An Algebraic Analysis of Clitic Pronouns in Italian

An Algebraic Analysis of Clitic Pronouns in Italian Claudia Casadio1 and Joachim Lambek2 1 Dipartimento di Filosofia, Università di Bologna Via Zamboni 38, 40126 Bologna, Italy [email protected] 2 Dept. of Mathematics and Statistics, McGill University 805 Sherbrooke St. West, Montreal QC Canada H3A 2K6 [email protected] Abstract. We1 analyze both pre- and post-verbal clitics in Italian in the context of infinitival phrases and declarative sentences, using a new form of categorial grammar in which one assigns to each morpheme a type, an element of the free “pregroup” generated by a partially ordered set of basic types. 1 Introduction Clitic pronouns in French have been analyzed in [1], with the help of a technique known as “compact bilinear logic”, essentially by performing calculations in a partially ordered monoid, not quite a group, elsewhere ([6], [7]) called a “pregroup”. Like other Romance languages, Italian has clitic pronouns. Unlike French, it has not only preverbal clitics but, like Spanish and Portuguese, also postverbal ones in declarative sentences2 . The main idea, which is shared with other categorial grammars, is this: we assign to each word of the language a compound type and hope that a calculation on the types associated with a string of words will recognize whether the latter is a well formed sentence. While one of the authors has written a book [2] on cliticization in Italian, the other does not speak Italian and is completely dependent on such computations. Here is how the compound types are constructed: we begin with a partially ordered set (A, →) of basic types, the partial order being denoted by the arrow. We form the free pregroup generated by A, whose elements are products of simple types. Each simple type has the form: ... , aℓℓ , aℓ , a, ar , arr , ... 1 2 The first author acknowledges support from Italian CNR and the second author from the Social Sciences and Humanities Research Council of Canada. French differs in this respect from the other Romance languages. where a ∈A is a basic type. The only computations required are contractions x ℓx → 1 , x x r → 1 , and expansions 1 → x x ℓ , 1 → x rx , where x is a simple type. Fortunately, for the purpose of sentence verification the expansions are not needed, but only the contractions, combined with some rewriting induced by the partial order of A, as we proved in [6]. Still, for theoretical purposes, the expansions are useful, for example to prove x r ℓ = x = x ℓr , x → y ⇒ (y ℓ → x ℓ ∧ y r → x r ). One can also extend the operation (. . . )ℓ and (. . . )r to compound types, by defining 1ℓ = 1 = 1r , (x · y)ℓ = y ℓ · x ℓ , (x · y)r = y r · x r . The symbol 1 here stands for the empty string of types in the free pregroup and will usually be omitted, as will be the dot that stands for multiplication. 2 Verbs and their types We start introducing the following basic types: i, i∗ o ω λ for for for for complete infinitive or infinitival verb phrase , direct object , indirect object , locative phrase . The following basic and compound types will be assigned to a few representative verbs 3 : vedere : obbedire : dare : mettere : arrivare : i , i oℓ ; i , i ωℓ ; i ω ℓ oℓ , i oℓ ω ℓ ; i λℓ oℓ , i oℓ λℓ ; i ∗ , i ∗ λℓ . The star on i ∗ will act as a reminder that the perfect tense is to be formed with the auxiliary essere rather than avere (see Sect. 6). 3 See the Appendix for English translations of the Italian expressions and sentences considered in this paper. These type assignments will justify the derivation of the following infinitival phrases of type i or i ∗ : vedere , vedere (un libro) ; i (i oℓ ) o → i obbedire , obbedire (a Mario) ; i (i ω ℓ ) ω → i dare (un libro) (a Mario) , (i ω ℓ oℓ ) o ω → i dare (a Mario) (un libro) ; (i oℓ ω ℓ ) ω o → i mettere (un libro) (sul tavolo) , (i λℓ oℓ ) o λ → i mettere (sul tavolo) (un libro) ; (i oℓ λℓ ) λ o → i arrivare , arrivare (a Roma) ; i∗ (i ∗ λℓ ) λ → i∗ We note that each of the above verbs has more than one type. For example, vedere can be used intransitively as an item of type i and transitively as an item of type i oℓ . The two types for dare are required to allow for the less common but admissible expression dare a Mario un libro. We have not analyzed the direct object phrase un libro, the indirect object phrase a Mario and the locative prepositional phrase sul tavolo. In principle, types can also be assigned to the separate words un, libro, a, Mario, sul and tavolo; but we desist from doing so in the present paper, since our main concern is the analysis of verb phrases and their subcategorization properties within the context of cliticization. When combined with a postverbal clitic, the final letter of the infinitive is dropped and the type is changed as follows: veder : obbedir : dar : metter : arrivar : ıoℓ ; ıωℓ ; ı ω ℓ o ℓ , ı oℓ ω ℓ ; ı λℓ o ℓ , ı oℓ λℓ ; ı∗ λℓ . The purpose of the bar will become clear later, but it is inspired by the standard conventions of X-theory; we postulate in the partially ordered set of basic types i → ı , o → o , ω → ω , λ → λ , i ∗ → ı∗ . We do not postulate x → x , and here we assume x 6→ x . 3 Postverbal clitics Italian has both preverbal and postverbal clitic pronouns. Here is a list of the clitic pronouns together with their types when they are used postverbally: mi, ti, ci, vi, si : o Accusative lo, la, li, le : ob mi, ti, ci, vi, si, gli, le : ω Dative me, te, ce, ve, se, glie : ω o ob ℓ ci, vi : λ Locative ce, ve : λ o ob ℓ We shall ignore the partitive clitic ne in this article (see [2]). We have introduced a new basic type ob for the clitics lo, etc., which require special treatment, and we postulate ob → o, but o 6→ ob. We illustrate these type assignments with a representative set of infinitival phrases of type ı or ı∗ : veder . lo , (ı o ℓ ) ob → ı obbedir . mi , (ı ω ℓ ) ω → ı dar . mi (ı oℓ ω ℓ ) ω (un libro) , o → ı dar . lo (a Mario) , ω → ı (ı ω ℓ o ℓ ) ob dar . me . lo , (ı oℓ ω ℓ ) (ω o ob ℓ ) ob → ı arrivar . ci , (ı∗ λℓ ) λ → ı∗ Note that o ℓ ob → o ℓ o → 1 . The dots were introduced to separate the morphemes, but the usual spelling is darmelo, etc. . We observe that two consecutive clitics only contract when the first is a dative such as me or a locative such as ce and the second is an accusative such as lo : me . lo (ω o ob ℓ ) ob , ce . lo (λ o ob ℓ ) ob . However, no contraction takes place with mi . lo , lo . mi , lo . me , ω ob ob ω ob (ω o ob ℓ ) mi . ti , mi . lo , mi . ti ω o o ob o ω , me . ti , (ω o ob ℓ ) o / the last because o 6→ ob, and similarly with λ in place of ω. The following ungrammatical examples show why such pairings are useless: *dar . mi . lo (ı oℓ ω ℓ ) ω ob , *dar . lo . mi , (ı ω ℓ o ℓ ) ob ω / (b o 6→ o , ω 6→ ω) / The bar on the initial i or i∗ of the short infinitives will ensure that these cannot be preceded by clitics (see Sect. 4). 4 Preverbal clitics We now list the types of the same clitics when used preverbally: Accusative Dative mi, ti, ci, vi : j oℓℓ i ℓ si : j ∗ oℓℓ i ℓ lo, la, li, le : j oℓℓ i ℓ mi, ti, ci, vi, gli, le : j ω ℓℓ i ℓ , j ∗ ω ℓℓ i ∗ℓ si : j ∗ ω ℓℓ i ℓ me, te, ce, ve, se, glie : j ω ℓℓ j ℓ se : j ∗ ω ℓℓ j ℓ ci, vi : j λℓℓ i ℓ , j ∗ λℓℓ i ∗ℓ Locative ce, ve : j λℓℓ j ℓ We have introduced four new basic types j , j ∗ , j and j ∗ for infinitives and we postulate j → j , j ∗ → j ∗ , but i 6→ j 6→ i etc. . It follows that infinitives of type j cannot be preceded by any clitics and infinitives of type j only by clitics such as me and ce. We have double clitics such as me . lo , (j ω ℓℓ j ℓ ) ( j oℓℓ i ℓ ) → j ω ℓℓ oℓℓ i ℓ me . lo dare , (j ω ℓℓ oℓℓ i ℓ ) (i oℓ ω ℓ ) → j ce . lo mettere , (j λℓℓ oℓℓ i ℓ ) (i oℓ λℓ ) → j Other pairs of clitics will not contract. Here are some illustrations of preverbal clitics: lo vedere , (j oℓℓ i ℓ ) (i oℓ ) → j ci arrivare , (j ∗ λℓℓ i ∗ℓ ) (i ∗ λℓ ) → j ∗ mi dare (un libro) (j ω ℓℓ i ℓ ) (i ω ℓ oℓ ) o → j , lo dare (a Mario) , (j oℓℓ i ℓ ) (i oℓ ω ℓ ) ω → j but *mi dar . lo (j ω ℓℓ i ℓ ) (ı ω ℓ o ℓ ) ob / , *mi dare lo (j ω ℓℓ i ℓ ) (i ω ℓ oℓ ) ob / . These results follow from the assumptions ı 6→ i , ob → o , ob 6→ o . Therefore our type assignments never allow preverbal and postverbal clitics to occur together, expressing a fundamental property of Italian pronominal cliticization. 5 The perfect tense with avere The perfect infinitive, corresponding to Latin amavisse, is formed in Italian from the past participle with the help of the auxiliary verbs avere or essere. The latter applies only to a set of intransitive verbs and in this section we shall concentrate on the former. We adopt the type assignments avere : i p ℓ2 , aver : ı p ℓ2 where p 2 is the type of the past participle of a verb with infinitive of type i, e.g. the type of visto when vedere is used intransitively. The role of the bar on p 2 will become clear later. We shall construct the past participle of a verb with the help of a so-called “inflector” (see [6]). Here we use the inflector Perf of type p 2 i ℓ so that Perf (vedere) = visto (p 2 i ℓ ) i p2 and Perf (vedere) = visto (p 2 i ℓ ) (i oℓ ) p 2 oℓ In the following examples, we require a preliminary analysis of the past participles before the types are assigned: avere visto un libro , = avere Perf (vedere) (un libro) (i p ℓ2 ) (p 2 i ℓ ) (i oℓ ) avere = avere dato Perf (dare) (i p ℓ2 ) (p 2 i ℓ ) (i ω ℓ oℓ ) o → i un libro a Mario , (un libro) (a Mario) o ω → i It must be pointed out that visto and dato may change the final vowel o to a, i or e, depending on the gender and number of any preceding accusative clitic pronoun. This is something that we shall ignore in our preliminary type theoretic analysis. In general, the inflector Perf is used in connection with “extended infinitives”(see [1] for this notion for French). By an extended infinitive we shall mean the infinitive of a verb together with all the preverbal clitics that happen to precede it. In the presence of preverbal clitics the type of Perf should be p2 j ℓ , as in the following examples: aver.lo visto , = aver Perf (lo (ı pℓ2 ) ℓ (p2 j ) ( j o vedere) ℓℓ i ℓ ) (i oℓ ) → ı where Perf (lo vedere) gives lo visto which combined with aver becomes aver.lo visto 4 . The verb avere can also be a transitive verb like vedere, so we must allow avere avuto un libro , = avere Perf (avere) (un libro) (i p ℓ2 ) (p 2 i ℓ ) (i oℓ ) o → i where avuto has type p 2 oℓ . Unfortunately, our analysis also admits the ungrammatical string * avere avuto visto un libro = avere Perf (avere) Perf (vedere) (un libro) (i p ℓ2 ) (p 2 i ℓ ) (i p ℓ2 ) (p 2 i ℓ ) (i oℓ ) o → i To avoid this, we would have to modify our type assignments in such a way as to prevent avuto = Perf (avere) from receiving the type p 2 p ℓ2 , or we could stipulate that the auxiliary verb avere does not possess a past participle. 6 The auxiliary verb essere 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 ∗ λℓ . We account for this by judicious applications of the star, assigning types as follows: essere : i ∗ p ∗2 ℓ , esser : ı ∗ p ∗2 ℓ , p ∗2 → p ∗2 being new basic types, and by giving two new types to the inflector Perf, in summary: Perf : p 2 i ℓ , p2 j ℓ , p ∗2 i ∗ℓ , p ∗2 j ∗ℓ . 4 The alternative infinitives aver.la vista, aver.li visti, aver.le viste are not explained by the uniform type j oℓℓ i ℓ for the preverbal clitics lo, la, li and le. A possible way out would be to replace the basic type o by on , where n = 1, 2, 3 and 4 in the four cases respectively. If this strategy were adopted, mi and ti would require n = 1 or 2, ci and vi would require n = 3 or 4, but si would go with any n. The following examples will illustrate how the perfect infinitive of the starred verbs is formed: essere = essere arrivato , Perf (arrivare) (i ∗ p ∗2 ℓ ) (p ∗2 i ∗ℓ ) i ∗ esser . ci = esser → i∗ arrivato , Perf ( ci arrivare ) (ı ∗ p∗2 ℓ ) (p∗2 j ∗ℓ )(j ∗ λℓℓ i ∗ℓ )(i ∗ λℓ ) → i∗ ci ci = essere essere arrivato , Perf (arrivare) (j ∗ λℓℓ i ∗ℓ )(i ∗ p ∗2 ℓ ) (p ∗2 i ∗ℓ )(i ∗ λℓ ) → j ∗ Unfortunately, our analysis also allows the ungrammatical string * essere stato = essere Perf (essere) arrivato , Perf (arrivare) (i ∗ p ∗2 ℓ ) (p ∗2 i ∗ℓ )(i ∗ p ∗2 ℓ ) (p ∗2 i ∗ℓ ) i ∗ → i ∗ To avoid this, as remarked at the end of Section 5, we would have to modify our type assignments in such a way as to prevent stato = Perf (essere) from receiving the type p 2 p ℓ2 , or to stipulate that the auxiliary verb essere does not possess a past participle. The auxiliary verb essere can also be used to form the passive infinitive, corresponding to Latin amari, again from the past participle; but it has a different type: essere : i ∗ oℓℓ p ℓ2 , esser : ı ∗ oℓℓ pℓ2 as illustrated by the following examples: essere = essere visto , Perf(vedere) (i ∗ oℓℓ p ℓ2 ) (p 2 i ℓ )(i oℓ ) → i ∗ essere = essere stato Perf(essere) visto , Perf(vedere) (i ∗ p ∗2 ℓ ) (p ∗2 i ∗ℓ )(i ∗ oℓℓ p ℓ2 ) (p 2 i ℓ )(i oℓ ) → i ∗ esser . mi dato , = esser Perf ( mi dare ) mi essere dato , mi essere Perf ( dare ) = (ı ∗ oℓℓ pℓ2 )(p2 j ℓ )(jω ℓℓ i ℓ )(i ω ℓ oℓ ) → ı ∗ (j ∗ ω ℓℓ i ∗ℓ )(i ∗ oℓℓ p ℓ2 )(p 2 i ℓ )(i oℓ ω ℓ ) → j ∗ 7 The modal verbs We now consider the modal verbs potere, volere, dovere with types as follows: potere : i ı ℓ , i ∗ ı ∗ℓ ; poter : ı j ℓ , ı ∗ j ∗ℓ and here are a few illustrative examples: potere obbedire (a Mario) , ℓ (i ı ) mi ℓ (i ω ) potere poter . mi obbedire , (ı j ℓ ) (jω ℓℓ i ℓ ) (i ω ℓ ) → ı ω → i obbedire , potere dar . me . lo , (j ω ℓℓ i ℓ ) (i ı ℓ ) (i ω ℓ ) → j (i ı ℓ ) (ı oℓ ω ℓ ) (ω o ob ℓ ) ob → i poter . me . lo me . lo ℓ ℓℓ ℓℓ (ı j ) (j ω o dare , ℓ ℓ ℓ ℓℓ ℓℓ i ) (i o ω ) → ı (j ω o potere arrivar . ci , (i ı ∗ ∗ℓ ci ℓ ) (ı λ ) (λ) → i ∗ potere potere ℓ dare , i ) (i ı ) (i oℓ ω ℓ ) → j poter . ci ℓ arrivare , (ı j ) (j λ i ) (i ∗ λℓ ) → ı ∗ ∗ ∗ ∗ℓ ∗ ℓℓ ∗ℓ arrivare , ∗ ℓℓ ∗ℓ (j λ i ) (i ∗ ı ∗ℓ ) (i ∗ λℓ ) → j ∗ . Modal verbs allow repetition, not only potere volere , ℓ ℓ (i ı )(i ı ) → i ı but even potere potere , ℓ ℓ ℓ (i ı )(i ı ) → i ı and potere poter, ℓ (i ı ℓ )(ı j ℓ ) → i j ℓ . In the last three examples we can replace i by i ∗ and j by j ∗ . How do the modal verbs interact with the auxiliary verbs avere and essere? The following examples will answer the question. potere avere visto un libro , = potere avere Perf (vedere) (un libro) (i ı ℓ ) (i p ℓ2 ) (p 2 i ℓ ) (i oℓ ) o → i avere potuto vedere un libro , = avere Perf (potere) vedere (un libro) (i p ℓ2 ) (p 2 i ℓ ) (i ı ℓ ) (i oℓ ) = o → i potere potere esser . ci arrivato esser Perf ( ci arrivare ) (i ∗ ı ∗ℓ ) (ı ∗ p∗2 ℓ ) (p ∗2 j ∗ℓ )(j ∗ λℓℓ i ∗ℓ )(i ∗ λℓ ) → i ∗ essere = essere potuto arrivar . ci Perf (potere) arrivar . ci (i ∗ p ∗2 ℓ ) (p ∗2 i ∗ℓ ) (i ∗ ı ∗ℓ ) (ı∗ λℓ ) λ → i ∗ esser = esser . ci potuto arrivare Perf ( ci potere ) arrivare (ı ∗ p∗2 ℓ )(p ∗2 j ∗ℓ )(j ∗ λℓℓ i ∗ℓ ) (i ∗ ı ∗ℓ )(i ∗ λℓ ) → i ∗ Up to now we have introduced eight different basic types i → ı , j → j , i ∗ → ı∗ , j ∗ → j ∗ all representing complete infinitival phrases, that is, infinitival phrases that do not require a direct or indirect object or a locative phrase for their completion. It would be convenient to subsume all of them under a single basic type ı , by postulating ı , j , ı∗ , j ∗ → ı 8 Finite verb-forms and declarative sentences So far we have looked only at infinitives and infinitival verb phrases, but sentences are constituted from finite verb-forms. To each Italian verb V there is associated a matrix Vj k of 7 × 6 = 42 finite verb-forms, 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 , π k for k-th person subject . For purpose of illustration, we shall confine our attention to the cases j = 1 (present tense) and k = 1 (first person) or k = 3 (third person). For example, the present tense sentence (io) vedo (un libro) , π 1 (π r1 s1 oℓ ) o → s1 where the optional pronoun io has type π 1 , leads us to assign to the first verbform vedo the type (π r1 s1 oℓ ) . We shall assume that Italian speakers are familiar with the whole conjugation matrix Vj k and we shall not discuss rules for constructing this matrix, although there certainly are such rules. For our purposes Vj k is obtained from the infinitive V by an inflector Cj k such that Cj k (V) = Vj k . We intend to apply Cj k not only to plain infinitives such as vedere, but also to extended infinitives, following the procedure adopted for French in [1]. We assign the following types to this inflector: Cj k : π rk sj ı ℓ , sj ı ℓ , the former if the optional subject is present, as we shall assume from now on. We conclude by looking at some examples of declarative sentences in the present tense involving pre- or post-verbal cliticization. (io) te . lo do , = io C11 ( te . lo dare) π 1 (π r1 s1 ı ℓ ) (j ω ℓℓ oℓℓ i ℓ ) (i oℓ ω ℓ ) → s1 (j→ı) Mario lo vuole vedere , = Mario C13 ( lo volere ) vedere π 3 (π r3 s1 ı ℓ ) (j oℓℓ i ℓ ) (i ı ℓ )(i oℓ ) → s1 (j →j→ı) Mario vuole vedere lui , = Mario C13 ( volere ) vedere lui π 3 (π r3 s1 ı ℓ ) (i ı ℓ ) (i oℓ ) o → s1 (io) devo dar . te . lo , = io C11 ( dovere ) dar . te . lo π 1 (π r1 s1 ı ℓ ) (i ı ℓ )(ı oℓ ω ℓ ) (ω o ob ℓ ) ob → s1 (io) = io te . lo devo dare , C11 ( te . lo dovere ) dare π 1 (π r1 s1 ı ℓ ) (j ω ℓℓ oℓℓ i ℓ )(i ı ℓ ) (i oℓ ω ℓ ) → s1 * (io) devo te . lo dare , = io C11 ( dovere ) te . lo π1 (io) = io π1 (π r1 ℓ ℓ ℓℓ ℓℓ s1 ı ) (i ı )(j ω o / lo ho visto , C11 ( lo avere ) (π r1 ℓ ℓℓ s1 ı ) (j o i ℓ (j→ı) dare ℓ i ) (i oℓ ω ℓ ) ( j 6→ ı ) Perf (vedere) )(i p ℓ2 )(p 2 i ℓ )(i oℓ ) → s 1 (j→j→ı) * (io) ho lo visto , = io C11 ( avere ) Perf (lo vedere) π 1 (π r1 s1 ı ℓ ) (i p ℓ2 )(p 2 i ℓ )(j oℓℓ i ℓ )(i oℓ ) / (j 6→ i ) (Had we picked Perf : p2 j ℓ , we would have been led to p ℓ2 p2 6→ 1 . ) (io) = io ci metto un libro , C11 ( ci mettere ) (un libro) π 1 (π r1 s1 ı ℓ ) (j λℓℓ i ℓ )(i λℓ oℓ ) o → s1 (io) = io lo metto sul tavolo , C11 ( lo mettere ) (sul tavolo) π 1 (π r1 s1 ı ℓ ) (j oℓℓ i ℓ )(i oℓ λℓ ) λ → s1 (io) = io (j→ı) (j→j→ı) ce lo metto , C11 ( ce lo mettere ) π 1 (π r1 s1 ı ℓ ) (j λℓℓ oℓℓ i ℓ )(i oℓ λℓ ) → s1 (j→ı) 9 Appendix vedere obbedire dare mettere arrivare : : : : : to see to obey to give to put to arrive vedere un libro obbedire a Mario dare un libro a Mario mettere un libro sul tavolo arrivare a Roma 10 avere : to have essere : to be potere : may, can volere : shall, will dovere : must : : : : : to to to to to see a book obey Mario give a book to Mario put a book on the table arrive at Rome Concluding remarks As far as we know, this is the first attempt to give an accurate account of Italian clitics by means of an algebraic formalism. The advantages of the approach via pregroups as compared with earlier forms of categorial grammars have been discussed elsewhere, e.g. in [4]. Still, the following remarks should help the present reader 5 . In analyzing a sentence, we go from left to right and mimic the way a human hearer might proceed: recognizing the type of each word as it is received and rapidly calculating the type of the string of words up to that point, never overburdening the temporary storage capacity of the human mind beyond Miller’s [8] seven chunks of information. (This simplified account neglects the necessity of parallel computation .) Earlier kinds of categorial grammar, in spite of their mathematical elegance, require much more complicated calculations, such as Gentzen-style proofs in tree-form. While these can be carried out on paper or on the blackboard, it is hard to imagine that they lend themselves to rapid mental computations. Proof nets of linear logic provide a nice visual representation of the kind of deductions or derivations that are required; but again it is not likely (at least in the opinion of one of us) that they correspond to anything in the human mind. Nonetheless, we have not ignored them: in the compact bilinear logic we employ here, the proof nets are just the links indicating contractions, expansions having been shown to be unnecessary for sentence recognition in [6]. We are indebted to A. K. Joshi for pointing out that such links were first introduced by Z. Harris [5] in a different context. 5 We are grateful to Mario Fadda for his careful reading of the manuscript and for spotting a couple of mistakes. He also pointed out that the symbol j ∗ never actually occurs as a type in the present fragment. Some readers may wonder why we need so many basic types, where the earliest categorial grammars used only two (s for sentences and n for names). We introduced the smallest number of basic types which allowed us to account for the linguistic data, recognizing all and only grammatically well-formed sentences. Moreover, many of such basic types are related, e.g. the types for infinitives ı , j , ı∗ , j ∗ → ı . We hope that our grammar, with the finely tuned order on the set of basic types, will not accept non-grammatical strings as sentences; if it does, we must revise it by introducing yet more basic types. In selecting basic types, we are primarily interested in their algebraic effectiveness and not so much in whether they correspond to traditional grammatical categories. References [1] Bargelli, D., Lambek, J.: An algebraic approach to French sentence structure (to appear). [2] Casadio, C.: A Categorial Approach to Cliticization and Agreement in Italian, Saggi di Filosofia del Linguaggio e della Scienza, CLUEB Editrice, Bologna, (1993). [3] Casadio, C.: Non-commutative linear logic in linguistics, submitted to Grammar. [4] Casadio, C., Lambek, J.: A tale of four grammars, to appear in Studia Logica. [5] Harris, Z.: A cyclic cancellation-automaton for sentence well-formedness, International Computation Centre Bulletin 5 (1966), 69-94. [6] Lambek, J.: Type grammars revisited, in A. Lecomte, F. Lamarche and G. Perrier (eds.), Logical Aspects of Computational Linguistics, Springer LNAI 1582, (1999) 1-27. [7] Lambek, J.: Pregroups: a new algebraic approach to sentence structure, in C. Martin-Vide and G. Paun (eds), Recent Topics in Mathematical and Computational Linguistics, Editura Academici Romb ane, Bucharest (2000). [8] Miller, G.A.: The magical number seven plus or minus two: some limits on our capacity for processing information, Psychological Review 63 (1956), 81-97. [9] Ragusa, O.: Essential Italian Grammar, Dover Publications, New York 1963.