An Algebraic Approach to Latin Grammar

An Algebraic Approach to Latin Grammar

C. Casadio and J. Lambek

Contents First Last Prev Next J Contents

1 Introduction ...... 3 2 Conjugation ...... 4 3 Declension ...... 7 3.1 Adjectives ...... 8 3.2 Nouns ...... 9 3.3 Relative pronouns ...... 10 3.4 Personal pronouns ...... 11 4 The algebraic machinary ...... 12 5 The verb phrase ...... 14 5.1 Verb forms ...... 15 5.2 Verb types ...... 16 6 Variations in type ...... 21 7 The noun phrase ...... 24 8 The auxiliary verb sum ...... 31 9 The perfect passive ...... 35 10 Relative clauses ...... 38 11 Concluding remarks ...... 42 12 Appendix: templates and rules ...... 45

Contents First Last Prev Next J 1. Introduction We present a type theoretic analysis of Latin grammar, which pays as much attention to inflectional morphology as to syntax. We assign different types to the finite forms of Latin verbs as well as to their infinitives.

Contents First Last Prev Next J 1. Introduction We present a type theoretic analysis of Latin grammar, which pays as much attention to inflectional morphology as to syntax. We assign different types to the finite forms of Latin verbs as well as to their infinitives. The rich repertory of agreement information exhibited by Latin is accounted for by a system of numerical indexes (superscripts and subscripts) attached to the types.

Contents First Last Prev Next J 1. Introduction We present a type theoretic analysis of Latin grammar, which pays as much attention to inflectional morphology as to syntax. We assign different types to the finite forms of Latin verbs as well as to their infinitives. The rich repertory of agreement information exhibited by Latin is accounted for by a system of numerical indexes (superscripts and subscripts) attached to the types. Agreement coherence and control of sentencehood for strings of words are obtained by means of calculations performed on the corresponding strings of types, in accor- dance with the grammar of pregroups (a refinement of classical bilinear logic.)

Contents First Last Prev Next J 1. Introduction We present a type theoretic analysis of Latin grammar, which pays as much attention to inflectional morphology as to syntax. We assign different types to the finite forms of Latin verbs as well as to their infinitives. The rich repertory of agreement information exhibited by Latin is accounted for by a system of numerical indexes (superscripts and subscripts) attached to the types. Agreement coherence and control of sentencehood for strings of words are obtained by means of calculations performed on the corresponding strings of types, in accor- dance with the grammar of pregroups (a refinement of classical bilinear logic.)

Casadio, C. and J. Lambek, A tale of four grammars, Studia Logica, vol. 71, 2 (2002). • Lambek, J., A mathematician looks at Latin conjugation, Theoretical Linguistics, 6 • (1979), 221-234. Lambek, J., Type grammar meets German word order, Theoretical Linguistics, 26 • (2000), 19-30.

Contents First Last Prev Next J 2. Conjugation

The Latin verb V has 3 5 6 finite verb forms Ca (V), where × × tp

Contents First Last Prev Next J 2. Conjugation

The Latin verb V has 3 5 6 finite verb forms Ca (V), where × × tp a = 1, 2, 3 represents three ASPECTS :

Contents First Last Prev Next J 2. Conjugation

The Latin verb V has 3 5 6 finite verb forms Ca (V), where × × tp a = 1, 2, 3 represents three ASPECTS : basic ( = active non perfect), perfect and passive;

Contents First Last Prev Next J 2. Conjugation

The Latin verb V has 3 5 6 finite verb forms Ca (V), where × × tp a = 1, 2, 3 represents three ASPECTS : basic ( = active non perfect), perfect and passive;

t = 1, ..., 5 represents five TENSE-MOODS : present indicative, present subjunctive, imperfect indicative, imperfect subjunctive, and future;

Contents First Last Prev Next J 2. Conjugation

The Latin verb V has 3 5 6 finite verb forms Ca (V), where × × tp a = 1, 2, 3 represents three ASPECTS : basic ( = active non perfect), perfect and passive;

t = 1, ..., 5 represents five TENSE-MOODS : present indicative, present subjunctive, imperfect indicative, imperfect subjunctive, and future;

p = 1, ..., 6 represents six PERSONS : 1st, 2nd, 3rd person singular and 1st, 2nd, 3rd person plural.

Contents First Last Prev Next J It is custumary to refer to the verb V by

a C11 (V)

Contents First Last Prev Next J It is custumary to refer to the verb V by

a C11 (V) where a represents the basic aspect a = 1 or another aspect if the basic aspect does not exist; t = 1 is the present indicative tense-mood; p = 1 indicates the first person.

Contents First Last Prev Next J It is custumary to refer to the verb V by

a C11 (V) where a represents the basic aspect a = 1 or another aspect if the basic aspect does not exist; t = 1 is the present indicative tense-mood; p = 1 indicates the first person.

For V = amo, we have all three aspects, represented by the following three 5 x 6 matrices:

Contents First Last Prev Next J am.o am.em am.abam¯ am.arem¯ am.abo¯ 1 . . . . . C (amo) = " . . . . . # am.ant am.ent am.abant¯ am.arent¯ am.abunt¯

amav.i amav.erimˇ amav.eramˇ amav.¯issem amav.eroˇ 2 . . . . . C (amo) = " . . . . . # amav.erunt¯ amav.erintˇ amav.erantˇ amav.¯issent amav.erintˇ

am.or am.er am.abar¯ am.arer¯ am.abor¯ 3 . . . . . C (amo) = " . . . . . # am.antur¯ am.entur¯ am.abantur¯ am.arentur¯ am.abuntur¯

Contents First Last Prev Next J 3. Declension Latin nouns, adjectives and pronouns are subject to similar inflectional modification, called “declension”. Thus, for each adjective A, one has the 3 2 5 forms Dg (A), where × × nc

Contents First Last Prev Next J 3. Declension Latin nouns, adjectives and pronouns are subject to similar inflectional modification, called “declension”. Thus, for each adjective A, one has the 3 2 5 forms Dg (A), where × × nc g = 1, 2, 3 stands for the gender : masculine, feminine and neuter;

Contents First Last Prev Next J 3. Declension Latin nouns, adjectives and pronouns are subject to similar inflectional modification, called “declension”. Thus, for each adjective A, one has the 3 2 5 forms Dg (A), where × × nc g = 1, 2, 3 stands for the gender : masculine, feminine and neuter; n = 1, 2 for the number : singular and plural;

Contents First Last Prev Next J 3. Declension Latin nouns, adjectives and pronouns are subject to similar inflectional modification, called “declension”. Thus, for each adjective A, one has the 3 2 5 forms Dg (A), where × × nc g = 1, 2, 3 stands for the gender : masculine, feminine and neuter; n = 1, 2 for the number : singular and plural; c = 1, ..., 5 for the case, where nominative : 1, genitive : 2, dative : 3, accusative : 4, and ablative : 5.

We will ignore a sixth case, the vocative, in this paper.

Contents First Last Prev Next J 3.1. Adjectives

bon.us bon.i bon.o bon.um bon.o D1(bonus) =  bon.i bon.orum bon.is bon.os bon.is 

bon.a bon.ae bon.ae bon.am bon.a D2(bonus) =  bon.ae bon.arum bon.is bon.as bon.is 

bon.um bon.i bon.o bon.um bon.o D3(bonus) =  bon.a bon.orum bon.is bon.a bon.is 

Contents First Last Prev Next J 3.2. Nouns Nouns have only one matrix each, the gender being fixed.

Contents First Last Prev Next J 3.2. Nouns Nouns have only one matrix each, the gender being fixed. g We write Dnc (N), for a noun N of gender g, e.g.

Contents First Last Prev Next J 3.2. Nouns Nouns have only one matrix each, the gender being fixed. g We write Dnc (N), for a noun N of gender g, e.g.

D1(puer) = puer puer.i puer.o puer.um puer.o h puer.i puer.orum puer.is puer.os puer.is i

D2(puella) = puell.a puell.ae puell.ae puell.am puell.a h puell.ae puell.arum puell.is puell.as puell.is i

Contents First Last Prev Next J 3.3. Relative pronouns The relative pronoun qui has three matrices, like an adjective, e.g.

Contents First Last Prev Next J 3.3. Relative pronouns The relative pronoun qui has three matrices, like an adjective, e.g.

1 D11 (qui) = qui , masculine, singular, nominative

2 D11 (qui) = quae , feminine, singular, nominative

3 D11 (qui) = quod . neuter, singular, nominative

Contents First Last Prev Next J 3.4. Personal pronouns The personal pronoun P has forms depending not only on gender, number and case, but also on person p 0 = 1, 2, 3.

Contents First Last Prev Next J 3.4. Personal pronouns The personal pronoun P has forms depending not only on gender, number and case, but also on person p 0 = 1, 2, 3. The gender is only marked in the third person singular or plural.

Contents First Last Prev Next J 3.4. Personal pronouns The personal pronoun P has forms depending not only on gender, number and case, but also on person p 0 = 1, 2, 3. The gender is only marked in the third person singular or plural.

As for the verb, we may combine p 0 and n into a single index g p = p 0 + 3 (n-1). Thus we have the forms Ppc. For example,

ego me P2 = " tu te # ea eam

Contents First Last Prev Next J 4. The algebraic machinary The kind of algebra we have in mind is a pregroup, a partially ordered monoid in which each element a has a left adjoint a` such that

a`a 1 a a` → → and a right adjoint ar such that

a ar 1 ara → →

Contents First Last Prev Next J A pregroup is freely generated by a partially ordered set of basic types. From each basic type a one can form simple types :

..., a`, a, ar,... .

Contents First Last Prev Next J A pregroup is freely generated by a partially ordered set of basic types. From each basic type a one can form simple types :

..., a`, a, ar,... .

We assign to each Latin word-form a type :

α1, ..., αn

consisting in a finite string of basic types αi.

Contents First Last Prev Next J 5. The verb phrase

Basic types for analyzing the verb phrase :

Contents First Last Prev Next J 5. The verb phrase

Basic types for analyzing the verb phrase :

st for sentences in the t-th tense-mood s for sentences when the tense does not matter πp for the subject in the p-th person oc for the object in the c-th case

We postulate st s . →

Contents First Last Prev Next J 5.1. Verb forms

A finite Latin verb form requires a subject of type πp, although this is frequently omitted, and a number of (optional) complements in different cases.

Contents First Last Prev Next J 5.1. Verb forms

A finite Latin verb form requires a subject of type πp, although this is frequently omitted, and a number of (optional) complements in different cases.

morior no complement, ⇒

Contents First Last Prev Next J 5.1. Verb forms

A finite Latin verb form requires a subject of type πp, although this is frequently omitted, and a number of (optional) complements in different cases.

morior no complement, ⇒ amo one accusative complement, ⇒

Contents First Last Prev Next J 5.1. Verb forms

A finite Latin verb form requires a subject of type πp, although this is frequently omitted, and a number of (optional) complements in different cases.

morior no complement, ⇒ amo one accusative complement, ⇒ do an accusative and a dative complement, ⇒

Contents First Last Prev Next J 5.1. Verb forms

A finite Latin verb form requires a subject of type πp, although this is frequently omitted, and a number of (optional) complements in different cases.

morior no complement, ⇒ amo one accusative complement, ⇒ do an accusative and a dative complement, ⇒ appello two accusative complements, ⇒

Contents First Last Prev Next J 5.1. Verb forms

A finite Latin verb form requires a subject of type πp, although this is frequently omitted, and a number of (optional) complements in different cases.

morior no complement, ⇒ amo one accusative complement, ⇒ do an accusative and a dative complement, ⇒ appello two accusative complements, ⇒ amo (in the passive aspect) a nominative and an ablative complement. ⇒

Contents First Last Prev Next J 5.2. Verb types

3 r Ctp (morior): πp st a r r Ctp (amo) : o4 πp st (a = 1, 2) a r r r Ctp (do) : o4 o3 πp st ” a r r r Ctp (appello) : o4 o4 πp st ” 3 r r Ctp (amo) : o5 πp st

Contents First Last Prev Next J The following metarule is required to relate the basic and perfect aspect of transitive verbs (a = 1, 2) with their passive aspect (a = 3)

Contents First Last Prev Next J The following metarule is required to relate the basic and perfect aspect of transitive verbs (a = 1, 2) with their passive aspect (a = 3)

Metarule 1. 1 r r 2 If Ctp(V) has type ... o4 πp st, then Ctp(V) has the 3 r r same type, but Ctp(V) has type ... o5 πp st, r provided o5 occurs only once.

3 r r r The proviso is required for appello, since Ctp(appello) : o5 o1 πp st .

Contents First Last Prev Next J (1) (the horse dies) equˇus moriatur r π3 (π s1) s1 3 →

Contents First Last Prev Next J (1) (the horse dies) equˇus moriatur r π3 (π s1) s1 3 →

(2) (the boy had loved the girl) puer puellam amaverat r r π3 o4 (o π s3) s3 4 3 →

Contents First Last Prev Next J (3) (the girl will give the boy an apple) puella puero ma¯lum dabit r r r π3 o3 o4 (o o π s5) s5 4 3 3 →

Contents First Last Prev Next J (3) (the girl will give the boy an apple) puella puero ma¯lum dabit r r r π3 o3 o4 (o o π s5) s5 4 3 3 →

(4) (the boys have named the girl queen) pueri puellam reginam appellaverunt r r r π6 o4 o4 (o o π s1) s1 4 4 6 →

Contents First Last Prev Next J (5) (the girl was loved by the boy) puella puero amabatur r r π3 o5 (o π s3) s3 5 3 →

Contents First Last Prev Next J (5) (the girl was loved by the boy) puella puero amabatur r r π3 o5 (o π s3) s3 5 3 →

(6) (the girl is named queen by the boys) puella regina pueris appellatur r r r π3 o1 o5 (o o π s1) s1 5 1 3 →

Contents First Last Prev Next J 6. Variations in type While the normal word order might be puer puella amat , all six permutations of these words are possible, thus leading to six different types for amat. Furthermore, the object puellam can be omitted, and so can the subject puer. In fact, it is customary to omit the subject with first and second person singular or plural. Thus, we also have sentences like puer amat , puellam amat , amat , and the permutations of the first two. Altogether then, we must assign 11 different types to amat .

Contents First Last Prev Next J We account for these variations in type by a metarule :

Contents First Last Prev Next J We account for these variations in type by a metarule :

Metarule 2. a If the verb form Ctp(V) has been assigned r r type x1 ... xn st , then r (1) any xi can be omitted , r r (2) xi and xj can be permuted , r ` (3) any xi on the left of st can be moved to the right as xi .

Contents First Last Prev Next J Thus, amat can, in principle, have the following types :

Contents First Last Prev Next J Thus, amat can, in principle, have the following types :

r r o4 π3 s1 ,(puer puellam amat) r r π3 o4 s1 ,(puellam puer amat) ` ` s1 π3 o4 ,(amat puellam puer) ` ` s1 o4 π3 ,(amat puer puellam) r ` π3 s1 o4 ,(puer amat puellam) r ` o4 s1 π3 ,(puellam amat puer) r o4 s1 ,(puellam amat) r π3 s1 ,(puer amat) ` s1 o4 ,(amat puellam) ` s1 π3 ,(amat puer) s1 (amat).

Contents First Last Prev Next J 7. The noun phrase Nouns N and adjectives A display gender, number, and case.

Contents First Last Prev Next J 7. The noun phrase Nouns N and adjectives A display gender, number, and case. We assign basic types accordingly

Contents First Last Prev Next J 7. The noun phrase Nouns N and adjectives A display gender, number, and case. We assign basic types accordingly

g Dnc (N) : n gnc ,

g Dnc (A): a gnc , where g is fixed for nouns but variable for adjectives.

Contents First Last Prev Next J Nouns may occur as subjects or objects and adjectives may be used as nouns. Taking advantage of the fact that the set of basic types is partially ordered, we therefore postulate :

Contents First Last Prev Next J Nouns may occur as subjects or objects and adjectives may be used as nouns. Taking advantage of the fact that the set of basic types is partially ordered, we therefore postulate :

n g11 π3 , → n g21 π6 , → n gnc oc , → a gnc n gnc . →

Contents First Last Prev Next J Personal pronouns may dispaly gender, person, and case :

Contents First Last Prev Next J Personal pronouns may dispaly gender, person, and case :

g Ppc : πgpc

Contents First Last Prev Next J Personal pronouns may dispaly gender, person, and case :

g Ppc : πgpc we postulate

πgp1 πp , πgpc oc . → →

ng1c πg3c , ng2c πg6c . → →

Contents First Last Prev Next J These new postulates are consistent with the earlier ones, since we can infer

Contents First Last Prev Next J These new postulates are consistent with the earlier ones, since we can infer

ng11 πg31 π3 , → → ng2c πg6c π6 , → → ng1c πg3c oc , → → ng2c πg6c oc . → →

Contents First Last Prev Next J (7) (he loved the girl) is puellam amabat r r π131 n214 (o π s2) s2 4 3 →

Contents First Last Prev Next J (7) (he loved the girl) is puellam amabat r r π131 n214 (o π s2) s2 4 3 →

because n214 o4 and π131 π3 ; → →

Contents First Last Prev Next J (7) (he loved the girl) is puellam amabat r r π131 n214 (o π s2) s2 4 3 →

because n214 o4 and π131 π3 ; → →

(8) puellae te amabunt (girls will love you) r r n221 π124 (o π s5) s5 4 6 →

Contents First Last Prev Next J (7) (he loved the girl) is puellam amabat r r π131 n214 (o π s2) s2 4 3 →

because n214 o4 and π131 π3 ; → →

(8) puellae te amabunt (girls will love you) r r n221 π124 (o π s5) s5 4 6 →

because π124 o4 and n221 π6 . → →

Contents First Last Prev Next J To Justify compound noun phrases such as puella bona , bonarum puellarum we require a metarule :

Metarule 3.

An adjective of type agnc, also has types

r ` (ngnc ngnc) and (ngnc ngnc).

Contents First Last Prev Next J (9) puellam bonam amo r r n214 (n n214)(o s1) s1 214 4 →

r Here Metarule 2 is used to delete πi from the type of amo, r and Metarule 3 is invoked to assign the new type (n214 n214) to bonam of type a214 .

Contents First Last Prev Next J 8. The auxiliary verb sum We know that Descartes used sum without complement, but usually it requires a nominative complement of type o1 or an adjectival complement of type agn1, but then the gender of the subject becomes relevant. Thus we assign types

Contents First Last Prev Next J 8. The auxiliary verb sum We know that Descartes used sum without complement, but usually it requires a nominative complement of type o1 or an adjectival complement of type agn1, but then the gender of the subject becomes relevant. Thus we assign types

a r r r r Ctp(sum): o1 πp st , agn1 πgp1 st ,

Contents First Last Prev Next J 8. The auxiliary verb sum We know that Descartes used sum without complement, but usually it requires a nominative complement of type o1 or an adjectival complement of type agn1, but then the gender of the subject becomes relevant. Thus we assign types

a r r r r Ctp(sum): o1 πp st , agn1 πgp1 st , where a = 1 or 2 and n = 1 when p = 1, 2 or 3 ,

Contents First Last Prev Next J 8. The auxiliary verb sum We know that Descartes used sum without complement, but usually it requires a nominative complement of type o1 or an adjectival complement of type agn1, but then the gender of the subject becomes relevant. Thus we assign types

a r r r r Ctp(sum): o1 πp st , agn1 πgp1 st , where a = 1 or 2 and n = 1 when p = 1, 2 or 3 , but n = 2 when p = 4, 5 or 6 .

Contents First Last Prev Next J 8. The auxiliary verb sum We know that Descartes used sum without complement, but usually it requires a nominative complement of type o1 or an adjectival complement of type agn1, but then the gender of the subject becomes relevant. Thus we assign types

a r r r r Ctp(sum): o1 πp st , agn1 πgp1 st , where a = 1 or 2 and n = 1 when p = 1, 2 or 3 , but n = 2 when p = 4, 5 or 6 . We invoke Metarule 1 when πp is omitted.

Contents First Last Prev Next J (10) consul fui (I have been consul) r o1 (o s1) s1 1 →

(11) puer consul erat (the boy was consul) r r n111 o1 (o π s3) s3 1 3 →

(12) puella consul erat (the girl was consul) r r n211 o1 (o π s3) s3 1 3 →

Contents First Last Prev Next J (13) puella bona est (the girl is good) r r n211 a211 (a π s1) s1 211 231 →

Contents First Last Prev Next J (13) puella bona est (the girl is good) r r n211 a211 (a π s1) s1 211 231 →

(14) puer persona bona erat (the boy was a good person) r r r n111 n211 (n n211)(o π s3) s3 211 1 3 →

Contents First Last Prev Next J Here we have an example of feature conflict

(15) * puella bonus erat (the girl was good[masculine]) r r n211 a111 (agn1 πgp1 s1) s1 ? 6→ ?

Contents First Last Prev Next J Here we have an example of feature conflict

(15) * puella bonus erat (the girl was good[masculine]) r r n211 a111 (agn1 πgp1 s1) s1 ? 6→ ?

r To cancel a111 a 1 we require g = 1 and n = 1, gn1 →

Contents First Last Prev Next J Here we have an example of feature conflict

(15) * puella bonus erat (the girl was good[masculine]) r r n211 a111 (agn1 πgp1 s1) s1 ? 6→ ?

r To cancel a111 a 1 we require g = 1 and n = 1, gn1 → r but to cancel n211 πgp1 1 we require g = 2 and p = 3, which leads to a contradiction.→

relatives

Contents First Last Prev Next J 9. The perfect passive

There is a compound aspect, as in

amatus sum (I have been loved) combining the passive with the perfect. Consider a transitive verb V 3 r r when Ctp (V) has type o5 πp s1 , say amatur in

(16) puella puero amatur (the girl is loved by the boy) r r n211 n115 (o π s1) s1 5 3 →

Contents First Last Prev Next J To transform this into the perfect aspect, we require

(17) (the girl has been loved by the boy) puella puero amata est r r r n211 n115 (o a211)(a π s1) s1 5 211 231 →

r Here amata has type o5 a211 . In general the passive participle amatus is like an adjective, yet it inherits a possible complement from the r passive verb amor, hence has type o5 agnc . There may be more than one complement. For example, consider

Contents First Last Prev Next J (18) (an apple was given to the girl by the boy) malum puellae puero dabatur r r r n311 n213 n115 (o o π s3) s3 5 3 3 →

Contents First Last Prev Next J (18) (an apple was given to the girl by the boy) malum puellae puero dabatur r r r n311 n213 n115 (o o π s3) s3 5 3 3 →

In the perfect aspect this becomes

(19) (an apple had been given to the girl by the boy) malum puellae puero datum erat r r r r n311 n213 n115 (o o a311)(a π s3) s3 5 3 311 331 →

Contents First Last Prev Next J 10. Relative clauses g The relative pronoun Dnc (qui) introduces a relative clause which usually modifies a noun, although this noun may be omitted. It agrees with the noun in gender and number, but not in case:

g r ` D (qui): n ngnc s ngnc , nc0 gnc 0 ` ngnc s ngnc0

Contents First Last Prev Next J (20) (to the boy who loves the girl) puero qui amat puellam r ` r ` n113 (n n113 s n111)(π s1 o ) n214 n113 113 3 4 →

Contents First Last Prev Next J (20) (to the boy who loves the girl) puero qui amat puellam r ` r ` n113 (n n113 s n111)(π s1 o ) n214 n113 113 3 4 →

(21) (to the boy whom the girl loves) puero quem amat puella r ` r ` n113 (n n113 s n114)(o s1 π ) n211 n113 113 4 3 →

Contents First Last Prev Next J (22) (I love the girl whom I have given the apple) amo puellam cui dedi malum ` r ` r ` (s1 o ) n214 (n n214 s n213)(o s1 o ) n314 s1 4 214 3 4 →

Contents First Last Prev Next J (22) (I love the girl whom I have given the apple) amo puellam cui dedi malum ` r ` r ` (s1 o ) n214 (n n214 s n213)(o s1 o ) n314 s1 4 214 3 4 →

(23) (I love whom I have given the apple) amo cui dedi malum ` ` r ` (s1 o )(n214 s n213)(o s1 o ) n314 s1 4 3 4 →

Contents First Last Prev Next J (24) (the girl whom I have given the apple loved the boy) puella cui dedi malum amabat puerem r ` r ` r ` n211 (n n211 s n213)(o s1 o ) n314 (π s3 o ) n114 s3 211 3 4 3 4 →

Contents First Last Prev Next J 11. Concluding remarks We have investigated a small part of Latin grammar, in the expecta- tion that it would suffice for an analysis of Caesar’s De Bello Gallico. Unfortunately, it won’t suffice even for the very first sentence :

Gallia est omnis divisa in partes tres, quarum unam incolunt Belgae [et quarum] aliam [incolunt] Aquitani [et quarum] tertiam [incolunt] qui ipsorum lingua [et] institutis [et] legibus inter se differunt.

To make our algebraic approach at all feasible, we had to insert the words in square brackets. Moreover, we had not considered verbal complements which are prepositional phrases such as in partes tres of divisa, and inter se of differunt. In time, we hope to look at this and to other matters as well.

Contents First Last Prev Next J In the mean time, all we can say is that the only recently developed pregroup grammars show some promise of applying to classical Latin, as they do to its modern descendents, although some differences in approach should be emphasized.

In the small fragment we have investigated so far, there was no apparent need for double adjoints, such as a``, which had proved so useful for handling traces in English [15], and clitics in modern Ro- mance languages [3][9] .

Contents First Last Prev Next J On the other hand, multiple numerical indexes had to be attached to the basic types in order to account for agreement in gender, number and case. In particular, this system of indexes allows us to control matching and percolation of features.

The relatively free word order in Latin necessitates multiple type assignments to each Latin word. Rather than assuming that these types are all listed in the dictionary, we account for them by a small number of metarules of the form : if a word as a type such and such, then it also has a type so and so.

Contents First Last Prev Next J 12. Appendix: templates and rules

a = ASP ECT t = TENSE p = P ERSON   Basic : 1 pres. ind. : 1 first : 1   a  P erfect : 2 pres. subj. : 2 second : 2  Ctp(V) =    P assive : 3 imperf. ind. : 3 third : 3   imperf. subj. : 4 fourth : 4     future : 5 fifth : 5   sixth : 6 

Contents First Last Prev Next J g = GENDER n = NUMBER c = CASE   Masculine : 1 singular : 1 nominative : 1 g   Dnc(A) =  F eminine : 2 plural : 2 genitive : 2   Neuter : 3 dative : 3     accusative : 4   ablative : 5 

g = GENDER n = NUMBER c = CASE   Masculine : 1 singular : 1 nominative : 1 g   Dnc(N) =  F eminine : 2 plural : 2 genitive : 2   Neuter : 3 dative : 3     accusative : 4   ablative : 5 

Contents First Last Prev Next J 1 r r 2 Metarule 1. If Ctp(V) has type ... o4 πp st, then Ctp(V) 3 r r has the same type, but Ctp(V) has type ... o5 πp st, r provided o5 occurs only once.

a Metarule 2. If the verb form Ctp(V) has been assigned r r type x1 ... xn st , then r (1) any xi can be omitted , r r (2) xi and xj can be permuted , r ` (3) any xi on the left of st can be moved to the right as xi .

Metarule 3. An adjective of type agnc, also has types r ` (ngnc ngnc) and (ngnc ngnc).

Contents First Last Prev Next J References [1] Abrusci, V. M., Sequent calculus and phase semantics for pure noncommutative classical propositional logic, Journal of Symbolic Logic 56, (1991), 1403-1451. [2] Baker, M. C., The Atoms of Language, Basic Books, New York, 2001 . [3] Bargelli, D. and J. Lambek, An algebraic approach to French sen- tence structure, in P. de Groote, G. Morrill and C. Retor´e(eds.), Logical Aspects of Computational Linguistics, 62-78, Springer- Verlag, Berlin, 2001. [4] Caesar, J., The Gallic War, Loeb Classical Library, Harvard Uni- versity Press, Cambridge Mass., 1917/2000.

REFERENCES Contents First Last Prev Next J [5] Calboli, G. (ed.), Rethorica ad Erennium, with introduction and comment to the text, Patron, Bologna, 1993. [6] Calboli, G., Varrone e la teoria dei casi, in Papers on Grammar, VI, CLUEB, Bologna, 2001. [7] Cardinal, K., An algebraic study of Japanese grammar, Master’s Thesis, McGill University, Montreal 2002. [8] Casadio, C., Logic for Grammar, Bulzoni Editore, Roma, 2002. [9] Casadio, C. and J. Lambek, An algebraic analysis of clitic pronouns in Italian, in P. de Groote, G. Morrill and C. Retor´e(eds.), Logi- cal Aspects of Computational Linguistics, 110–124, Springer-Verlag, Berlin, 2001. [10] Casadio, C. and J. Lambek, A tale of four grammars, Studia Log- ica, vol. 71, 2 (2002). Special Issue edited by W. Buszkowski.

REFERENCES Contents First Last Prev Next J [11] Kislak, A., Pregroups versus English and Polish grammar, in V. M. Abrusci and C. Casadio (eds.), New Perspectives in Logic and Formal Linguistics, Bulzoni Editore, Roma, 2002 . [12] Klyve, G., Latin Grammar, Teach Yourself Books, Hodder Head- link Plc, London, 2002 . [13] Lambek, J., A mathematician looks at Latin conjugation, Theo- retical Linguistics, 6 (1979), 221-234. [14] Lambek, J., Type grammar meets German word order, Theoreti- cal Linguistics, 26 (2000), 19-30. [15] Lambek, J., Type grammars as pregroups, Grammars 4, 21-39, 2001. [16] Wohlbery, J., 201 Latin Verbs, Barron’s Educational Series, Woodbury N. Y., 1964 .

REFERENCES Contents First Last Prev Next J