The quantifier store will be used for creating scope ambiguities. We will assume that for lexical entries the quantifier store is empty.

Transcription

1 CHAPTER SEVEN: QUANTIFYING-IN AND STORAGE 7.1 Scope ambiguities, quantifying-in and storage. We will now add a mechanism for creating scope ambiguities. Following Cooper 1975, 1983, the mechanism we will use will be a storage mechanism. We will first introduce the basic mechanism and show its working in several examples. Up to now our grammar generates pairs of syntactic structures and semantic representation. We will extend this now. The grammar will now generate triple <A,A',S> where: -A is a syntactic tree. -A', the translation of A, is an IL expression. -S, the quantifier store, is a set of pairs <n,α> where n is an index and α an IL expression. The quantifier store will be used for creating scope ambiguities. We will assume that for lexical entries the quantifier store is empty. We assume that stores are inherited upward in the following way: <A,B',S B > B For unary branching, both the meaning and the store inherit up. < A,APPLY[f,a],S B S C > B C For binary branching, the stores corresponding to the daughters are united. We now introduce two operations STORE n and RETRIEVE n STORE n [<,',S >] = <,x n,s {<n,'>}> STORE n replaces the interpretation ' of the by a variable x n (of type e), and stores the interpretation ' under index n in the quantifier store. Thus, STORE n tells you to use a variable x n for the time being as the interpretation of the, instead of the translation that we have just stored. 176

2 Example: STORE 1 [ <, λp. x[boy(x) P(x)],Ø > ] = D NP every boy <, x 1, {<1,λP. x[boy(x) P(x)]>} > D NP every boy Our second new rule is quantifying in, RETRIEVE n. Let <IP,IP',S IP > be a triple, with IP a syntactic tree, IP' its interpretation, S IP the corresponding store. Let <n,α> S IP, and assume that no other interpretation is stored under index n in S IP. RETRIEVE n [<IP,IP',S IP >] = <IP,APPLY[λx n.ip',α],s IP {<n,α>}> RETRIEVE n turns the IP meaning IP'of type t into a predicate λx n.ip' of type <e,t>, by abstracting over variable x n of type e. It takes the meaning α, which was stored under index n, out of the quantifier store S IP and it applies the predicate λx n.ip' to α with APPLY. We assume that we always want to end up with an empty store. With this mechanism, we are able to derive translations of sentences where the meaning of an NP takes wider semantic scope than its syntactic scope. EXAMPLES (1) Every girl kisses. I will use this example to demonstrate the effects of the scope mechanism. This sentence has five types of derivations. I will go through all five, and show that as a result, the sentence will be generated by the grammar with two meanings: one where the meaning of every girl takes scope over the meaning of, and one where the meaning of takes scope over the meaning of every girl. All derivations start out with the same structure, interpretation and store for every girl and some boy: 177

3 <, λp u[girl(u) P(u)],Ø > D NP every girl <, λp z[boy(z) P(z)],Ø > D a NP boy (1) Every girl kisses. DERIVATION 1 From the lexical item kiss we generate: <V,KISS,Ø > kiss The V takes the as a complement, we apply the V interpretation to the interpretation of [ ], and we unite the stores (giving Ø ). This gives: < VP, APPLY[KISS,λP. z[boy(z) P(z)]],Ø > V APPLY[KISS,λP. z[boy(z) P(z)]] = LIFT[KISS] (λp. z[boy(z) P(z)]) [def APPLY] = λtλx.t(λy.[kiss(y)](x)) (λp. z[boy(z) P(z)]) [def LIFT] = λx.λp. z[boy(z) P(z)](λy.[KISS(y)](x)) [λ-con] = λx. z[boy(z) (λy.[kiss(y)](x))(z)] [λ-con] = λx. z[boy(z) [KISS(z)](x))] [λ-con] = λx. z[boy(z) KISS(x,z)] [rel notatiom] 178

4 So we get: < VP, λx. z[boy(z) KISS(x,z)],Ø > V We have: <I,λP.P,Ø > Ø The I takes the VP as a complement, and we get: < I', λx. z[boy(z) KISS(x,z)],Ø > I VP Ø V The I' takes [ every girl] as a specifier, and we get: < IP, APPLY[λx. z[boy(z) KISS(x,z)], λp. x[boy(x) P(x)],Ø > I' every girl Ø V APPLY[λx. z[boy(z) KISS(x,z)],λP u[girl(u) P(u)]] = LIFT[λx. z[boy(z) KISS(x,z)]] (λp u[girl(u) P(u)]) [def APPLY] = λt.t(λx. z[boy(z) KISS(x,z)]) (λp u[girl(u) P(u)]) [def LIFT] = λp u[girl(u) P(u)] (λx. z[boy(z) KISS(x,z)]) [λ-con] = u[girl(u) [λx. z[boy(z) KISS(x,z)]](u)] [λ-con] = u[girl(u) z[boy(z) KISS(u,z)] [λ-con] 179

5 So we get: < IP, u[girl(u) z[boy(z) KISS(u,z)]],Ø > I' every girl Ø V (1) Every girl kisses. DERIVATION 2 We build up the same I' as under derivation 1: < I', λx. z[boy(z) KISS(x,z)],Ø > I VP Ø V We apply STORE 1 to every girl: STORE 1 [<, λp. u[girl(u) P(u)],Ø >] = D every NP girl <, x 1, {<1,λP u[girl(u) P(u)]>} > D every NP girl The I' takes this as a specifier, applies I' interpretation to the interpretation of the, which is x 1, and unites the stores. We get: 180

6 < IP, APPLY[λx. z[boy(z) KISS(x,z)],x 1 ], {<1,λP u[girl(u) P(u)]>}> I' every girl Ø V APPLY[λx. z[boy(z) KISS(x,z)],x 1 ] = λx. z[boy(z) KISS(x,z)] (x 1 ) [def APPLY] = z[boy(z) KISS(x 1,z)] [λ-con] So we get: < IP, z[boy(z) KISS(x 1,z)], {<1,λP u[girl(u) P(u)]>}> I' every girl Ø V To this we apply RETRIEVE 1, which gives: < IP, APPLY[λx 1. z[boy(z) KISS(x 1,z)], λp u[girl(u) P(u)]],Ø > I' every girl Ø V 181

7 APPLY[λx 1. z[boy(z) KISS(x 1,z)],λP u[girl(u) P(u)]] =LIFT[λx 1. z[boy(z) KISS(x 1,z)]] (λp u[girl(u) P(u)]) [def APPLY] =λt.t(λx 1. z[boy(z) KISS(x 1,z)]) (λp u[girl(u) P(u)]) [def LIFT] = [λp u[girl(u) P(u)]] (λx 1. z[boy(z) KISS(x 1,z)])[λ-con] = u[girl(u) [λx 1. z[boy(z) KISS(x 1,z)]](u)] [λ-con] = u[girl(u) z[boy(z) KISS(u,z)]] [λ-con] So we get the same as in derivation 1: < IP, u[girl(u) z[boy(z) KISS(u,z)]],Ø > I' every girl Ø V (1) Every girl kisses. DERIVATION 3 This time we start by applying STORE 2 to and get: <,x 2, {<2,λP z[boy(z) P(z)]>} > D a NP boy The V kiss takes this as a complement, and we get: < VP, APPLY[KISS,x 2 ], {<2,λP. z[boy(z) P(z)]>} > V APPLY[KISS,x 2 ] = KISS(x 2 ) [def APPLY] 182

8 So we get: < VP, KISS(x 2 ), {<2,λP. z[boy(z) P(z)]>} > V The empty I takes this as a complement, and we get: < I', KISS(x 2 ), {<2,λP. z[boy(z) P(z)]>} > I VP Ø V This I' takes every girl as a specifier, and we get: < IP, APPLY[KISS(x 2 ),λp u[girl(u) P(u)]], {<2,λP. z[boy(z) P(z)]>}> I' every girl Ø V APPLY[KISS(x 2 ),λp u[girl(u) P(u)]] = LIFT[KISS(x 2 )] (λp u[girl(u) P(u)]) [def APPLY] = λt.t(kiss(x 2 )) (λp u[girl(u) P(u)]) [def LIFT] = λp u[girl(u) P(u)] (KISS(x 2 )) [λ-con] = u[girl(u) [KISS(x 2 )](u)] [λ-con] = u[girl(u) KISS(u,x 2 )] [rel notation] So we get: 183

9 < IP, u[girl(u) KISS(u,x 2 )], {<2,λP. z[boy(z) P(z)]>}> I' every girl Ø V To this we apply RETRIEVE 2 and we get: < IP, APPLY[λx 2. u[girl(u) KISS(u,x 2 )], λp z[boy(z) P(z)]],Ø > I' every girl Ø V APPLY[λx 2. u[girl(u) KISS(u,x 2 )],λp z[boy(z) P(z)]]] = LIFT[λx 2. u[girl(u) KISS(u,x 2 )]](λp z[boy(z) P(z)]) [def APPLY] = λt.t(λx 2. u[girl(u) KISS(u,x 2 )])(λp z[boy(z) P(z)]) [def LIFT] = λp z[boy(z) P(z)] (λx 2. u[girl(u) KISS(u,x 2 )]) [λ con] = z[boy(z) [λx 2. u[girl(u) KISS(u,x 2 )]](z)] [λ-con] = z[boy(z) u[girl(u) KISS(u,z)]] [λ-con] So we get: < IP, z[boy(z) u[girl(u) KISS(u,z)]],Ø > I' every girl Ø V 184

10 This time the derivation produces the sentence with a different meaning: the object NP takes wide scope over the subject NP. (1) Every girl kisses. DERIVATIONS 4 and 5 In both derivations 4 and 5 we apply STORE 1 to every girl and STORE 2 to, so these derivations use: <, x 1, {<1,λP u[girl(u) P(u)]>} > D NP every girl and <, x 2, {<2,λP z[boy(z) P(z)]>}> D a NP boy DERIVATION 4 We build up the I' kiss in the same way as in derivation 3 and get: < I', KISS(x 2 ), {<2,λP. z[boy(z) P(z)]>} > I VP Ø V This I' takes every girl as a specifier, and we get: 185

11 < IP, APPLY[KISS(x 2 ),x 1 ], {<1,λP u[girl(u) P(u)]>, <2,λP. z[boy(z) P(z)]>} > I' every girl Ø V APPLY[KISS(x 2 ),x 1 ] = [KISS(x 2 )](x 1 ) [def APPLY] = KISS(x 1,x 2 ) [rel notation] So we get: < IP, KISS(x 1,x 2), {<1,λP u[girl(u) P(u)]>, <2,λP. z[boy(z) P(z)]>} > I' every girl Ø V To this we apply RETRIEVE 2 and we get: < IP, APPLY[λx 2.KISS(x 1,x 2 ), λp. z[boy(z) P(z)]], {<1,λP u[girl(u) P(u)]>} > I' every girl Ø V 186

12 APPLY[λx 2.KISS(x 1,x 2 ), λp. z[boy(z) P(z)]] = LIFT[λx 2.KISS(x 1,x 2 )](λp. z[boy(z) P(z)]) [def APPLY] = λt.t(λx 2.KISS(x 1,x 2 ))(λp. z[boy(z) P(z)]) [def LIFT] = [λp. z[boy(z) P(z)]](λx 2.KISS(x 1,x 2 )) λ con] = z[boy(z) [λx 2.KISS(x 1,x 2 )](z)] [λ-con] = z[boy(z) KISS(x 1,z)] [λ-con] So we get: < IP, z[boy(z) KISS(x 1,z)], {<1,λP u[girl(u) P(u)]>} > I' every girl Ø V At this moment, we have reached the same stage as we did in derivation 1, RETRIEVE 1 will produce, after reduction: < IP, u[girl(u) z[boy(z) KISS(u,z)]],Ø> I' every girl Ø V 187

13 DERIVATION 5 Derivation 5 proceeds in exactly the same way as derivation 4 and produces: < IP, KISS(x 1,x 2), {<1,λP u[girl(u) P(u)]>, <2,λP. z[boy(z) P(z)]>} > I' every girl Ø V This time, instead of applying RETRIEVE 2 we apply RETRIEVE 1 and get: < IP, APPLY[λx 1.KISS(x 1,x 2 ),λp u[girl(u) P(u)] ], {<2,λP. z[boy(z) P(z)]>} > I' every girl Ø V APPLY[λx 1.KISS(x 1,x 2 ), λp u[girl(u) P(u)]] = LIFT[λx 1.KISS(x 1,x 2 )](λp u[girl(u) P(u)]) [def APPLY] = λt.t(λx 1.KISS(x 1,x 2 ))(λp u[girl(u) P(u)]) [def LIFT] = λp u[girl(u) P(u)] (λx 1.KISS(x 1,x 2 )) [λ con] = u[girl(u) [ λx 1.KISS(x 1,x 2 )](u)] [λ-con] = u[girl(u) KISS(u,x 2 )] [λ-con] So we get: 188

14 < IP, u[girl(u) KISS(u,x 2 )], {<2,λP. z[boy(z) P(z)]>} > I' every girl Ø V We have now joined derivation 3, applying RETRIEVE 2 gives after reduction: < IP, z[boy(z) u[girl(u) KISS(u,z)]],Ø> I' every girl Ø V We conclude that we derive every girl kisses with two meanings: for every girl there is that she kisses, and: there is such that every girl loves him. -comments on scope islands and the differences between propositional attitude complements and relative clauses -functional readings of relative clauses 189

15 7.1. Some thoughts about wide scope. I am concerned here with scoping out of the complement of propositional attitude verbs. As is well-known, while indefinite noun phrases in the complements of propositional attitude verbs easily get de re-interpretations, such interpretations are generally much harder to get for quantificational noun phrases. (1) a. Every schoolboy believes that a mathematician wrote Through the Looking Glass. -de dicto: this is the kind of stuff mathematicians write. -de re, widest scope: they all believe of Charles Dodgson that he wrote Through the Looking glass under the pseudonym of Lewis Caroll. -de re, narrow scope: Buck believes Hilbert wrote it, Chuck believes Hardy wrote it, b. Buck believes that every linguist knows many languages. Several kinds of mechanisms have been proposed to deal the scoping of indefinites (for instance, mechanisms of 'specific interpretation' - not always with a clear understanding of what specific means), mechanisms of choice functions (See the L&P volume with papers by Tanya Reinhart and by Yoad winter), skolem functions, individual concepts, etc. The scope mechanism provided above is unrestricted, allows wide scope over the intensional context for any noun phrase. Other scope mechanisms, like movement, are clause bound and do not extend beyond the IP level. Even if we assume that this is a good idea as a mechanism that applies to scope of true quantifiers, and hence explain why they don't scope as freely as indefinites, we do have to face the fact that de re interpretations of quantificational elements are in fact possible: (2) a. Buck thinks that every girl in Dafna's class is pretty. Situation: Buck sees a group of girls in Park HaYarkon. He doesn't know them, but he thinks all of them are pretty, and says so (mumbling). You ask me: what did Buck say? Now I know that this is a party of all the girls in Dafna's class. You don't know those girls either, but you know Dafna, so I say: Buck thinks that every girls in Dafna's class is pretty. This is felicitous and it shows a de re reading: it's not literally what he said, and there is no reason to assume that he thinks de dicto that the girls in Dafna's class are pretty. What can we do about this? Here is a first suggestion: Retrieval assumption (Default): Retrieval of noun phrases in store takes place as soon as the types fit for retrieval. 190

16 With this principle, the store is by default emptied when the derivation reaches type t, i.e. at the IP level. With this principle, you will not expect wide scope readings. How do we get them when we do? Well, let us first look at the following de re construction: (2) b. Buck believes of every girl in Dafna's class that she is pretty. Here there is an adjunct on the higher verb which takes the quantificational noun phrase as a complement, and the CP-complement of the propositional attitude verb contains a variable (a pronoun). Note too that the variable is obligatory, (4) is infelicitous: (2) c. Buck believes of every girl in Dafna's class that Mary is pretty. IP I' Buck I VP -s PP VP P n V CP of every girl believe C IP that n I' she I PRED[AP] is This structure only allows for a de re reading. How does it come about? pretty Binding in the de re structure: 1. We interpret the pronoun in the CP-complement as a variable x n 2. We require that this variable is bound by the complement of the higher adjunct. Quantifiers don't bind variables, so this is interpreted in the following way: 3. We store the -complement in the higher adjunct under the same variable x n as the variable in the CP-complement: x n {<n,every girl in Dafna's class>} This is all. Now when we retrieve the stored meaning at the matrix level, abstraction will bing both variables. Obviously, this gives the correct de re reading. 191

17 Suggestion: What if we assume that (2a) allows a structure along the lines of the de re structure. (2) a. Buck thinks that every girl in Dafna's class is pretty. IP I' Buck I VP -s PP VP P n V CP Ø Ø believe C IP that n I' every girl I is PRED[AP] pretty For concreteness: 1. Assume a null P with the same meaning as of, and assume that it triggers the conditions of the de re structure. 2. Assume that the null complement of P is a variable x n. 3. The de re structure requires a variable in the CP complement. This condition is satisfied by storing the -subject inside the CP under the same variable x n {<n,every girl in Dafna's class>}. We have now created a situation in which there is grammatical pressure on the retrieve operation: -By default, every girl comes out of store at the lower IP level. -There is pressure to keep every girl in store to resolve the higher variable. We assume that this pressure can override the default. [Note that if this is represented in the syntax, we are creating a weak cross-over violation. But then, weak cross-over violations are so common anyway that there is little reason to assume a principle forbidding weak cross-over in the syntax. cf: (3) The string that ties it to its mother keeps every foetus alive. ] 192

18 We can go one step further, and observe that for corresponding propositional attitude nouns the de re structure cannot be formed with the preposition of but requires about: (4) a. Mary is bothered by the belief of every girl in Dafna's class that she is pretty. b. Mary is bothered by the belief about every girl in Dafna's class that she is pretty. c. Mary is bothered by the belief that every girl in Dafna's class is pretty. In (4a) every girl in Dafna's class expresses the believer. The de re structure here can only be expressed as (4b). Let us now make the assumption that the null-preposition is a variant of a lexically realized preposition, and in fact, a null-variant of a flexible preposition, one that easily switches between multiple interpretations, like of: Null of-assumption: the null-pp is a null-version of of. Since of allows the about-interpretation in the verbal domain, its null-version is available for de re interpretations of propositional attitude verbs; since of does not allow the aboutinterpretation in the nominal domain, its null-version is not available for de re interpretations of propositional attitude nouns. It follows that cases like (4c) do not allow the same wide scope interpretations as do the corresponding propositional attitude verb cases. Note finally, that the escape strategy suggested here for propositional attitude verbs is unavailble for relative clauses: (5) I want you to make me a list of all cases of that invited every girl in the class to his birthday party. (5) invites you to write down cases of all-girl inviting boys, not a list of girl-boy pairs, specifying per girl the boys that invited her. every girl in the class does not scope out of the relative clause. The suggestions made here provide no reason to expect otherwise. 193

19 7.3. Some remarks on dislocation in the syntax and in the semantics. The X-bar theory I have introduced provides us with a theory of syntactic trees. For the present discussion it is useful to give the semantic interpretations associated with these syntactic trees the form of a tree as well. This is done along the following lines: Let: := λqλp. x[q(x) P(x)] : = λqλp. x[q(x) P(x)] A := APPLY [ A α, β ] := APPLY[α,β] IP A I' A A λp.p A BOY every boy Ø V KISS A GIRL a girl Thus, the semantic tree consists of the meanings of the basic expressions at the leaves, and the semantic operations labeling the non-leaves, under the convention that the function is on the left branch. Now, we are interested in two kinds of dislocation: syntactic dislocation and semantic dislocation: Syntactic dislocation: e.g. topicalization: the topic is sitting at the top of the syntactic tree while its interpretation is (or can be) sitting in its normal place down in the semantic tree: TopP A Top' A A D NP Top n IP λp.p A BOY a girl Ø I' KISS A GIRL every boy Ø V n kiss Ø 194

20 Semantic dislocation: e.g. inverse scope: the meaning of a girl is sitting at the top of the semantic tree while the a girl is sitting in its normal place down in the syntactic tree: Let: [ λ x n, β] := λx n.β IP A I' λ A x n A GIRL every boy Ø V A A λp.p A BOY a girl KISS x n When you calculate the meaning, you'll get out: APPLY[λx n. x[boy(x) KISS(x,x n )], λp. y[girl(y) P(y)]] = y[girl(y) x[boy(x) KISS(x,y)]] For each of these two dislocation situations there are three strategies to deal with them: 1. Syntactic dislocation: a syntactic expression is sitting at the top of the syntactic tree, while its meaning is sitting at the bottom of the semantic tree: STRATEGY 1.1: Generate both [ a girl] and its meaning A[,GIRL] at the bottom of the respective trees, and move [ a girl] up to the top of the syntactic tree. This is syntactic movement. STRATEGY 1.2: Generate both [ a girl] and its meaning A[,GIRL] at the top of the respective trees, and move the meaning A[,GIRL] down to the bottom of the semantic tree. This is what is called reconstruction. STRATEGY 1.3: Generate both [ a girl] and its meaning A[,GIRL] at the top of the respective trees, but assume a different semantic tree, which will have the effect of movement, without movement: This is a non-movement function composition analysis: (e.g. GPSG, Categorial Grammar, Jacobson) 195

21 TopP A Top' LC A D NP Top n IP RC A GIRL a girl Ø I' λp.p RC BOY KISS λt.t every boy Ø V n kiss Ø Here: [ RC α, β ] := RC[α,β] and RC is right function composition: Right Function Composition: RC[α,β] = λx.apply[α,β(x)] [ LC α, β ] := LC[α,β] and LC is left function composition: Left Function Composition: LC[α,β] = λx.apply[α(x),β] We discuss this below. 2. Semantic dislocation: a meaning is sitting at the top of the semantic tree, while the syntactic expression of which it is the meaning is sitting at the bottom of the syntactic tree: STRATEGY 2.1: Generate both [ a girl] and its meaning A[,GIRL] at the bottom of the respective trees, and move A[,GIRL] up to the top of the semantic tree. This is semantic movement: Cooper's storage mechanism, and also Quantifier Raising. STRATEGY 2.2: Generate both [ a girl] and its meaning A[,GIRL] at the top of the respective trees, and move the [ a girl] down to the bottom of the syntactic tree. An example of this strategy is (in essence) Montague's original quantifying-in rule. STRATEGY 2.3: Generate both [ a girl] and its meaning A[,GIRL] at the bottom of the respective trees, but assume a different semantic tree, which will have the effect of movement, without movement: This is a non-movement type shifting analysis (Hendriks in Categorial Grammar): 196

22 IP A I' A A λp.p A BOY every boy Ø V INV[KISS] A GIRL a girl Here: INV[α] = λtλu.t(λy.u(λx.α(x,y))) This means that you shift the verb meaning to the function which will put the arguments it receives in inverse scope order: INV[KISS] = λtλu.t(λy.u(λx.kiss(x,y))) APPLY[INV[KISS],λP. y[girl(y) P(y)]] = λu. y[girl(y) U(λx.KISS(x,y))] λp.p stands for the identity function at the required type (which is a higher type in this case). We get: APPLY[λU. y[girl(y) U(λx.KISS(x,y))], λp. x[boy(x) P(x)]] = y[girl(y) x[boy(x) KISS(x,y)] 7.4. Scope ambiguities via type shifting (Hendriks 198?) An n-place relation is a relation of type <a n,<a n 1,<,<a 2,<a 1,t>> >>> Let R be an n-place relation and for place i (i n) let a i =e. R = λx n λx i+1 λx i λx i 1 λx 1.R(x 1,,x i,,x n ) Let x i VAR e, T i VAR <<e,t>,t> LIFTi[R] = λx n λx i+1 λt i λx i 1 λx 1. T i (λx i.r(x 1,,x i,,x n )) Assume we allow LIFTi[R] as a free typeshift rule. (1) Every girl kissed. One derivation is as before: LIFT 1 [λpp (LIFT 2 [kiss]( (boy))] ( (girl)) 197

23 But another derivation is possible: (λpp (LIFT 2 [LIFT 1 [kiss]]( (boy))) ( (girl)) LIFT 1 [kiss] = λyλt 1. T 1 (λx 1 kiss(x,y)) LIFT 2 [LIFT 1 [kiss]] = λt 2 λt 1. T 2 (λy.t 1 (λx 1 kiss(x,y)) This interpretation combines with (boy) to give: λt 1. y[boy(y) T 1 (λx 1 kiss(x,y)) Applying this to (girl) gives: y[boy(y) x[girl(x) kiss(x,y)]] Thus we derive the inverse scope reading. In the type shifting framework scope ambiguities concern co-arguments of a relation (in fact, in a way similar to what is assumed for the binding theory by Reinhart and Reuling). But the relevant relation is more extended than what the verb gives: (2) a. Every boy kissed a girl in two cities on three occasions Kissed in on Some of the scopal arguments are in positions that are arguments of the verb, some are in PP-adjuncts. Thus we need to access semantically the four-place relation kiss in on We are facing now the problem of verbal polyadicity (McConnell-Ginet on adverbs, also Parsons 1990, Landman 2000): the shiftability of the number of argument in the light of the unclear semantic distinction between arguments and adjuncts. I will say more about this later, but suggest here a typeshifting approach to the relevant polyadicity which allows us to extend the above analysis to cases like (2). I will, for the sake of these examples, create a mismatch between syntax and semantics for prepositions in the verbal domain: I will assume that prepositions combine with objects to form verb modifiers: lexical item: in category: P interpretation: in of type <e p, <<<e,<e,t>>,<<e,<e,t>> > lexical item: on category: P interpretation: on of type <e t, <<<e,<e,t>>,<<e,<e,t>> > We assume here for ease that places and times are a subsort of type e, e p and e t. Prepositions take a complement of type e, and form modifiers of transitive verbs. 198

24 Grammatically, however, I assume that they are VP-modifiers: VP VP PP VP PP P V P on sunday kiss Ronya in Amsterdam And that means that their type should have been <e,<<e,t>,<e,t>>>. But it isn't, and there we have a mismatch. The mismatch is in this case resolved in a unorthodox way, by 're-interpreting' the verb via type-shifting as a relation 'so to say' incorporating the prepositions. The following concept is tailored to the current analysis. An n+m place relation is a relation: R = λx n λx 2 λy m λy 1 λx 1.R(x 1, x n ) (where m 0) Think about this as a relation that is derived from an n-place relation by sticking in some extra arguments. Let R be an n+m place relation and π VAR <e,<<e,<e,t>>,<e,<e,t>>> n-shift[r] = λx n λx 2 λy m λy 1 λπλz λx 1. ((π(z))(r))(x 1,,x n ) The output relation is an n+(m+2) place relation. In relation λx n λx 2 λy m λy 1 λπλz λx 1. ((π(z))(r))(x 1,,x n ) π(z) semantically directly modifies relation R. But, as the λ-prefix shows, it will get its semantic content from the VP level (after material closer to the verb and before the external subject corresponding to λx 1. Let's apply this operation to the case at hand. We 2-shift kiss: λyλx.kiss(x,y) λyλπ 2 λz 2 λx.[(π 2 (z 2 ))(kiss)](x,y) Then, we 2-shift this relation: 199

25 λyλπ 2 λz 2 λx.[(π 2 (z 2 ))(kiss)](x,y) λyλπ 2 λz 2 λπ 1 λz 1 λx. [(π 1 (z 1 ))[(π 2 (z 2 ))(kiss)]](x,y) With this relation we start working our way up the tree: We apply this shifted meaning of kiss with Ronya: λπ 2 λz 2 λπ 1 λz 1 λx. [(π 1 (z 1 ))[(π 2 (z 2 ))(kiss)]](x,ronya) We come to the PP, we do not at this point apply the PP-meaning to the VP meaning: The shift in essence re-analyzes adjunction as application to arguments (although semantically it is just adjunction. So we apply to the preposition to fill in the content of the role the argument to be adjoined plays, and then we apply to the argument of the preposition: λz 2 λπ 1 λz 1 λx. [(π 1 (z 1 ))[(in(z 2 ))(kiss)]](x,ronya) λπ 1 λz 1 λx. [(π 1 (z 1 ))[(in(amsterdam))(kiss)]](x,ronya) And this repeats with the next PP: λz 1 λx. [(on(z 1 ))[(in(amsterdam))(kiss)]](x,ronya) λx. [(on(sunday))[(in(amsterdam))(kiss)]](x,ronya) Finally, we apply to the subject: [(on(sunday))[(in(amsterdam))(kiss)]](fred, Ronya) Now we come to scope. We have shifted the verb to a six-place relation: λyλπ 2 λz 2 λπ 1 λz 1 λx. [(π 1 (z 1 ))[(π 2 (z 2 ))(kiss)]](x,y) Four of these places are arguments of type e: λyλπ 2 λz 2 λπ 1 λz 1 λx. [(π 1 (z 1 ))[(π 2 (z 2 ))(kiss)]](x,y) The scope shift operation that we introduced earlier can lift any of these place from type e to type <<e,t>,t> and in that way create any scope order between them. Here we see the importance of adjoining to the verb, rather than the VP. If we adjoin to the VP instead, the object argument is already processed and cannot take scope in between two adjuncts, because either you haven't lifted the object and it takes scope under both adjuncts, or you have lifted the object, and it will take scope over both adjuncts. 200

26 Note that it is not claimed here that PPs must be analyzed this way and must be semantically adjoined to the verb rather than the VP. It is claimed that this scope mechanism requires the possibility of doing so. Of course, if you allow adjunction to the VP level as well, you will need to have meaning principles that relate readings: [(on(sunday))[(in(amsterdam))(kiss)]](fred, Ronya) = [(on(sunday))[(in(amsterdam)) (kiss(ronya))]](fred) These are general problems of polyadicity, though. These issues will be addressed (succesfully) in a later chapter Categorial grammar and function composition We add to the syntactic theory developed sofar a treatment of gaps in categorial grammar. Our main interest here is to introduce function composition and small and big lambdas. (Our discussion here is indepted to many works, but I single out the work on surface compositionality by Polly Jacobson as the main inspiration for the current discussion.) Let A stand for a tree with topnode A. A/ stands for a tree with topnode A, and with node missing in it, i.e. with a -gap: A A/ mary Ø On this model, a tree of the form / is a with a missing in it. The smallest kind of this is a with itself missing, this is a -gap: / Ø The normal X-bar rules that we have dictate the natural principles for percolating slashes up the tree: X-bar theory allows a tree of the form: 201

27 V' V If we replace the by a -gap, then the top V' is itself no longer a V', but a V' with a missing in it: V'/ V / Ø Following this strategy, we can build up an IP/ structure containing a chain: IP/ I'/ / every boy Ø V / kiss Ø At this point we can allow an operation that provides the head of the chain, and we get a topicalization structure. (all this can, of course, be made syntactically sophisticated). IP IP/ D NP I'/ some girl / every boy Ø V / kiss Ø We are interested in the semantic interpretation. We start with the gap. /. Standardly, we would interpret a -gap as a free variable x. Here we think of the gap as a tree with 'itself' missing in it. We carry over that idea into the semantics: We would like to think of the meaning of the gap as a 202

28 variable meaning, but it must really be a variable meaning with itself missing it, i.e. something that would be a variable if you applied it to a variable. This is just a complex way of describing the identity function, and since we are concerned with a variable of type e, we interpret the gap as: / λx.x with x a variable of type e <e,e> V KISS <e,<e,t>> We would get at the VP level: VP APPLY[KISS, ] <e,t> But we don't get that, we get: VP/ KISS + And the reasoning about types tells us that VP/ is not of type <e,t>, like VP, but has a meaning missing in it, i.e. is a function from -meanings to VP-meanings: VP/ KISS + / <e,<e,t>> <e,<e,t>> <e,t> The operation involved is function composition: α β = λx.α(β(x)) -Apply β to a variable x -Apply α to the result β(x) -Abstract over x. You get the function that maps every x onto the result of first applying β to x, and then applying α to the result. Instead of simple function application, we write APPLY: Right Function Composition: RC[α,β] = λx.apply[α, (β(x))] VP/ KISS + / <e,<e,t>> <e,<e,t>> <e,t> 203

29 RC[KISS, λx.x] = λz.apply[kiss, λx.x(z)] = λz.apply[kiss, z] = λz.kiss(z) = KISS Thus, we derive, correctly, a relational meaning for the VP/. I'/ λp.p + VP/ <e,<e,t>> <<e,t>,<e,t>> <e,<e,t>> The same operation of RC builds the correct meaning for I'/: (the gap is, as before, inside the argument, and we use the same operation of RC) RC[λP.P, KISS] = λz.apply[λp.p, KISS(z)] = λz.λp.p(kiss(z)) = λz.kiss(z) = KISS At the next stage, the gap is in the function, rather than in the argument, since we chose the I' to be a function on the specifier. We pay the price for that decision here. If we had switched the function-argument structure around, we could have continued with R-C. Now we must use L-C (since the gap is inside the function) and do a lot of type shifting: λp. x[boy(x) P(x)] IP/ L-C[KISS, λp. x[boy(x) P(x)]] L-C[KISS, λp. x[boy(x) P(x)]] = λz.apply[ KISS(z), λp. x[boy(x) P(x)]] = LIFT λz. λt.t(kiss(z)) (λp. x[boy(x) P(x)]) = λz. λp. x[boy(x) P(x)]( KISS(z)) = λz. x[boy(x) KISS(x,z)] IP/ λz. x[boy(x) KISS(x,z)] <e,t> The property that you have if every boy kisses you What function composition does in each stage of the derivation is: -Temporarily apply the thing with the gap to a variable (of type type of the gap). -Then do the operation that you would do if there were no gap (i.e. APPLY). -After that, abstract over the variable again. This way you keep the lambda-operation λz. on the outside of the expression all the time during the derivation. So, not surprisingly, since IP is of type t, IP/ is of type <e,t>. This combines as a function to the -top of the chain: 204

30 Now the final operation is APPLY rather than compose. The types get balanced again at the top of the chain (normality is re-established, anything else is just your own problem): IP APPLY[λz. x[boy(x) KISS(x,z)]. λp y[girl(y) P(y)] This gets resolved in standard way, and we derive: IP y[girl(y) x[boy(x) KISS(x,y)]] We have derived one interpretation. However, the topicalization sentence Some girl every boy kissed is scopally ambiguous, it allows both scope readings. The problem is: how do we get the narrow scope reading? We know that the gap / has to have the interpretation of an identity function. We also know that its interpretation will have to take narrow scope. Above, we interpreted the gap as low as we could as λx.x, with an individual variable missing in it. But for narrowest scope we want to hide inside the gap not an interpretation of type e, but a full interpretation at type <<e,t,>,t>, because that is the expression that takes narrow scope. So we allow an interpretation of the gap with an interpretation at the highest relevant type missing: / λt.t with T a variable of type <<e,t,>,t> <<<e,t>,t>,<<e,t>,t>> We follow the derivation: VP/ RC[KISS, λt.t] RC[KISS, λu.u] = [alphabetic variable] λt.apply[kiss, λu.u(t)] = λt.apply[kiss, T] = [lift KISS] λt ([λuλx.u(λy.kiss(x,y))](t)) = λtλx.t(λy.kiss(x,y) It will not come as a surprise that this interpretation inherits with R-C up to I'/. At the next step L-C will give: IP/ L-C[λTλx.T(λy.KISS(x,y)), λp. x[boy(x) P(x)]] -we apply the function to a variable T, and get λx.t(λy.kiss(x,y)) of type <e,t>. This applies in the normal way to the meaning, deriving: x[boy(x) T(λy.KISS(x,y))] 205

31 Finally, we abstract over T and get: IP/ λt. x[boy(x) T(λy.KISS(x,y))] This applies to the topicalized : IP λt. x[boy(x) T(λy.KISS(x,y))] (λp. y[girl(y) P(y)]) λ-conversion gives: IP x[boy(x) y[girl(y) KISS(x,y)]] Thus we derive the narrow scope interpretation by giving a higher interpretation to the gap (with a 'big lambda'). This technique has also been applied to cases involving internal interpretations of external material in relative clauses involving intensional contexts (e.g. Sharvit 20??). In the next section we extend the scope theory to quantifying into intensional contexts Intensional verbs In this section we add to the grammar the intensional transitive verb seek, the complementizer that, and the propositional attitude verb believe. We add to the category-type assignment: V <<s,<<e,t>,t>>,<e,t>>, <<s,t>,<e,t>> C <<s,t>,<s,t>> And we add: COMP[C,I] = E l Lexical item: seek Category: V Interpretation: SEEK CON <<s,<<e,t>,t>>,<e,t>> Lexical item: believe Category: V Interpretation: BELIEVE CON <<s,t>,<e,t>> Lexical item: that Category: C Interpretation: λp.p with p VAR <s,t> 206

32 We extend the type shifting theory with two operations: Intensionalization: LIFT: a <s,a> LIFT[α] = α Intensional argument lift: LIFT: e <s,<<e,t>,t>> LIFT[α] = λp.p(α) We define the following relation: SEEK * = λyλx.[seek( λp.p(y))](x) or in relational notation: SEEK * = λyλx.seek(x, λp.p(y)) The complementizer that takes a proposition (type <s,t>) and turns it into the same proposition, it is the identity function on propositions. Believe is interpreted as a relation between individuals and propositions. The transitive verb seek is a relation between individuals and the intensions of generalized quantifiers. As we have argued before, unlike find, seek is not defined in terms of a lower relation FIND between individuals. Seek is intensional in object position: it is a relation between an individual and a function from worlds into sets of properties of individuals (extensional properties, i.e. sets of sets). In certain cases, a function from worlds into sets of properties corresponds uniquely with an individual. That is precisely the case when that function assigns to every world the set of properties that some individual y has in that world. The ones that do are precisely the functions of the form: λp.p(y), where y is a rigid designator (a constant of type e or a variable). If F is of the form λp.p(y), with y a rigid designator, then we know that to the relation SEEK(x,F) between an individual and F, there corresponds uniquely a relation between the individuals x and y. This relation between individuals we call SEEK *, and it is defined as: SEEK * is the relation that x and individual y stand in iff x and λp.p(y) stand in the relation SEEK. (in other words j SEEK * m iff j stands in the relation SEEK to the function that assigns to every world the set of properties that Mary has in that world.) 207

33 This means that in the special case that a function F from worlds into sets of properties corresponds uniquely to an individual y, and this function is the object of SEEK, we can simplify a meaning representation like SEEK(x,F) to a relation SEEK * (x,y) between individuals. Example: Suppose we get to a derivation with meaning SEEK(j, λp.p(m)) We know that m is a rigid designator, hence the fact that it is in an intensional context (under ) does not block λ-conversion. We can apply backwards λ-conversion and get: SEEK(j,λP.P(m)) = [λx.seek(x,λp.p(m))] (j) [backward λ-conversion] = [ [λyλx.seek(x, λp.p(y))] (m) ](j) [backward λ-conversion, m is rigid] The functional expression we get now is precisely the definition of SEEK *, so we substitute it: = SEEK * (j,m) So in these cases we can represent the relation as a relation between individuals. We can not do that if the function does not correspond in this way to an individual. For example, we can not in the same way reduce the following to a relation between individuals: SEEK(j, λp. x[u(x) P(x)]) If John stands in the relation SEEK to the function that assigns to every world the set of properties that at least one unicorn in that world has, there is no guarantee that John stands in any relation to any actual individual, hence the above expression can not be reduced further. We argued before, that we could not give a translation of seek as λtλx.seek(x, T) of type <<e,t>,t>, to make it closer to find. λ-conversion into the complement of seek would not be valid for non-rigid NP translations. We see exactly the same with the meaning of that. We have assumed that it is λp.p of type <<s,t>,<s,t>>. Wouldn't it be nicer to interpret that as: λφ. φ of type <t,<s,t>>? Yes, but again we cannot reduce λφ. φ (WALK(j) to WALK(j) through λ-conversion for the same reason. We still want to end up with WALK(j), and the strategy we apply here for getting that is a very general one: We cannot convert in α for a variable of extensional type a under an intensional operator: λx a... (...x a...)...(α a ) 208

34 We want to get α a there nevertheless. What we do is, abstract over a variable of type <s,a> instead and apply this to α a : λx <s,a>... (... x <s,a>...)...( α a ) This will reduce with λ-conversion to:... (... α a...) and the latter with cup-cap elimination to the required result:... (...α a...) The above interpretation of that as λp.p is really an instance of this, because, since variable p is a rigid designator, λp. ( p) = λp.p Thus, as a general strategy for dealing with the problem of converting something into an intensional context, we will make sure that λ-abstraction that binds a variable for which we want to convert α a is of type <s,a>. Then α a is not of the right type to be converted in, but APPLY will lift α a to α a, which can be converted in. 209

35 EXAMPLES (1) John seeks Mary. SYNTAX IP SEMANTICS A I' A JOHN john I VP λp.p A Ø V SEEK MARY seek mary APPLY[SEEK,MARY] = SEEK(LIFT[MARY]) [def. APPLY] = SEEK( λp.p(mary)) [def. LIFT] APPLY[λP.P,SEEK( λp.p(mary))] = λp.p(seek( λp.p(mary))) [def. APPLY] = SEEK( λp.p(mary)) APPLY[SEEK( λp.p(mary)),john] = SEEK( λp.p(mary)) (JOHN) [def. APPLY] = SEEK(JOHN, λp.p(mary)) [rel. notation] = [λx.seek(x, λp.p(mary)](john) [backwards λ-con] = [[λyλx.seek(x, λp.p(y))](mary) ] (JOHN) [backwards λ-con, MARY is rigid] = SEEK * (JOHN,MARY) [def SEEK * ] So we get: SYNTAX SEMANTICS IP A = SEEK * (j,m) I' A JOHN john I VP λp.p A Ø V SEEK MARY seek mary 210

36 (2) John seeks a unicorn. DERIVATION 1. IP A I' A JOHN john I VP λp.p A Ø V SEEK A seek D NP UNICORN a unicorn APPLY[SEEK,λP. z[unicorn(z) P(z)]] = SEEK(LIFT[λP. z[unicorn(z) P(z)]]) [def APPLY] = SEEK( λp. z[unicorn(z) P(z)]) APPLY[λP.P,SEEK( λp. z[unicorn(z) P(z)])] = SEEK( λp. z[unicorn(z) P(z)]) APPLY[SEEK( λp. z[unicorn(z) P(z)]),JOHN] = SEEK( λp. z[unicorn(z) P(z)]) (JOHN) [def. APPLY] = SEEK(JOHN, λp. z[unicorn(z) P(z)]) [rel. notation.] So we get: IP A = SEEK(j, λp. z[unicorn(z) P(z)]) I' A JOHN john I VP λp.p A Ø V SEEK A seek D NP UNICORN a unicorn 211

37 (2) John seeks a unicorn. DERIVATION 2. IP A, Ø I' λ,,<{<3,[ A,UNICORN]>},3> A john I VP x 3 A UNICORN Ø V 3 A JOHN seek D NP λp.p A a unicorn SEEK x 3, {<3,[ A,UNICORN]>} APPLY[SEEK,x 3 ] = SEEK(LIFT[x 3 ]) [def. APPLY] = SEEK( λp.p(x 3 )) [def. LIFT] APPLY[λP.P,SEEK( λp.p(x 3 ))] = SEEK( λp.p(x 3 )) APPLY[SEEK( λp.p(x 3 )),JOHN] = SEEK( λp.p(x 3 )) (JOHN) [def. APPLY] = SEEK(JOHN, λp.p(x 3 )) [rel. notation.] = [λx.seek(x, λp.p(x 3 )) ](JOHN) [backwards λ-con] [[λyλx.seek(x, λp.p(y))](x 3 ) ] (JOHN) [backwards λ-con, x 3 is rigid] = SEEK * (JOHN,x 3 ) [def SEEK * ] [ λ x 3,SEEK * (JOHN,x 3 )] = λx 3.SEEK * (JOHN,x 3 ) [def. λ] = λx.seek * (JOHN,x) [alphabetic var.] APPLY[λx.SEEK * (JOHN,x), λp. z[unicorn(z) P(z)]] = LIFT[λx.SEEK * (JOHN,x)](λP. z[unicorn(z) P(z)]) [def. APPLY] = λt.t(λx.seek * (JOHN,x))(λP. z[unicorn(z) P(z)]) [def. LIFT] = λp. z[unicorn(z) P(z)] (λx.seek * (JOHN,x)) [λ-con] = z[unicorn(z) [λx.seek * (JOHN,x)](z)] [λ-con] = z[unicorn(z) SEEK * (JOHN,z)] [λ-con, z is rigid] 212

38 IP z[unicorn(z) SEEK * (JOHN,z)] I' john I VP Ø V 3 seek D NP a unicorn (3) Bill believes that a unicorn danced. DERIVATION 1. IP A I' A BILL Bill I VP λp.p A Ø V CP BELIEVE A believe C IP λp.p A that I' A A λp.p DANCE UNICORN a unicorn Ø dance APPLY[APPLY[λP.P,DANCE],APPLY[,UNICORN]] = x[unicorn(x) DANCE(x)] APPLY[λp.p, x[unicorn(x) DANCE(x)]] = λp.p (LIFT[ x[unicorn(x) DANCE(x)]]) [def APPLY] = λp.p( x[unicorn(x) DANCE(x)]) [def LIFT] = x[unicorn(x) DANCE(x)] APPLY[BELIEVE, x[unicorn(x) DANCE(x)]] = BELIEVE( x[unicorn(x) DANCE(x)]) [def APPLY] APPLY[BELIEVE( x[unicorn(x) DANCE(x)]),BILL] = BELIEVE( x[unicorn(x) DANCE(x)])(BILL) [def APPLY] = BELIEVE(BILL, x[unicorn(x) DANCE(x)]) 213

39 So we get: IP, BELIEVE(BILL, x[unicorn(x) DANCE(x)]) I' Bill I VP Ø V CP believe C that IP I' a unicorn Ø dance (3) Bill believes that a unicorn danced. DERIVATION 2. IP, A,Ø I' λ,<{<4, [ A,UNICORN]>},4> A Bill I VP x 4 A, UNICORN Ø V CP A BILL believe C IP λp.p A that 4 I' BELIEVE A, λp.p A a unicorn Ø dance A x 4,{<4, A [,UNICORN]>} λp.p DANCE 214

40 APPLY[APPLY[λP.P,DANCE],x 4 ] = DANCE(x 4 ) APPLY[λp.p,DANCE(x 4 )] = DANCE(x 4 ) APPLY[APPLY[λP.P,APPLY[BELIEVE, DANCE(x 4 )]],BILL] = BELIEVE(BILL, DANCE(x 4 )) [ λ x 4,BELIEVE(BILL, DANCE(x 4 ))] = λx.believe(bill, DANCE(x)) APPLY[λx.BELIEVE(BILL, DANCE(x)),APPLY[,UNICORN]] = x[unicorn(x) BELIEVE(BILL, DANCE(x))] So we get: IP, x[unicorn(x) BELIEVE(BILL, DANCE(x))] I' Bill I VP Ø V CP believe C that IP I' a unicorn Ø dance A more sophisticated analysis of quantifying-in involving individual concepts and dealing more adequantely with de re and de se readings is given in the next chapter. 215