Abstract I argue that standard definitions of quantifiers are inadequate and offer a new one. The new definition categorizes expressions as quantifiers in accordance with our pre-theoretical judgments, it is broadly applicable to both formal and natural languages, and it eschews unnecessary theoretical commitments about the details of the syntax and semantics of these expressions. Many debates in semantics revolve around questions whether certain expressions – descriptions, numerals, the verb ‘exist’, the auxiliary ‘ought’, etc. – are quantifiers or something else.1 On the face of it, these questions are perplexing: they ask how to fit certain natural language expressions into syntactic categories of formal languages. Why would we want to do that? Because the relevant formal languages were, by and large, set up to make deductive inferences more perspicuous, and accordingly, they base their syntactic categorization of expressions on their inferential profile. Thus, while ‘quantifier’ is officially a syntactic label, it is used as a proxy for a semantic one. Discovering that an expression of Swahili, Polish, or Japanese is a quantifier we learn that it shares a kind of meaning with various expressions across languages. To make progress in semantic categorization we need an account of what makes something a quantifier that is not wedded to idiosyncratic features of particular languages. In this paper, I argue that the accounts currently available are insufficient for this purpose and then propose an alternative account that finesses these challenges. 1. The standard characterization of quantifiers in formal languages Quantifiers in formal languages are basic or defined. Membership on the short list of basic quantifiers is taken as primitive; other quantifiers are defined using these plus other logical constants. All quantifiers share the following characteristics: they are associated with a variable, they combine with a formula, and they yield another formula in which the free occurrences of the variable in the original formula are bound. The quantifiers in a given formal language can thus be characterized by providing the list of basic quantifiers, a list of logical constants, and a specification of the syntax. The category of quantifiers thus understood is too narrow. Consider, for example, the Rescher quantifier, added to the language of first-order logic with the following syntactic and semantic clauses:2 (Syn r) If ϕ is a formula and x a variable, r x.ϕ is a formula (Sem r) Rx.ϕM,g is true iff for more than half of the o∈ DM, ϕM,g[x:o]is true Being introduced this way should intuitively guarantee that R is a quantifier. But it is well known that R is not definable in first-order logic (see Herre et al. 1991), so unless it is a basic quantifier of the expended language, by the standard formal characterization, it would not count as a quantifier at all. Another problem with the standard characterization is that to ensure that ‘every’ and ‘some’ count as quantifiers in English, we would need to posit unpronounced variables in (1) and (2), and claim that ‘every’ and ‘some’ combine not with ‘philosopher’ to form a determiner phrase, but rather, with some formula obtained by first combining ‘philosopher’ and ‘is snub nosed’: (1) Every philosopher is snub nosed. (2) Some philosopher is snub nosed. (1′) Every x [philosopher is snub nosed x] (2′) Some x [philosopher is snub nosed x] Moreover, ‘philosopher’ and ‘snub nosed’ must combine in different ways within (1′) and (2′), given that (1) is true just in case everything is snub nosed if it is a philosopher, and (2) just in case something is both a philosopher and snub nosed. One way to mark the difference is to posit different unpronounced connectives that link two open formulas: (1′′) Every x [philosopher x ⊃ is snub nosed x] (2′′) Some x [philosopher x & is snub nosed x] While in the end we may have to acknowledge some discrepancy between the syntax and the semantics of English,3 we should not posit entirely different syntactic and semantic forms for sentences simply because we decided to adopt a characterization for quantifiers that is a bad fit for natural languages. We should look for a different characterization instead. 2. The standard characterization of quantifiers in natural languages The standard definition of quantifiers employed in natural language semantics avoids these pitfalls. Instead of stipulating that quantifiers bind variables, it states that they belong to a certain semantic type. And instead of stipulating that they are definable in terms of basic quantifiers plus logic, it states that they satisfy certain requirements of logicality.4 On this account, determiners with intuitively quantificational meanings, like ‘every’, ‘some’, ‘most’, ‘many’, ‘eight’, ‘at least five but no more than seven’, etc. are standardly taken to be of type 〈〈e, t〉,〈〈e, t〉, t〉〉, i.e. their semantic values are functions taking sets to sets of sets. This might be also the semantic type of possessives like ‘my’, ‘Jill’s’, or ‘the tallest person’s’, and of paradigmatically referring expressions like ‘she’, ‘Jack’, or ‘this yellow pencil’. These are intuitively not quantifiers, which is usually explained by pointing to the fact that they are not logical expressions: their semantic values can vary with permutations of the domain of the model. But saying that quantifiers are logical expressions of type 〈〈e, t〉,〈〈e, t〉, t〉〉 doesn’t capture the category we are after. Quantifiers occur in multiple semantic types – ‘someone’, ‘everywhere’, ‘always’, certainly don’t have the semantic type 〈〈e, t〉,〈〈e, t〉, t〉〉. Which are the semantic types that contain quantifiers? Presumably not all of them – we don’t want to collapse the notion of a quantifier to that of a logical expression. Another problem is that many intuitively quantificational determiners exhibit vagueness (e.g. ‘many’, ‘few’, ‘rarely’, etc.) and that vague expressions are arguably non-logical.5 One might try to relax the logicality requirement on quantifiers but it is hard to see how to do that without dropping it altogether, which would allow proper names, complex demonstratives, and definite descriptions into the category. Finally, there is a concern about applying the definition to most formal languages. Since the universal and existential quantifiers can take any open or closed formula as input, assigning a semantic type to them is not straightforward. (This is one good reason why they are usually interpreted syncategorematically.) If we stipulate that all formulae belong to type t, the type of quantifiers must be 〈t, t〉. This would be a problem, given that we would have to assign the same type to negation, a logical expression that is not a quantifier. Thus the standard definition of quantifiers for natural languages fits formal languages as poorly as the standard definition of quantifiers used for formal languages fits natural languages. To try to say what all quantifiers have in common, we should opt for a new approach. 3. Cardinality quantifiers defined Substitutional clauses specify the truth-conditions of a quantified sentence through a generalization over the truth-conditions of its instances. For example, ‘Everyone was bored’ is true just in case the number of its false instances (i.e. the number of false sentences of the form ‘ a was bored’) is zero and ‘Someone thought everyone was bored’ is true just in case the number of its true instances (i.e. the number of true sentences of the form ‘ b thought everyone was bored’) is larger than zero. In the remainder of this paper, I argue that, quantifiers in all languages, formal or natural, are devices of generalization over instances. What varies across languages is only how instances are best defined. The instances of ∀x.ϕ and ∃x.ϕ are usually taken to be formulae of the form ϕ[x/a]. If we wish to give the truth-conditions of ∀x.ϕ and ∃x.ϕ in terms of their instances, we have to make sure that each value x can take is associated with exactly one constant in the language. Since this is usually not the case (and when the language has a finite lexicon and an infinite model it cannot be the case), it is better to define instances by appealing directly to the entities we quantify over: Instances in formal languages: The instances of Qx.ϕ relative to the unembedded occurrence of the quantifier Q are ordered pairs of the form 〈 ϕ, a〉, where a is an entity in the domain of the model M that can be a value of x. 〈 ϕ, a〉 is positive with respect to the model M and the assignment g just in case ϕM,g[x:a] is true; otherwise it is negative with respect to that model and assignment. Given this definition, the following equivalences hold: (3) ∀x.ϕM,g is true iff the number of negative instances of ∀x.ϕ relative to the leftmost occurrence of ∀ with respect to M and g is zero. (4) ∃x.ϕM,g is true iff the number of positive instances of ∃x.ϕ relative to the leftmost occurrence of ∃ with respect to M and g is larger than zero. (3) and (4) are not intended to replace the usual semantic clauses of ∀x.ϕ and ∃x.ϕ. All they do is make plain one of the consequences of those clauses, given how we decided to characterize positive and negative instances. They suggest the following definition: Cardinality quantifiers: An expression ε is a cardinality quantifier iff whenever it occurs unembedded in a sentence σ, the truth-value of σ (with respect to some parameters) depends only on the cardinalities of the sets of its positive and negative instances relative to this occurrence of ε (with respect to those parameters). ∀ and ∃ are cardinality quantifiers, and the same holds of quantifiers that can be defined in terms of them (e.g. ∃5, ∃<8, etc.). So is R, since the following clause captures the truth-conditions of Rx.ϕ: (5) Rx.ϕM,gis true iff the number of positive instances of Rx.ϕ relative to the leftmost occurrence of R with respect to M and g is larger than the number of negative instances relative to the leftmost occurrence of R with respect to M and g. 4. Quantifiers defined Natural languages (at least on the surface) do not contain variables. The closest thing they have are demonstratives. Staying as close as possible to the definition of instances for formal languages, we get the following: Instances in natural languages: Let the expression ε occur unembedded in sentence σ and let δ be a demonstrative intersubstituible with ε. Then the instances of σ relative to ε are ordered pairs of the form 〈 σ[ε/δ], a〉 where a is some entity or other. An instance is positive with respect to the context c just in case σ[ε/δ]c[δ:a] is true, and negative with respect to the context c just in case σ[ε/δ]c[δ:a] is false. ( c[δ:a] is a context that differs from c only in assigning a as a semantic value to δ.) This definition has demonstratives where the earlier has variables and contexts where the earlier had assignment functions. The syntax of a typical formal language guarantees that the scope of an unembedded quantifier is always the first component of its instances but the syntax of natural languages does not work this way – we need to allow some flexibility in identifying instances. For example, the first component of the instances of ‘Two coins are in my pocket’ relative to the occurrence of ‘two’ should be ‘That coin is in my pocket’, not ‘That coins are in my pocket’; the first component of the instances of ‘Several of the dogs fell asleep’ relative to the occurrence of ‘several’ should be ‘That dog fell asleep’ or ‘That one of the dogs fell asleep’, not ‘That of the dogs fell asleep’. These are relatively minor differences in the distribution of quantifiers and demonstratives; we assume that the relevant quantifiers and demonstratives are actually intersubstitutable, but substitution requires syntactic adjustment.6 We can now use this definition of instances and check whether ‘every’ fits the definition of cardinality quantifiers given in the previous section. The instances of ‘Every philosopher is snub nosed’ relative to the occurrence of ‘every’ are ordered pairs of the form 〈‘That philosopher is snub nosed’, a〉, where a is some entity or other. (We are ignoring models.) In a positive instance a is a philosopher who is snub nosed; in a negative instance a is a philosopher who is not snub nosed. (We assume, as seems plausible, that in contexts where the demonstrative does not refer to a philosopher ‘That philosopher is snub nosed’ is semantically deviant, and hence, neither true nor false.) Then ‘Every philosopher is snub nosed’ is true in the context c just in case the number of its negative instances relative to the occurrence of ‘every’ with respect to c is zero. So, ‘every’ plausibly counts as a cardinality quantifier.7 The same holds for expressions usually categorized as quantificational determiners (‘some’, ‘no’, ‘most’, etc.) – provided they combine with count nouns.8 When quantificational determiners combine with mass nouns, the situation is different. The instances of ‘Jack spent most time Saturday with his friends’ relative to the occurrence of ‘most’ are ordered pairs of the form 〈‘Jack spent that time Saturday with his friends’, a〉; among these the positive ones are the ones where a is a time Saturday Jack spent with his friends, the negative ones are the ones where a is a time Saturday Jack spent without his friends. The truth-value of the sentence is not fixed by the number of positive and negative instances (both of which would be infinite, assuming time is infinitely divisible and Jack spent some but not all of the time Saturday with his friends). What matters instead is the duration of the fusion of the second components of all positive instances and the duration of the fusion of the second components of all negative instances – the sentence is true just in case the former is larger than the latter. This shows that ‘most’ is not a cardinality quantifier. Yet, it is obviously a quantifier. What we need is a way to generalize the definition of cardinality quantifiers. Let μ be a partial function that assigns some numerical value to entities and let these numerical values measure the quantity of those entities, in some intuitive sense. (In the simplest case, μ assigns 1 to each entity indicating that it is one thing, but it could also assign to entities their mass in pounds, their duration in seconds, etc.) Let μ′ be a function that assigns some numerical value to sets of entities aggregating the values μ assigns to their members. (In the simplest case, μ′ assigns to a set the sum of the values μ assigns to their members, but it could also assign the maximum of those values, or the value μ assigns to their fusion, etc.) Finally, let μ′′ be a function that assigns to a set of instances (which are ordered pairs) what μ′ assigns to the set of their second components. Here, then is our general definition of quantifiers: Quantifiers: An expression ε is a quantifier iff whenever it occurs unembedded in a sentence σ there is a contextually salient measure function μ such that the truth-value of σ (with respect to some parameters, including context) depends only on the values μ′′ assigns to the set of its positive and negative instances relative to this occurrence of ε (with respect to those parameters). Quantificational determiners that can combine with mass nouns (‘some’, ‘most’, ‘much’, etc.) and adverbs of quantification (‘sometimes’, ‘usually’, ‘often’, etc.) are plausibly quantifiers according to this definition. The instances of a sentence like ‘Marie visited Paris exactly twice’ are ordered pairs of the form 〈‘Mary visited Paris in that case’, a〉.9 Problems of how to ‘count’ cases of Marie visiting Paris are bypassed – if there is some contextually salient measure function μ such that μ′′ assigns 2 to the set of positive instances of this sentence, it comes out true in the context. 5. Some consequences The proposed definition has a number of welcome consequences. First, we can explain why ‘or’ and ‘not’ are not quantifiers. There is no demonstrative that can take their syntactic position, and so, sentences where they occur unembedded do not have instances. We also have no problem granting that quantifiers can be vague. Whether ‘Many books are on the table’ is true or false in a context depends on more than the quantity of books on the table and the quantity of books not on the table – the standard of precision matters too. Still, the truth-value of sentence with respect to a context depends only on the number of its true and false instances relative to the occurrence of ‘many’ with respect to that context – the standard of precision is the same for the sentence and its instances. The account predicts that while ‘every’ is a cardinality quantifier ‘every book’ is not. The demonstrative ‘that’ can be substituted for either, generating two sets of instances for the sentence ‘Every book is on the table’: the set of pairs 〈‘That book is on the table’, a〉 and the set of pairs 〈‘That is on the table’, a〉, where a is some entity or other. The sentence expresses a cardinality generalization over the first set: it is true with respect to a context just in case none of them is negative with respect that context. But it does not express a cardinality generalization over the second: whether ‘Every book is on the table’ is true with respect to a context cannot be determined just by counting instances from the second set that are positive and negative with respect to that context – we also need to know how many of the demonstrated things are books. Assuming, as seems plausible, that English determiners that combine exclusively with count nouns are either cardinality quantifiers or not quantifiers at all, we can conclude that ‘every book’ is not a quantifier.10 For analogous reasons, a possessive phrase like ‘Jack’s’ or a proper name like ‘Jill’ are also not quantifiers. What about ‘only’? ‘Only books are on the table’ is true just in case the number of things on the table that are not books is zero, ‘Only water is in the pitcher’ is true just in case the amount of stuff in the pitcher that is not water is zero.11 But these truth-conditions cannot be captured as generalizations over positive and negative instances relative to the occurrence of ‘only’: information about how many books are on the table and how many are not on the table does not settle the truth-value of the first sentence and information about how much water is in the pitcher and how much water is not in the pitcher does not settle the truth-value of the second. Thus, on the proposed account ‘only’ is not a quantifier. Semanticists often reach the same conclusion through an argument which shows that ‘only’ is not a determiner, and so, it does not have the semantic type 〈〈e, t〉,〈〈e, t〉, t〉〉 (see, for example, Keenan and Stavi 1986). But couldn’t there be a determiner that works as ‘only’ seems to in this sentence? The standard view is that there could not: if ‘only’ were a quantifier, it would not be conservative and it appears to be a semantic universal that natural language quantifiers are conservative.12 But a quantifier absent from natural languages is still a quantifier. Better to say that ‘only’ is not a quantifier at all.13 Footnotes 1 In generalized quantifier theory, the term ‘quantifier’ is reserved for the semantic values of quantifier expressions. What I am calling quantifiers are expressions of formal or natural languages. 2 This device was introduced in Rescher 1962. The notation used is standard: ·M,g is a function that assigns semantic values to expressions of the language relative to the model M and the assignment function g. DM is the domain of the model M, a non-empty set of objects. g[x:o] is an assignment function that assigns to x the object o∈ DM, and assigns to all other variables what g does. Here and throughout the paper italics is used for variables in the object language, and boldface for variables in the metalanguage. 3 For a useful discussion of whether grammatical form is a misleading guide to logical form, see Chapter 6 of Sainsbury 1991. 4 Actually, in semantics it is customary to call all expressions that belong to certain semantic types quantifiers. For a detailed discussion of the logicality requirements, see Chapter 9 of Peters and Westerståhl 2006. For some considerations in favor of the claim that natural language quantifiers are all logical, see Section 1.7 of Glanzberg 2006. 5 This point served as one of the principal motivations for developing the supervaluationst account of vagueness; see Fine 1975. 6 The so-called floating quantifiers (e.g. ‘all’ in ‘The students have all passed their exams’) clearly can occur in positions where demonstratives cannot. Brisson 2000 argues that floating occurrences of ‘all’ are not quantificational. Rather, they eliminate exceptions to the universal quantification provided by the definite plural. 7 The caveat is necessary because not all unembedded occurrences of ‘every’ share syntactic form with ‘Every philosopher is snub nosed’. The claim that ‘every’ is a cardinality quantifier is an empirical hypothesis about English that cannot be conclusively proved until we have a semantics for the entire language. 8 A referee has suggested that we could define instances in natural languages without mentioning demonstratives. This would be desirable, since it would align the definitions used for natural and formal languages. Unfortunately, the suggestion fails. Here is the alternative definition: Let the expression ε occur unembedded in sentence σ; the instances of σ relative to ε are ordered pairs of the form 〈 σε, a〉, where σε is obtained from σ by deleting ε (and perhaps changing the number features on some constituents within σ) and a is some entity or other; 〈 σε, a〉 is positive if a satisfies σε and negative otherwise. Now, consider 〈‘philosopher is snub nosed’, Donald Trump〉. According to the alternative definition, this is an instance of ‘Most philosophers are snub nosed’ relative to the occurrence of ‘most’ and it must be either positive or negative. But this is wrong: to correctly predict that ‘most’ is a cardinality quantifier we want to be able to say that ‘Most philosophers are snub nosed’ is true iff it has more positive than negative instances, which in turn requires that instances whose second component is a non-philosopher should be neither positive nor negative. This is precisely what the appeal to demonstratives ensures in the definition adopted in this paper. 9 I am assuming, contra Lewis (1975), that adverbs of quantification are not unselective binders. For support of the view that they quantify over cases (a.k.a. situations), see Berman 1987, Heim 1990 and von Fintel 2004. 10 We could introduce the notion of a restricted quantifier as follows: ε is a restricted quantifier with respect to domain D iff sentences where it takes widest scope are generalizations over instances whose second components are taken from D. ‘Every book’ is a restricted quantifier with respect to any domain consisting solely of books, but not a restricted quantifier with respect to any other domain. 11 There is also a presupposition that there are some books on the table, which I am ignoring. 12 A quantificational determiner Q is conservative just in case [S[DPQF]G] is equivalent to [S[DPQF]F and G]. That this is a semantic universal was first proposed in Barwise and Cooper (1981). 13 Thanks to Tamar Szabó Gendler, Eric Guindon, Anna Szabolcsi, Rich Thomason and Tim Williamson for discussion and comments. The manuscript has improved substantially thanks to comments by two anonymous reviewers for Analysis. References Barwise J., Cooper R.. 1981. Generalized quantifiers and natural language. Linguistics and Philosophy 4: 159– 219. Google Scholar CrossRef Search ADS Berman S. 1987. Situation-based Semantics for Adverbs of Quantification. In J. Blevins and A. Vainikka eds University of Massachusetts Occasional Papers in Linguistics 12: 45– 58. Amherst: University of Massachusetts Press. Brisson C. 2000. Floating quantifiers as adverbs. In Proceedings of the Fifteenth Eastern States Conference on Linguistics, eds R. Daly and A. Riehl, 13–24. Ithaca, NY: CLC Publications. Fine K. 1975. Vagueness, truth, and logic. Synthese 30: 265– 300. Google Scholar CrossRef Search ADS Glanzberg M. 2006. Quantifiers. In The Oxford Handbook of Philosophy of Language , eds Lepore E., Smith B., 794– 821. Oxford: Oxford University Press. Heim I. 1990. E-type pronouns and donkey-anaphora. Linguistics and Philosophy 13: 137– 77. Google Scholar CrossRef Search ADS Herre H., Krynicki M., Pinus A., Väänänen J.. 1991. The Härtig quantifier: a survey. Journal of Symbolic Logic 56: 1153– 83. Google Scholar CrossRef Search ADS Keenan E., Stavi J.. 1986. A semantic characterization of natural language determiners. Linguistics and Philosophy 9: 253– 326. Google Scholar CrossRef Search ADS Lewis D. 1975. Adverbs of quantification. Reprinted in Papers in Philosophical Logic 5– 20. Cambridge: Cambridge University Press. 1998. Peters S., Westerståhl D.. 2006. Quantifiers in Language and Logic . Oxford: Clarendon Press. Rescher N. 1962. Plurality quantification. Journal of Symbolic Logic 27: 373– 74. Google Scholar CrossRef Search ADS Sainsbury M. 1991. Logical Forms: An Introduction to Philosophical Logic . Oxford: Basil Blackwell. von Fintel K. 2004. A minimal theory of adverbial quantification. In Context Dependence in the Analysis of Linguistic Meaning , eds Partee B., Kamp H., 137– 75. Amsterdam: Elsevier. © The Author 2017. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: firstname.lastname@example.org
Analysis – Oxford University Press
Published: Nov 15, 2017
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