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)](KM)}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 pronouns 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: properties 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 interpretation,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] [ Op [ [some cat] [Op [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 onlyλx. bindsome variables cat λx that. chased allow themxtox 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] [ Op [ [every dog] [Op [ t1 chased t2 ]]]] b. SS: Op2 Op1 [every dog] t1 [ chased [some cat] t2 ] ,g ,g ,g c. Interpretation: some cat λx. every dog λx. chased x x 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 [ Op [ [some cat] [Op [ t1 chased t2 ]]]] ,g ,g ,g c. Interpretation: every dog λx. some cat λx. chased x x J KM J KM J KM (5) Inverse: ( ( ( )( ))) a. SS: [every dog] [ chased [some cat] ]

b. LF: [some cat] [ Op [ [every dog] [Op [ t1 chased t2 ]]]] ,g ,g ,g c. Interpretation: some cat λx. every dog λx. chased x x J KM J KM J KM Note how extremely similar this is to (our( reinterpretation of)( Quantiﬁer Lowering:( )( we’ve))) mainly traded the labels “DS” and “LF”, in the process reversing assumptions about which level is derived from which. Sometimes QR is required because the derivation would be ruled out otherwise (unless, of course, we have other semantic mechanisms at our disposal — see below).

• Nothing forced us to move the subject in (5); but we could not have avoided moving the object, because the verb requires an object of type e rather than e, t , t .

• Another case in which interpretation has been claimed to be impossible⟨⟨ without⟩ ⟩ QR (or something with the same eﬀect) is antecedent-contained deletion: sentences like Mary saw every movie that Bill did (Sag 1976; Heim & Kratzer 1998; Jacobson 2014). • Next time we’ll consider in more detail arguments for QR from scope restrictions and their similarity with restrictions on overt wh-movement.

Dispensing with traces creates problems. Fox 2002 replaces the trace interpretation rule with a rule that the lower copy is interpreted as a bound deﬁnite description: Mary saw every movie is interpreted as “For every movie x, Mary saw the movie x”.

5 Hendriks’s (1993) Flexible Types Many theorists dislike the idea of covert syntactic movement intensely (presumably, a superset of those who dislike the idea of syntactic movement of any kind intensely). Can we deal with quantifier scope ambiguities without covert movement? Absolutely. Hendriks(1993) show how to derive the semantic effect of QR in a directly compositional way, with no syntactic structures that aren’t directly motivated by the surface string. Theories of quantifier scope ambiguities share the goal of building a pair of semantic representations like these for the surface string Every dog chased some cat — and doing so in a syntactically and semantically respectable way.

,g ,g ,g (6) a. every dog λx. some cat λx. chased x x J KM,g J KM,g J KM,g b. some cat λx. every dog λx. chased x x J K ( J K ( J K ( )( ))) Abstracting away fromM the precise choice of verbM and QPs, thisM gives the target schemata: ( ( ( )( )))

5 (7) a. QPA λx1.QPB λx2.V x2 x1 (direct scope)

b. QPB λx2.QPB λx1.V x2 x1 (indirect scope) ( ( ( )( ))) Note that ( ( ( )( )))

• QPA and QPB are both objects of type e, t , t , and V is an object of type e, e, t .

• By the definition of scope given on page⟨⟨ 1, QP⟩ ⟩A takes scope over QPB in (⟨7a),⟨ and⟩⟩ QPB takes scope over QPA in (7b). Now we need to make our first conceptual shift: Think of these unnamed generalized quantifiers, not as names for specific GQs, but as variables over GQs. Letting 1, 2, ... be variables over objects of type e, t , t : Q Q (8) a. 1 λx1. 2 λx2.V ⟨⟨x2 ⟩x1 ⟩ b. 2 λx2. 1 λx1.V x2 x1 Q ( Q ( ( )( ))) We can thenQλ(-abstractQ ( over( these)( GQ-type))) variables. Importantly, there is no particular connection between the order in which abstract over the GQ meanings, and the order in which the corresponding arguments appear as arguments to the verb.

(9) a. λ 2λ 1 1 λx1. 2 λx2.V x2 x1

b. λ 2λ 1 2 λx2. 1 λx1.V x2 x1 Q Q [Q ( Q ( ( )( )))] Second conceptualQ Q shift:[Q ( insteadQ ( of thinking( )( of)))] these as schemata for sentence meanings that are underdetermined by the sentence’s phonological form, think of them as schemata for the verb meaning that are underdetermined by the verb’s phonological form. 2 will be the ﬁrst syntactic argument, and 2 will be the second; but their scope will diﬀer between (9a) and (9b). What this means is that — as long as we have a way of ensuring thatQ both of these interpretations are availableQ for V — the order in which the verb takes the arguments in the syntax does not necessarily determine the relative scope of the arguments. In other words, syntactic hierarchy does not necessarily determine semantic scope. The representations in (7) are of course equivalent to the less brief ones in (10):

(10) a. λ 2λ 1 1 λx1. 2 λx2.V x2 x1 QPB QPA λ 1 1 λx1.QPB λx2.V x2 x1 QPA [ QPQ AQλx[Q1.QP( B Qλx(2.V x2( x)(1 )))]]( )( ) (direct scope, = (7a)) = Q [Q ( ( ( )( )))]]( ) b. =λ 2λ (1 2 λx2.( 1 λx1(.V)(x2 )))]x1 QPB QPA λ 1 QPB λx2. 1 λx1.V x2 x1 QPA [ QPQ BQλx[Q2.QP( A Qλx(1.V x2( x)(1 )))]]( )( ) (indirect scope, = (7b)) = [ Q [ ( Q ( ( )( )))]]( ) but this time= we’ve( generated( both scopes( )( while)))] introducing the generalized quantiﬁers in the same order. That is, we’ve introduced them into the interpretation in a way that conforms to the surface syntax — no covert mechanisms needed.

6 OK, this all works if we have some reason to think that a verb like see is ambiguous in this way. Is there a way to generate both meanings from the basic meaning (type e, e, t ) without simply stipulating it in the lexicon? Hendriks proposes that all expressions, including verbs like chase, should be⟨ assigned⟨ ⟩⟩ ﬂexible types. Hendriks shows that the full versions of these two rules — together with a third rule (Value raising) designed to deal with intensional contexts — can be used to generate meanings that are equivalent to all possible applications of the syntactic rule QR. Let’s turn to Barker(2005) for explication (appendix).

6 Constraints on quantiﬁer scope The question remains — for Hendriks and also (e.g.) Barker & Shan(2014) — whether there is a well-motivated way to restrict its application in cases in which not all scopes are possible: (11) Most boys like more than 3 movies. ( direct, ?? inverse)

• Reinhart(1997) follows a long tradition✓ which argues that syntactic approaches are motivated by restrictions on the scope of certain quantiﬁcational expressions which resemble independently motivated syntactic constraints (roughly, island constraints).

(12) a. Somebody wondered whether I had considered all the facts. (# inverse) b. # What did somebody wonder whether I had considered?

• Defenders of theories inspired by Hendriks would presumably take heart – if Value Raising is not needed, so that variable scope is automativally clause- bounded(Jacobson 2014). Then we need a non-scope-based way of dealing with quantifying into intensional contexts; or – if non-syntactic, processing-based theories of islandhood turn out to be right (Hofmeister & Sag 2010). Perhaps a similar account of restrictions on the scope of certain quantiﬁers is possible ...

References Barker, Chris. 2005. Remark on jacobson 1999: Crossover as a local constraint. Linguistics and Philosophy 28(4). 447–472. Barker, Chris & Chung-Chieh Shan. 2014. Continuations and natural language. Oxford University Press. Chomsky, Noam. 1976. Conditions on rules of grammar. Linguistic analysis 2(4). 303–351. Fox, Danny. 2002. Antecedent-contained deletion and the copy theory of movement. Linguistic Inquiry 33(1). 63–96. Heim, Irene & Angelika Kratzer. 1998. Semantics in generative grammar. Blackwell. Hendriks, H.L.W. 1993. Studied ﬂexibility: Categories and types in syntax and semantics. Institute for Logic, Language and Computation.

7 Hofmeister, Philip & Ivan A Sag. 2010. Cognitive constraints and island effects. Language 86(2). 366. Jackendoff, Ray. 1972. Semantic interpretation in generative grammar. MIT Press. Jacobson, Pauline. 2014. Compositional semantics: An introduction to the syntax/semantics interface. Oxford University Press. Lakoff, George. 1971. On generative semantics. In D. D. Steinberg & L. A. Jakobovits (eds.), Semantics: An interdisciplinary reader in philosophy, linguistics and psychology, Cambridge University Press. May, Robert. 1977. The grammar of quantification: Massachusetts Institute of Technology dissertation. May, Robert. 1985. Logical form: Its structure and derivation, vol. 12. The MIT Press. Montague, Richard. 1973. The proper treatment of quantification in ordinary English. In J. Hintikka, J. Moravcsik & P. Suppes (eds.), Approaches to natural language, vol. 49, 221–242. Reidel. Reinhart, Tanya. 1997. Quantifier scope: How labor is divided between QR and choice functions. Linguistics and philosophy 20(4). 335–397. Sag, Ivan A. 1976. Deletion and logical form: MIT dissertation. Szabolcsi, A. 2010. Quantification. Cambridge University Press.