quantificational operator Q and a complex formula consisting of immediate subformulas ϕ 1 and ϕ 2, in that order. Then: Dom(C+ ϕ ) = Dom(C); Sat(C+ ϕ ) = {<w,g> ∈Sat(C) | for Q-many pairs <w,g′> (where g′agrees with gon all elements in Dom(C)) such that <w,g′> ∈Sat(C+ ϕ 1), there is some <w,g′′> (where g′′ agrees with g′on all elements of Dom(C+ ϕ 1)) such that <w,g′′> ∈Sat((C+ ϕ 1)+ ϕ 2)}. 4. Let ϕ be a quantified formula consisting of a negator and the formula ψ . Then: Dom(C+ ϕ ) = Dom(C); Sat(C+ ϕ ) = {<w,g> ∈Sat(C) | there is no pair <w,g′> (where g′agrees with g on all elements in Dom(C)) such that <w,g′> ∈Sat(C+ ψ )}. References Alonso, Amado (1954). «Estilística y Gramática del Artículo en Español». In: Alsono, Amado. Estudios lingüísticas. Temas españoles. 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