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Semantics & Pragmatics I Semantics & Pragmatics I: 5/2/19 Handout Szabolcsi 2010 §2; Jacobson 2014, §14 Dan Lassiter 1 Scope A generalized quantifier is (the characteristic function of) a set of properties. A quantified (declarative) sentence asserts that some property is in the GQ denoted by the widest-scope quantifier in the sentence. For example, (1) a. every dog barks ;g = 1 iff barks ;g every dog ;g J KM J ;gKM J KM b. every dog 3 loves his3 master = 1 iff Jx loves ;g master ;g x KMx every∈ dog ;g (bound his) [ J KM] J KM ;g J KM c. every dog 3 loves his4 master = 1 iff J{x S loves ;g[( master ;g)(g)](4KM)}x∈ every dog ;g (free his) d. every[ J dogKM chased] J someKM cat ;g = 1 iffJ K (wideM scope every dog) J{x S y cat ;g[( y chased)(KM(;g ))](y x )} ∈ every dog ;g e. everyJ dogK chasedM someJ catKM ;g = 1 iffJ K (wideM scope some cat) J{y S ∃x[ dog ;g( x) ∧ chasedKM (;g )(y )]}x ∈ some cat ;g J KM J KM J KM Scope is a{ propertyS ∀ [ of expressions,( ) → which is defined( )( in)]} terms∈ of the way that the denotation of the expression is semantically combined with the denotations of other expressions in the sentence. The scope of a quantificational DP, on a given analysis of the sentence, is that part of the sentence which denotes a property that is asserted to be an element of the generalized quantifier denoted by DP on that analysis. (Szabolcsi 2010: 10) In the examples in (1a)-(1d), the scope of the expression every dog is whatever linguistic material denotes the sets that are claimed to be in the GQ every dog ;g. J KM • In (1a) the scope of every dog is barks. • In (1b) and (1c) the scope of every dog is loves his master • In (1d) the scope of every dog is chased some cat. • In (1e) the scope of every dog is just chased. In (1e) some cat takes scope over every dog chased. Note that this holds regardless of your syntactic commitments: the idea that semantic scope is determined by syntactic hierarchy is a hypothesis (due to Reinhart) and is controversial in formal semantics. 1 2.3 Scope and constituent structure 13 chosen below will make the relation between Montague’s, May’s, and Hen- driks’s methods the most transparent. We start by applying the verb to arguments interpreted as free individual variables and build a sentence without quantifiers. Montague notated such variables with indexed pro- nouns in the syntactic derivation; we employ indexed empty categories ec. Properties (of type e, t )arethenformedfromthissentencebyabstract- ing over the variables! one" by one. Abstraction is achieved by binding the variable by a λ-operator. (18) If α is an expression, λx[α]isanexpression.λx[α]denotesa function of type b, a ,whereb is the type of the variable x and a is the type of! the" function value α.Whenappliedtosome Althoughargument scope isβ a simpleof the concept, same type finding as outx,thevalueofthefunctionis how to derive a meaning for a sentence that deliverscomputed the appropriate by replacing scope compositionally every occurrence is often of notx simple.bound by λx in α Usefulby terminologyβ.E.g.λ (forx[x 2-quantifier2](3) = 32. sentences): • `Direct scope': The expression which occurs in a higher position in surface syntax also Eachhas time wider asemantic property scope. is formed, a quantifier can be introduced. The later a quantifier is introduced, the wider its scope: other operators may already• `Inverse be buried scope': in The the expression definition which of the occurs property in a lower that position it combines in surfacewith, syntax has and thereforewider semantic they scope. fall within its scope. The derivation of the reading where the subject existential scopes over the direct-object universalis 2 Montague 1973 given first. Recall that it is to be read bottom-up, starting with “build asentencewithtwofreevariables.”ThecardinalityquantifierMontague's approach was to build a sentence with free variables and choosemore the than order in whichone will we want be abbreviated to employ λ-abstraction using > over1.Thelaststepisspelledoutin(20). them, and then apply the generalized quantifiers in the appropriate order. Liberal use∃ of string-rewrite rules ensures that the DPs denoting (19)the relevant Subject quantifiers> Object get pronounced reading in the appropriate places. (20) λP z[dragon !(z) P (z)](λx y[man!(y) spot !(y)(x )]) = ∃>1 ∧ 2∀ → 2 z[dragon!(z) λx y[man!(y) spot !(y)(x )](z)] = ∃>1 ∧ 2∀ → 2 z[dragon!(z) y[man!(y) spot !(y)(z)]] ∃>1 ∧∀ → 2 14 Generalized quantifiers and their elements: operators and their scopes The derivation of the reading where the direct-object universal scopes over the subject existential differs from the above in just one respect: prop- erties are formed by λ-binding the subject variable first and the direct- object variable second, which reverses the order of introducing the two quantifier phrases. The last step that introduces the universal is in (22): (21) Object > Subject reading (22) λQ y[man!(y) Q(y)](λx z[dragon!(z) spot !(x )(z)]) = ∀ → 3∃>1 ∧ 3 y[man!(y) λx z[dragon !(z) spot !(x )(z)](y)] = ∀ → 3∃>1 ∧ 3 y[man!(y) z[dragon !(z) spot !(y)(z)]] ∀ →∃>1 ∧ Montague’s PTQ collapsed the two steps of λ-binding a free variable 3and Quantifier applying lowering a generalized (not in quantifier Szabolcsi to2010 the) property so formed into a single rule of quantifying-in. To make the derivation more transparent, Architecture of early Transformational Grammar and Generative Semantics: we disentangled the two steps, as do Heim and Kratzer (1998), who construeSemanticλ-abstraction interpretation as the reflexDeep of the Structure movementSurface of the Structure index on the variable.5 We followed PTQ in replacing the variable with the quantifier Adopting the assumption that Deep⇐Ô Structure is theÔ⇒ level of syntax which feeds inter- pretation,phrase in Lakoff the surface(1971) argued string. that This quantifiers feature are is syntactically syntactically attached unsophisticated at DS wherever theand interpretation need not be demands. taken too Their seriously; surface positions see May are and then Hendriks produced by below. transformational lowering,Before along turning with a deletion to other operation ways to that achieve erases the the bound same pronouns interpretiv thateresults are needed to getwe the take interpretation a brief look right. at quantifiers Anachronistically, binding the pronouns, two structures among associated other with reasons the string Everyin order dogchases to explain some catwhywould it makes be something sense for like: this book to set them aside. (2) Direct: a. DS: [every dog] [ Op1 [ [some cat] [Op2 [t1 chased t2]]]] 2.3.3 Interlude: quantifier phrases do not directly bind pronouns b. SS: Op1 Op2 [every dog] t1 [ chased [some cat] t2 ] ;g ;g ;g Predicate-logicalc. Interpretation: quantifiersevery do dog not onlyλx1: bindsome variables cat λx that2: chased allow themx2tox1 J K J K J K function as arguments of predicates.M (23), which canM be seen to trManslate (3) Inverse: ( ( ( )( ))) one reading of (24), contains three bound occurrences of the variable x, of which the one in room-of !(x)correspondstothepronounhis. 3 a. DS: [some cat] [ Op2 [ [every dog] [Op1 [ t1 chased t2 ]]]] b. SS: Op2 Op1 [every dog] t1 [ chased [some cat] t2 ] ;g ;g ;g c. Interpretation: some cat λx2: every dog λx1: chased x2 x1 J KM J KM J KM We generate the truth-conditions from the( DSs by supposing that( Opi is interpreted( )( as))) an instruction to λ-abstract over the variable xi in its syntactic sister, and that a ti denotes the variable xi. This approach is no longer used, for lots of reasons: e.g., even theorists who are otherwise free in their use of transformations generally don't like to make use of lowering operations. But it's not much different from Montague's more influential proposal, and: • It was presumably a major source of inspiration for May's (1977; 1985) influential Quantifier Raising proposal | though Lakoff and others who argued for this (McCawley, Lewis) are rarely cited in that tradition. (They were in the Bad Guys camp of the day.) • Semantically, the general form of the proposal is still what every theory of quantifier scope gives you: what varies in the two interpretations is the order in which the two variables are abstracted over, i.e., in how the syntax is supposed to create the required semantic structure. 4 Quantifier raising Interpretive semantics architecture (Jackendoff 1972; Chomsky 1976; May 1977; Heim & Kratzer 1998: etc.). Modifications associated with Minimalism are annotated \MP". DS (MP: numeration) SS (MP: Spell-out) LF Semantic interpretation Ô⇒ Ô⇒PF Ô⇒ ⇘ Assumptions (May 1977, 1985; Heim & Kratzer 1998, etc.): • Quantifier phrases can be raised and adjoined to any S node/node whose denotation has type t • This process is called \Quantifier raising" or \QR" • This is a \covert movement" process { one which is \silent", i.e., which is not associated with any overt phonetic realization. { creates an extra node | I'll call it Opi | which is interpreted as triggering λ- abstraction of some variable xi { leaves behind a trace which is obligatorily interpreted as the same variable xi that Opi triggers abstraction of. :::QP ::: S QP i Op :::ti::: S S [ ] ⇒ [ [ [ ] ]] 4 (4) Direct: a. SS: [every dog] [ chased [some cat] ] b. LF: [every dog]1 [ Op1 [ [some cat] [Op2 [ t1 chased t2 ]]]] ;g ;g ;g c. Interpretation: every dog λx1: some cat λx2: chased x2 x1 J KM J KM J KM (5) Inverse: ( ( ( )( ))) a. SS: [every dog] [ chased [some cat] ] b.
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