Quantitifcation in English Is Inherently Sortal

Within Linguistics the semantic analysis of natural languages (English, Swahili,...) has drawn extensively on semantical concepts first formulated and studied within classical logic, principally first order logic. Nowhere has this contribution been more substantive than in the domain of quantification and variable binding. As studies of these notions in natural language have developed they have taken on a life of their own, resulting in refinements and generalizations of the classical quantifiers as well as the discovery of new types of quantification which exceed the expressive capacity of the classical quantifiers. We refer the reader to Keenan & Westerståhl (1997) for an overview of results in this area. Here we focus on one property quantification in natural language – its inherently sortal nature – which distinguishes it from quantification in logic. §1 From Logic to Linguistic Analysis Within Linguistics a primary goal of semantics is to formulate a compositional semantic interpretation for the expressions of a given natural language. This of course presupposes a grammar which defines the set of expressions pretheoretically judged grammatical by native speakers. At time of writing grammars for English that have been proposed are incomplete – they fail to generate some expressions which speakers judge grammatical, and unsound – they generate some expressions not judged grammatical by native speakers. Nonetheless our understanding of the syntax of certain simple fragments of English is clear enough that it makes sense to ask for a compositional semantics for those fragments. We focus here on issues concerning the semantic analysis of quantificational structures in natural language. Classical Logic (CL) provides semantic representations like (1b) and (2b) for the English sentences (Ss) in (1a) and (2a) respectively. (1) a. All poets daydream b.∀x(Poet(x) → Daydream(x)) (2) a. Some poets daydream b. ∃x(Poet(x) & Daydream(x)) Ignoring the semantic properties associated with tense (present, past, ...) and aspect (generic, perfective, ...), this semantic analysis is correct in the sense that systematic use of such representations correctly captures certain judgements of semantic relatedness given by native speakers. E.g. an English sentence P is understood to entail a sentence Q iff their semantic representations P' and Q' are such that Q' is interpreted as True in all models in which P' is. Moreover this semantic analysis enables linguists to represent a variety of semantic distinctions which are difficult to understand and represent in the absence of a systematic representation of quantification. Here are three cases: First, English Ss like (3) are semantically ambiguous; they can be understood in two logically distinct ways, as represented in (3a) and (3b), using some obvious abbreviations. (3) Each student in the class read some play over the vacation a.∀x(Sx → ∃y(Py & xRy)) b. ∃y(Py & ∀x(Sx → xRy)) Of course (3a) can be true if no two students read a play in common, as long as for each student there is some play that that student read. In (3b) by contrast there must be at least one play which is such that each student read that play. Thus using representations which differ with respect to the relative scope of the quantifiers ∀x and ∃y permits a natural representation of the two ways of understanding (3). Second, in early work in generative grammar verb phrase (VP) coordination was derived from coordinate Ss by eliminating repeated occurrences of Noun Phrases (NPs) in the later conjuncts. It was understood that the derived Ss had the same meaning as those they were derived from, thus satisfying Compositionality. For example, (4b) was to be derived from (4a), with which it is logically synonymous. (4) a. John laughed and John cried b. John laughed and cried But obviously such a Conjunction Reduction transformation fails to be paraphrastic when the NPs are quantified and an appropriate coordinate conjunction is selected. Thus (5a) and (5b) are not logical paraphrases, as is clear from their CL representations. (5) a. Some student laughed and some student cried (∃x(Sx & Lx) & ∃x(Sx & Cx)) b.Some student laughed and cried ∃x(Sx & Lx & Cx) Of course if and is replaced by or in (5a,b) the resulting Ss are logically equivalent. Analogous claims hold for Either every student came early or every student left late and Every student either came early or left late. The two Ss are not paraphrases, though the results of replacing or by and are. Again, these semantic facts are clearly accounted for using standard CL representations. Thus CL helps us to see that the naive linguistic analysis of expressions like (5b) is problematic for Compositionality, as the semantic interpretation of the derived expression does not stand in a regular semantic relation to the one it is derived from. Sometimes it is logically equivalent to it and sometimes not. To get the correct truth conditions for the non-paraphrastic cases we need to see not only the identity of the conjunction used (and vs or) but also the choice of NP (every student vs some student). Third, in a similar vein, early work in generative grammar sometimes purported to derive expressions by replacing full (= non-pronominal) NPs by appropriate pronouns when the full NP was identical to another appropriate NP occurrence. But semantic problems comparable to those for Conjunction Reduction above arose. (6a,b) are not paraphrases, as is clear from their CL representations. (6) a. All poets admire all poets b. All poets admire themselves ∀x∀y((Px & Py) → xAy) ∀x(Px → xAx) So again, if a grammar of English generates (6b) by replacing the second occurrence of all poets in (6a) by the reflexive pronoun themselves we find that Compositionality is hard to satisfy. Merely knowing the models that satisfy (6a) is not sufficient to identify those that satisfy (6b) as the latter is a proper subset of the former. In all these ways then the representations of Classical Logic have proven insightful in the semantic analysis of natural language expressions. It might then seem surprising that the, often informally presented, semantic representations used by linguists for quantificational expressions in natural language differ from those of CL. some linguistic objections to the CL analysis All approaches to English syntax agree that in (1) the sequence all poets forms a syntactic constituent. It consists of the Determiner (Det) all and the (plural marked) noun poets. The VP daydream forms the other constituent of (1). We expect by Compositionality then that the semantic interpretation of the entire S is given in terms of the interpretation of all poets and that of daydream, and thus that these constituents have a semantic interpretation. But in (1b), the CL translation of (1a), there is no syntactic constituent which represents the meaning of the NP all poets. Rather the noun poet is ripped away from its Det all and is treated as a one place predicate. Moreover, tied to the linguist's respect for syntactic constituency here is the intuition that the semantic roles of the noun poet and that of the VP daydream are quite different. We can think of both as denoting properties that individuals may or may not have. But the noun property serves to limit the range of objects we are talking about, specifically those we are quantifying over, whereas the VP presents the property we are predicating of those objects (in accordance with the constraints determined by the Det all). By contrast in (1b) the variable x is understood to range over all the individuals in the universe of discourse. We may fairly read it in rough English as "For all individuals x, if x is a poet then x daydreams". So (1a) and (1b) differ in that in (1a) we are just talking about poets, whereas in (1b) we are talking about everything, though what we predicate of those objects is now expressed by a boolean compound of formulas built from the original noun and the original VP. It is something of an embarassment to this intuitive difference in meaning that, modulo tense and aspect, (1b) does adequately represent the truth conditions and entailment relations of (1a). But perhaps this is an accident of the example. (2a,b) suggest this may be the case. (2b), like (1b), quantifies over all objects in the universe of the model, but it incorporates the noun into its predicate differently, by using and rather than if then. Will yet different Determiners require yet further boolean connectives in combining the noun and the VP? Are there enough boolean connectives to accomodate the variety of English Dets? We see below that the answer is negative, and thus that natural languages, in distinction to standard first order languages, are inherently sortal. But we anticipate. Let us consider first the direct interpretation of NPs of the form Det+Noun. §2 From Linguistics to Logic Traditionally we think of subject-predicate Ss such as John daydreams as ones in which the predicate daydreams is the general term and the subject John the specific one. This is captured extensionally by treating a possible predicate denotation as a set of possible subject denotations, and we represent the truth in a model of John daydreams by saying that the object John denotes is an element of the set of objects daydreams denotes. But, as Frege realized, this general-specific distinction is cut the other way when we consider quantified NP subjects such as all poets, some poets, no poets, etc., rather than simple proper names. Now it is the subject phrase which denotes the general term and the predicate the more specific one. That is, extensionally, the set of possible quantified NP denotations corresponds to sets of one place predicate denotations. To see the idea behind this claim we take a simple example and show how to construct 2n extensionally distinct NP denotations, where n is the number of extensionally distinct VP denotations. In fact we can take the NPs to be proper nouns and just consider their logically distinct boolean compounds in and, or, not, and neither...nor.... Consider for example a universe with just 3 elements, a,b,c denoted say by Adam, Bill, and Chris. Now, adjusting number marking on the verb appropriately, consider the 8 Ss that result when X in (7a) is replaced by one of the 8 NPs in (7b). (7) a. X daydreams b. 1 Adam and Bill and Chris 2 Adam and Bill but not Chris 3 Adam and Chris but not Bill 4 Adam but neither Bill nor Chris 5 Bill and Chris but not Adam 6 Bill and neither Chris nor Adam 7 Chris and Adam but not Bill 8 Neither Adam nor Bill nor Chris For X = (b.1) we compute that (7a) is true iff daydreams denotes {a,b,c}. When X is (b.2) it is true iff daydreams denotes {a,b}, and so on to (b.8), where (7a) is true iff daydreams denotes the empty set. In this way then we see that the 8 NPs in (7b) are logically distinct, each one corresponding a single possible VP denotation. But now take any subset of the NPs in (7b) and form their disjunction: E.g. either Adam and Bill but not Chris, or both Bill and Chris but not Adam, or neither Adam nor Bill nor Chris. Clearly when X is such a disjunction (7a) is true iff daydreams denotes one of the sets denoted by one of the disjuncts. So disjunctions of distinct subsets of these NPs determine logically distinct NPs, so the number of logically distinct NPs corresponds to the number of sets of extensionally distinct VP denotations. In the case at hand we build 28 logically distinct NPs. (Note we are really just constructing NPs in disjunctive normal form, in analogy to the way this is done in propositional logic; see Keenan & Faltz, 1985)1 Of course in forming logically distinct NPs we can have recourse to ones that are not boolean compounds of proper nouns. Consider the NP like every student and no non-student. Setting X to be this NP, (7a) above is true iff the objects who daydream are exactly the students. So this NP can denote any of the eight possible denotations given by (b.1) – (b.8) above according to the set student denotes. Moreover interpreting student as {a,b,c} in the example above we can again form 8 logically distinct NPs using quantifiers and exception phrases, as in every student, every student but Adam, every student except Adam and Chris, ..., no student but Chris,..., no student. Now to say that NPs determine sets of VP denotations says that we can treat NPs semantically as functions mapping VP denotations into {True, False}. Call such functions generalized quantifiers. Consider for example all poets. Semantically it maps a set B, which we sometimes call the predicate set, to True iff each object in the set of poets is in B. That is, writing denotations in upper case, (ALL POET)(B) = True iff POET ⊆ B. More generally, for A,B any sets, (ALL A)(B) = True iff A ⊆ B. And this in turn says that we can interpret all as a function ALL which maps a set A to the generalized quantifier ALL(A). In this way we give a compositional interpretation to (1a) as in (8). (8) All poets daydream