Binding and Scope in Pregroup Grammar - Unich

Binding and Scope in Pregroup Grammar Evidence from Italian Claudia Casadio

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Contents 1 2 3 4 5

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Italian verb phrase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The auxiliary verbs avere and essere . . . . . . . . . . . . . . . . . . . Finite verb-forms and declarative sentences . . . . . . . . . . . . . . . . . . . Statements and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Subordinate clauses and indirect questions . . . . . . . . . . . . . Direct questions and wh-questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Metarule I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Metarule II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Subject vs. object reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Metarule III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.

Introduction

A pregroup {G, . , 1, ` , r , →} is a partially ordered monoid in which each element a has a left adjoint a` , and a right adjoint ar such that a` a → 1 → a a ` a a r → 1 → ar a the dot “.” stands for multiplication with unit 1, and the arrow denotes the partial order. In linguistic applications the symbol 1 stands for the empty string of types and multiplication is interpreted as concatenation.

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Adjoints are unique and we prove 1` = 1 = 1r , (a · b)` = b ` · a ` a→b b ` → a`

a→b , b r → ar

, (a · b)r = b r · a r , b` → a` , a`` → b``

b r → ar , arr → brr .

The following also hold ar` = a = a`r , a`` a` → 1 → a` a`` , ar arr → 1 → arr ar ,

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We work with the pregroup freely generated by a partially ordered set of basic types. From each basic type a we form simple types by taking single or repeated adjoints: . . . a`` , a` , a, ar , arr . . . Compound types or just types are strings of simple types. We assign to each word or word form in the dictionary of the language under investigation one (or more) types. The only computations required are contractions (C) and expansions (E) : (C)

a` a → 1 , a ar → 1 ,

(E)

1 → a a` , 1 → ar a ,

where a is a simple type. For the purpose of sentence verification expansions are not needed, but only contractions, combined with some rewriting induced by the partial order.

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Developing a pregroup grammar for a natural language, such as Italian, consists in two main steps: (i) assign one or more (basic or compound) types to each word in the dictionary; (ii) check the grammaticality and sentencehood of a string of words by a calculation on the corresponding types where the only rules involved are : contractions, ordering postulates taking the form α → β (α, β basic types) and appropriate conditions introduced in the lexicon, called metarules.

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2.

The Italian verb phrase

We introduce the free pregroup generated by the following partially ordered set of basic types: s, ˜s, s, σ q, q i, ˜ı, ı, i∗ , j, ı π o ω λ τ

declarative sentences and sentential complements , questions and indirect questions , infinitives or infinitival complements , subject , direct object , indirect object , locative phrase , temporal phrase .

The types o, ω, λ, τ correspond to the possible arguments subcategorized by a verb; the types ˜s, s are expansions of the type s; the type σ is defined as the type of sentential complements (including infinitive complements). The type ı is the maximal expansion of the different kinds of infinitival complements1 . 1

Details in Casadio and Lambek (2001). Contents

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Basic and compound types assigned to a few representative verbs: (to run)

correre :

i , i λ` ;

(to see)

vedere :

i , i o` ;

(to obey)

obbedire :

i , i ω` ;

(to give)

dare :

i ω ` o` , i o` ω ` ;

(to put)

mettere :

i λ` o` , i o` λ` ;

(to arrive) arrivare :

i ∗ , i ∗ λ` .

The star on i ∗ is a reminder that the perfect tense is to be formed with the auxiliary essere rather than avere, deriving infinitival phrases of type i ∗ , rather than of type i ;

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correre , i

correre (nel prato) ; (i λ` ) λ → i

vedere , i

vedere (un libro) ; (i o` ) o → i

obbedire , i

obbedire (a Mario) ; (i ω ` ) ω → i

dare (un libro) (a Mario) (i ω ` o` ) o ω → i mettere (un libro) (sul tavolo) (i λ` o` ) o λ → i arrivare , i∗

,

dare (a Mario) (un libro) ; (i o` ω ` ) ω o → i ,

mettere (sul tavolo) (un libro) ; (i o` λ` ) λ o → i arrivare (a Roma) ; (i ∗ λ` ) λ → i∗

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2.1.

The auxiliary verbs avere and essere

Italian has two auxiliary verbs: avere (to have) and essere (to be); differently from English, both are selected by active verbs, but essere is required in passive forms, like in English. The perfect infinitive is formed in Italian from the past participle with the help of the two auxiliary verbs; avere takes the type: avere : i p `2 where p 2 is the type of the past participle of a verb with infinitive of type i. The past participle of a verb is constructed by means of a perfect inflector : Perf : p 2 i `

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Perf (vedere) = visto (p 2 i ` ) i → p 2

Perf (vedere) = visto (p 2 i ` ) (i o` ) → p 2 o`

avere = avere (i p `2 )

visto Perf (vedere) (p 2 i ` ) (i o` )

un libro , (un libro) o → i

avere = avere (i p `2 )

dato un libro a Mario , Perf (dare) (un libro) (a Mario) (p 2 i ` ) (i ω ` o` ) o ω → i

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A number of intransitive verbs require the auxiliary verb essere for forming the perfect infinitive, e.g. the verb of motion arrivare of type i ∗ or i ∗ λ` . We account for this by assigning the type: essere : i ∗ p ∗2 ` , and by giving a new type to the inflector Perf : Perf : p 2 i ` , p ∗2 i ∗`

=

essere essere (i ∗ p ∗2 ` )

arrivato , Perf (arrivare) (p ∗2 i ∗` ) i ∗ → i∗

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3.

Finite verb-forms and declarative sentences

To each Italian verb V there is associated a matrix Vj k of 7 × 6 = 42 finite verbforms, where j = 1, ... ,7 denotes tenses (including moods) and k = 1, 2, 3 denotes the three persons singular, while k = 4, 5, 6 denotes the three persons plural. We shall use the types : sj

for declarative sentences in j-th tense ,

πk

for k-th person subject .

We confine our attention to the cases j = 1 (present tense), j = 2 (past tense), and k = 1 (first person) or k = 3 (third person).

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Consider the first person present tense sentence: (io) vedo (un libro) , π 1 (π r1 s1 o` ) o → s1 the optional pronoun io has type π 1 , and the first person verb-form vedo takes the type (π r1 s1 o` ). The properties of the inflectional system represented by the conjugation matrix Vj k play a crucial role in word order and constituent formation in Italian. To account for this we assume that Vj k is obtained from the infinitive V by an inflector Cj k (V) = Vj k . Being the source of assertions and declarative statements, we call this inflector declarative, with two possible types: Cj k : π rk sj ı ` , sj ı ` , we postulate i → ı → ı, i ∗ → ı∗ → ı Contents

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Italian can omit both subjects and full verbal complements, as a consequence of the interrelation of verbal inflectional morphology and pronominal cliticization : Maria d`a un libro a Piero (Maria gives a book to Piero) D`a un libro a Piero ([she] gives a book to Piero) Lo d`a a Piero ([she] it gives to Piero) Gli d`a un libro ([she] to-him gives a book ) Glielo d`a ([she] to-him-it gives) From top to bottom we notice a progressive disappearing of the lexicalized verb arguments (subject, direct object, indirect object) substituted by null subject and clitic pronouns. But, differently from English, also the last string is a perfectly correct Italian sentence. We therefore assign a single type to the declarative inflector: Cj k : (π rk ) sj ı ` Contents

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Maria corre (nel prato) , = Maria C13 (correre) (nel prato) π 3 (π r3 s1 ı ` ) (i λ` ) λ → s1 Maria obbedisce (a Piero) , = Maria C13 (obbedire) (a Piero) π 3 (π r3 s1 ı ` ) (i ω ` ) ω → s1 Maria d` a (un libro) (a Piero) , = Maria C13 (dare) (un libro) (a Piero) π 3 (π r3 s1 ı ` ) (i ω ` o` ) o ω → s1 Maria mette (un libro) (sul tavolo) , = Maria C13 (mettere) (un libro) (sul tavolo) , π 3 (π r3 s1 ı ` ) (i λ` o` ) o λ → s1 Maria arriva (a Roma) , = Maria C13 (arrivare) (a Roma) , π 3 (π r3 s1 ı ` ) (i ∗ λ` ) λ → s1 Contents

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4.

Statements and questions

In Italian the distinction between declarative and interrogative sentences is not clearly marked by specific items, such as the auxiliary verb do in English, but it is rather based on emphasis, intonation and word order. There are two types of interrogative sentences: direct questions and wh-questions; the former are structurally similar to statements, with possible word order variations, while the latter are introduced by the wh-pronouns chi (who), che cosa (what), quale (which), or the adverbials quando (when), dove (where), and are structurally similar to wh-questions in English. Also similarly to English, indirect questions are formally identical to declarative statements, while direct questions and wh-questions exhibit a distintictive pattern reflected in the type assignment.

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4.1.

Subordinate clauses and indirect questions

We introduce the simple types s, s, ˜s, σ q, q

for sentential complements,

for questions,

i, ı, ˜ı, ı

for infinitives;

and we postulate s j → s → ˜s, s → σ but s 6→ σ, ˜s 6→ σ; q → σ, but q 6→ q

and q 6→ σ;

i → ı → ı and ˜ı → σ, but i 6→ σ and ı 6→ σ.

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Compound types ˜s s` , sr ˜s ır ı s ˜s` q ˜s ` ˜ı ı ` i σ` i σ` i σ`

sentential adverbials infinitive clause adverbials sentential complementizer indirect question complementizer infinitive complementizer verbs taking infinitive complements verbs taking sentential complements verbs taking indirect questions

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domani (tomorrow) bene (well) che (that) se (wether) di (to) pensare (to think) dire (to say) chiedersi (to wonder)

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Maria corre oggi π 3 (π r3 s1 ) (sr ˜s) → ˜s1 Oggi Maria corre allo stadio (˜s s` ) π 3 (π r3 s1 λ` ) λ → ˜s1

(Maria is running today) since s 1 → ˜s

(Today Maria is running at the stadium) since s 1 → ˜s

Maria pensa di parlare bene π 3 (π r3 s1 σ ` ) (˜ı ı ` ) i (ı r ı) → s1

(Maria thinks to speak well ) since ˜ı → σ

Maria dice che Piero parla domani π 3 (π r3 s1 σ ` ) (s ˜s` ) π 3 (π r3 s1 ) (sr ˜s) → s1

(Maria says that Piero will speak

Maria si chiede se Piero parla domani π 3 (π r3 s1 σ ` ) (q ˜s` ) π 3 (π r3 s1 ) (sr ˜s) → s1

(Maria is wondering wether Piero

tomorrow )

since s → σ

will speak tomorrow ) since q → σ

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Maria pensa di parlare bene π 3 (π r3 s1 σ ` ) (˜ı ı ` ) i (ı r ı) → s1

Maria pensa parlare bene π 3 (π r3 s1 σ ` ) i (ı r ı) → s1 /

In the second case the calculation fails because the contraction (σ ` ı) is not allowed, having assumed ı 6→ σ. Maria dice che Piero parla domani π 3 (π r3 s1 σ ` ) (s ˜s` ) π 3 (π r3 s1 ) (sr ˜s) → s1

This calculation would fail if we omit the complementizer che of type (s ˜s` ), since we assume s → σ but ˜s 6→ σ .

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Maria si chiede se Piero parla domani π 3 (π r3 s1 σ ` ) (q ˜s` ) π 3 (π r3 s1 ) (sr ˜s) → s1

The calculation fails if we omit the complementizer se of type q s˜` , since we assume q → σ but ˜s 6→ σ.

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5.

Direct questions and wh-questions

A direct question is obtained in Italian from a declarative sentence just adding a question mark at the end of the sentence and changing the intonation pattern: Piero ama Maria. (Piero loves Maria) Piero ama Maria? (Does Piero love Maria? ) Lui ama Maria. (He loves Maria) Lui ama Maria? (Does he love Maria? ) La ama. ((He) loves her ) La ama? (Does he love her? )

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5.1.

Metarule I

This strict structural correspondence can be accounted for by introducing a new inflector C?j k and the metarule Interrogative Form Metarule I If Cj k (V) has type ((π rk ) sj ı ` ) , then C?j k (V) has type ((π rk ) qj ı ` ), where the assignment of type q to a string of words W results in adding the punctuation “ ? ” at the end of W; W? is the interrogative form of W, with type q.

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Piero ama Maria. π 3 C1k (i o` ) o

Lui la ama. π 3 C1k ( o`` i ` )(i o` )

= π 3 (π r3 s1 ı ` ) (i o` ) o → s1

= π 3 (π rk s1 ı ` ) ( o`` i ` )(i o` ) → s1

= π 3 (π rk s1 o` ) o → s1

= π 3 (π rk s1 o`` i ` )(i o` ) → s1

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Piero ama Maria. π 3 C1k (i o` ) o

Lui la ama. π 3 C1k ( o`` i ` )(i o` )

= π 3 (π r3 s1 ı ` ) (i o` ) o → s1

= π 3 (π rk s1 ı ` ) ( o`` i ` )(i o` ) → s1

= π 3 (π rk s1 o` ) o → s1

= π 3 (π rk s1 o`` i ` )(i o` ) → s1

Piero ama Maria? π 3 C?1k (i o` ) o

Lui la ama? π 3 C?1k ( o`` i ` )(i o` )

= π 3 (π r3 q1 ı ` ) (i o` ) o → q1

= π 3 (π rk q1 ı ` ) ( o`` i ` )(i o` ) → q1

= π 3 (π rk q1 o` ) o → q1

= π 3 (π rk q1 o`` i ` )(i o` ) → q1

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5.2.

Metarule II

Inversion is also possible in interrogative sentences: Ama Maria, Piero? (loves Maria, Piero? ) Domani arriva Piero? (tomorrow is Piero arriving? ) . Ha parlato Maria? (has spoken Maria? ) To treat these second interrogative form, we need a new metarule:

Interrogative Form Metarule II If Cj k (V) has type ((π rk ) sj ı ` ) , then C?j k (V) has type (qj (π `k ) ı ` ) where the assignment of type q to a string of words W results in adding the punctuation “ ? ” at the end of W; W? is the interrogative form of W, with type q.

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This rule allows one to introduce a second type of direct question in which the subject occurs in postverbal position with type π `k ; the inverted form is also the input of wh-questions introduced by interrogative pronouns2 such as: qπ ˆ 3`` q`

chi (who) subject, nominative ;

q oˆ`` q`

chi (whom) direct object, accusative ;

The basic types π ˆ 3`` , oˆ3`` occurring within the types assigned to the interrogative pronouns are related to a missing argument in the context following the pronoun (an unbounded dependency or a trace in the sense of Chomsky) and the contractions (ˆ π 3`` π 3` ) → 1 , (ˆ o `` o` ) → 1 , are licensed by the assumptions π ˆ 3 → π 3 , oˆ → o .

2

We take the types π ˆ 3`` , oˆ3`` from Lambek (2004) to distinguish interrogative pronouns from topicalized noun phrases, and to avoid cancellations with illicit arguments; but we introduce the type q in place of q ˜ for indirect questions. Contents

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So far we have two interrogative forms for direct questions of type q, introduced by Metarule I and Metarule II respectively, where the second form with the inverted subject is taken by the wh-pronouns to generate wh-questions of type q. The type q can also be used to form an indirect question and as such combines with σ ` since we postulate q → σ. Chi ha parlato? (Who has spoken? ) (q π ˆ 3`` q` ) (q1 π `3 ) → q Maria si chiede chi ha parlato ˆ 3`` q` ) (q1 π `3 ) → s1 π 3 (π r3 s1 σ ` ) (q π

(Maria is wondering who has spoken)

The verb form . . .ha parlato? takes the interrogative type (q1 π `3 ), where the subject is expected to occurr in postverbal position. The second example contracts to s1 because q → σ.

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Now we add a direct object to the interrogative statement; remind that incontrare is a transitive verb with type (i o` ) and past participle (p2 o` ) :

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Now we add a direct object to the interrogative statement; remind that incontrare is a transitive verb with type (i o` ) and past participle (p2 o` ) : Chi ha incontrato Piero? (Who has met Piero? ) (q π ˆ 3`` q ` ) (q1 π `3 p2` ) (p2 o` ) o → q Maria si chiede chi ha incontrato Piero (Maria is wondering who has met Piero) ˆ 3`` q ` ) (q1 π `3 p2` ) (p2 o` ) o → s π 3 (π r3 s1 σ ` ) (q π Chi dice che ha incontrato Piero? (Who says (he) has met Piero? ) (q π ˆ 3`` q ` ) (q1 π `3 σ ` ) (s ˜s` ) (s1 p2` ) (p2 o` ) o → q

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5.3.

Subject vs. object reading

In the last example the subject of the subordinate clause is omitted and the preferred interpretation is “there is somebodyi saying that hei has met Piero”. A second possible interpretation is “there is somebodyj that Pieroi says hei has met”, in this case the wh-pronoun is the object. We illustrate these two different analyses considering both a direct and an inderect question:

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5.3.

Subject vs. object reading

In the last example the subject of the subordinate clause is omitted and the preferred interpretation is “there is somebodyi saying that hei has met Piero”. A second possible interpretation is “there is somebodyj that Pieroi says hei has met”, in this case the wh-pronoun is the object. We illustrate these two different analyses considering both a direct and an inderect question: Chi dice che ha incontrato Piero?

(there is somebodyi saying that hei has met Piero)

(q π ˆ 3`` q ` ) (q1 π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → q

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5.3.

Subject vs. object reading

In the last example the subject of the subordinate clause is omitted and the preferred interpretation is “there is somebodyi saying that hei has met Piero”. A second possible interpretation is “there is somebodyj that Pieroi says hei has met”, in this case the wh-pronoun is the object. We illustrate these two different analyses considering both a direct and an inderect question: Chi dice che ha incontrato Piero?

(there is somebodyi saying that hei has met Piero)

(q π ˆ 3`` q ` ) (q1 π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → q

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5.3.

Subject vs. object reading

In the last example the subject of the subordinate clause is omitted and the preferred interpretation is “there is somebodyi saying that hei has met Piero”. A second possible interpretation is “there is somebodyj that Pieroi says hei has met”, in this case the wh-pronoun is the object. We illustrate these two different analyses considering both a direct and an inderect question: Chi dice che ha incontrato Piero?

(there is somebodyi saying that hei has met Piero)

(q π ˆ 3`` q ` ) (q1 π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → q

Chi dice che ha incontrato, Piero?

(there is somebodyj that Pieroi says hei has met)

(q oˆ3`` q ` ) (q1 σ ` ) (s ˜s` ) (s1 o` π `3 ) π 3 → q

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5.3.

Subject vs. object reading

In the last example the subject of the subordinate clause is omitted and the preferred interpretation is “there is somebodyi saying that hei has met Piero”. A second possible interpretation is “there is somebodyj that Pieroi says hei has met”, in this case the wh-pronoun is the object. We illustrate these two different analyses considering both a direct and an inderect question: Chi dice che ha incontrato Piero?

(there is somebodyi saying that hei has met Piero)

(q π ˆ 3`` q ` ) (q1 π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → q

Chi dice che ha incontrato, Piero?

(there is somebodyj that Pieroi says hei has met)

(q oˆ3`` q ` ) (q1 σ ` ) (s ˜s` ) (s1 o` π `3 ) π 3 → q

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(Maria is wondering who says he has met Piero) Maria si chiede chi dice che ha incontrato Piero π 3 (π r3 s1 σ ` ) (q π ˆ 3`` q ` ) (q π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → s

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(Maria is wondering who says he has met Piero) Maria si chiede chi dice che ha incontrato Piero π 3 (π r3 s1 σ ` ) (q π ˆ 3`` q ` ) (q π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → s

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(Maria is wondering who says he has met Piero) Maria si chiede chi dice che ha incontrato Piero π 3 (π r3 s1 σ ` ) (q π ˆ 3`` q ` ) (q π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → s

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(Maria is wondering who says he has met Piero) Maria si chiede chi dice che ha incontrato Piero π 3 (π r3 s1 σ ` ) (q π ˆ 3`` q ` ) (q π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → s

(Maria is wondering who Piero says he has met) Maria si chiede chi dice che ha incontrato, Piero π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q σ ` ) (s ˜s` ) (s1 o` π `3 ) π 3 → s

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(Maria is wondering who says he has met Piero) Maria si chiede chi dice che ha incontrato Piero π 3 (π r3 s1 σ ` ) (q π ˆ 3`` q ` ) (q π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → s

(Maria is wondering who Piero says he has met) Maria si chiede chi dice che ha incontrato, Piero π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q σ ` ) (s ˜s` ) (s1 o` π `3 ) π 3 → s

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(Maria is wondering who says he has met Piero) Maria si chiede chi dice che ha incontrato Piero π 3 (π r3 s1 σ ` ) (q π ˆ 3`` q ` ) (q π `3 σ ` ) (s ˜s` ) (s1 p2 ` ) (p2 o` ) o → s

(Maria is wondering who Piero says he has met) Maria si chiede chi dice che ha incontrato, Piero π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q σ ` ) (s ˜s` ) (s1 o` π `3 ) π 3 → s

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Comments:

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Comments:  the link connecting the types (ˆ o`` o` ) is longer than the link connecting `` ` the types (ˆ π 3 π 3 );

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Comments:  the link connecting the types (ˆ o`` o` ) is longer than the link connecting `` ` the types (ˆ π 3 π 3 );  a long distance dependency is in fact created between the wh-pronoun and the direct object of the subordinate clause. In this case we have an emphatic interpretation with the subject occurring in the last position in the string;

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Comments:  the link connecting the types (ˆ o`` o` ) is longer than the link connecting `` ` the types (ˆ π 3 π 3 );  a long distance dependency is in fact created between the wh-pronoun and the direct object of the subordinate clause. In this case we have an emphatic interpretation with the subject occurring in the last position in the string;  such interpretation does not follow either from the type ((π rk ) sj ı ` ) assigned to Cj k (V), or from the types ((π rk ) qj ı ` ) and (qj (π `k ) ı ` ) assigned to C?j k (V).

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Note that a simpler derivation is obtained when the subject of the subordinate clause is elliptical

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Note that a simpler derivation is obtained when the subject of the subordinate clause is elliptical (Maria is wondering who do you say to have met) Maria si chiede chi dici che hai incontrato π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 (π `2 ) σ ` ) (s ˜s` ) (s1 o` ) → s = π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 σ ` ) (s ˜s` ) (s1 o` ) → s

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Note that a simpler derivation is obtained when the subject of the subordinate clause is elliptical (Maria is wondering who do you say to have met) Maria si chiede chi dici che hai incontrato π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 (π `2 ) σ ` ) (s ˜s` ) (s1 o` ) → s = π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 σ ` ) (s ˜s` ) (s1 o` ) → s

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Note that a simpler derivation is obtained when the subject of the subordinate clause is elliptical (Maria is wondering who do you say to have met) Maria si chiede chi dici che hai incontrato π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 (π `2 ) σ ` ) (s ˜s` ) (s1 o` ) → s = π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 σ ` ) (s ˜s` ) (s1 o` ) → s

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Note that a simpler derivation is obtained when the subject of the subordinate clause is elliptical (Maria is wondering who do you say to have met) Maria si chiede chi dici che hai incontrato π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 (π `2 ) σ ` ) (s ˜s` ) (s1 o` ) → s = π 3 (π r3 s1 σ ` ) (q oˆ`` q ` ) (q1 σ ` ) (s ˜s` ) (s1 o` ) → s

where the subject tu (you) of type π `2 is omitted.

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5.4.

Metarule III

To account for the sentential forms with subjects occuring in postverbal and clausal final position we invoke a new metarule:

Subject Final Metarule If V has type (i α` ) , then !V has type (i α` π ` ) where α is a string, possibly empty, including one or more occurrences of the types o, ω, λ; !V is defined as the emphatic form of V.

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This metarule will also account for subject final declarative forms, such as the following, where (p o` π `3 ) is the past participle of (i o` π `3 ):

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This metarule will also account for subject final declarative forms, such as the following, where (p o` π `3 ) is the past participle of (i o` π `3 ): (Maria thinks that Piero has met him) Maria pensa che lo ha incontrato Piero = Maria pensa che Cj k (lo avere) incontrato Piero π 3 (π r3 s1 σ ` ) (s ˜s` ) ((π r3 ) s1 ı ` ) ( j o`` ı ` ) (i p` ) (p o` π `3 ) π 3 → s1 = π 3 (π r3 s1 σ ` ) (s ˜s` ) (s1 ı ` ) ( j o`` ı ` ) (i p` ) (p o` π `3 ) π 3 → s1

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This metarule will also account for subject final declarative forms, such as the following, where (p o` π `3 ) is the past participle of (i o` π `3 ): (Maria thinks that Piero has met him) Maria pensa che lo ha incontrato Piero = Maria pensa che Cj k (lo avere) incontrato Piero π 3 (π r3 s1 σ ` ) (s ˜s` ) ((π r3 ) s1 ı ` ) ( j o`` ı ` ) (i p` ) (p o` π `3 ) π 3 → s1 = π 3 (π r3 s1 σ ` ) (s ˜s` ) (s1 ı ` ) ( j o`` ı ` ) (i p` ) (p o` π `3 ) π 3 → s1

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This metarule will also account for subject final declarative forms, such as the following, where (p o` π `3 ) is the past participle of (i o` π `3 ): (Maria thinks that Piero has met him) Maria pensa che lo ha incontrato Piero = Maria pensa che Cj k (lo avere) incontrato Piero π 3 (π r3 s1 σ ` ) (s ˜s` ) ((π r3 ) s1 ı ` ) ( j o`` ı ` ) (i p` ) (p o` π `3 ) π 3 → s1 = π 3 (π r3 s1 σ ` ) (s ˜s` ) (s1 ı ` ) ( j o`` ı ` ) (i p` ) (p o` π `3 ) π 3 → s1

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6.

Summary

We have applied to Italian the free pregroup generated by a partially ordered set of basic types: s, ˜s, s, σ (declarative sentences and sentential complements), q, q (questions and indirect questions), i, ˜ı, ı, i∗ , j, ı (infinitives or infinitival complements), π, o, ω, λ (verb arguments). Infinitive forms and finite verb forms have then been derived on the basis of which the distinction between direct vs. indirect statements and between sentences vs. questions has been characterized. In this respect a crucial role is played by three inflectors that allow one to derive the appropriate inflected verb form: the perfect inflector Perf(V) of type (p 2 i ` ) , the declarative inflector Cj k of type ((π rk ) sj ı ` ) , and the interrogative inflector C?j k (V) of types ((π rk ) qj ı ` ) and (qj (π `k ) ı ` ). The last two types are introduced by two metarules needed to account for direct questions in Italian. Furthermore a metarule governing subject final forms is required for the analysis of emphatic forms and subject final sentences. We have considered two interrogative forms of type q, where the second form with the inverted subject is taken by the wh-pronouns to generate wh-questions of type q.

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References [1] Abrusci, V. M. and P. Ruet, Non-commutative logic I: the multiplicative fragment, Annals of Pure and Applied Logic, 101: 29–64, 2000. [2] Bargelli, D. and J. Lambek, An algebraic approach to French sentence structure. In P. de Groote, G. Morrill and C. Retor´e (eds.), Logical Aspects of Computational Linguistics, 62-78, Springer-Verlag, Berlin, 2001. [3] Buszkowski, W., Lambek grammars based on pregroups. In P. de Groote, G. Morrill and C. Retor´e (eds.), Logical Aspects of Computational Linguistics, 95109, Springer-Verlag, Berlin, 2001. [4] Casadio, C. and J. Lambek, An algebraic analysis of clitic pronouns in Italian. In P. de Groote, G. Morrill and C. Retor´e (eds.), Logical Aspects of Computational Linguistics, 110–124, Springer-Verlag, Berlin, 2001. [5] Casadio, C. and J. Lambek, A tale of four grammars, Studia Logica, 71: 2, 2002. [6] Casadio, C. and J. Lambek, An algebraic approach to Latin grammar, Ottawa Conference, 2003. REFERENCES

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[7] Graffi, G., Sintassi. Le strutture del linguaggio, Il Mulino, Bologna, 1994. [8] Kislak, A., Pregroups versus English and Polish grammar. In V. M. Abrusci and C. Casadio (eds.), New Perspectives in Logic and Formal Linguistics, 129-154, Bulzoni Editore, Roma, 2002 . [10] Lambek, J., Type grammars revisited. In A. Lecomte, F. Lamarche and G. Perrier (eds.), Logical Aspects of Computational Linguistics, Springer LNAI 1582, 1-27, 1999. [11] Lambek, J., Type grammar meets German word order, Theoretical Linguistics, 26, 19-30, 2000. [12] Lambek, J., Type grammars as pregroups, Grammars 4, 21-39, 2001. [13] Lambek, J., A computational algebraic approach to English grammar, Syntax, 2004. [16] Sanfilippo, A., Grammatical Relations, Thematic Roles and Verb Semantics, Phd Thesis, University of Edinburgh, 1990. [17] Rizzi, L., Issues in Italian Syntax, Foris, Dordrecht, 1982. REFERENCES

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