An Algebraic Approach to Latin Grammar C. Casadio and J. Lambek
Contents
First
Last
Prev
Next
J
Contents 1 2 3
4 5
6 7 8 9 10 11 12
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Declension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Adjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nouns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Relative pronouns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Personal pronouns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The algebraic machinary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The verb phrase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Verb forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Verb types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variations in type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The noun phrase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The auxiliary verb sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The perfect passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: templates and rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents
First
Last
Prev
3 4 7 8 9 10 11 12 14 15 16 21 24 31 35 38 42 45 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 accordance 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 accordance 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 Catp (V), where
Contents
First
Last
Prev
Next
J
2.
Conjugation
The Latin verb V has 3 × 5 × 6 finite verb forms Catp (V), where 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 Catp (V), where 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 Catp (V), where 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 Catp (V), where 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 Ca11 (V)
Contents
First
Last
Prev
Next
J
It is custumary to refer to the verb V by Ca11 (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 Ca11 (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
C1 (amo) =
C2 (amo) =
"
C3 (amo) =
"
am.o .. .
am.em am.¯ abam am.¯ arem am.¯ abo .. .. .. .. . . . . am.ant am.ent am.¯ abant am.¯ arent am.¯ abunt
#
amav.ˇ erim amav.ˇ eram amav.¯issem amav.ˇ ero .. .. .. .. . . . . amav.¯ erunt amav.ˇ erint amav.ˇ erant amav.¯issent amav.ˇ erint amav.i .. .
"
am.or .. .
am.er .. .
am.¯ abar .. .
am.¯ arer .. .
am.¯ abor .. .
#
#
am.¯ antur am.¯ entur am.ab¯ antur am.ar¯ entur am.ab¯ untur
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 Dgnc (A), where
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 Dgnc (A), where 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 Dgnc (A), where 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 Dgnc (A), where 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 1
bon.us bon.i bon.o bon.um bon.o bon.i bon.orum bon.is bon.os bon.is
2
bon.a bon.ae bon.ae bon.am bon.a bon.ae bon.arum bon.is bon.as bon.is
3
bon.um bon.i bon.o bon.um bon.o bon.a bon.orum bon.is bon.a bon.is
D (bonus) =
D (bonus) =
D (bonus) =
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. We write Dgnc (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. We write Dgnc (N), for a noun N of gender g, e.g. 1
D (puer) =
2
D (puella) =
h
h
puer puer.i puer.o puer.um puer.o puer.i puer.orum puer.is puer.os puer.is
i
puell.a puell.ae puell.ae puell.am puell.a puell.ae puell.arum puell.is puell.as puell.is
Contents
First
Last
Prev
i
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. D111 (qui) = qui , masculine, singular, nominative
D211 (qui) = quae , feminine, singular, nominative
D311 (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 p = p 0 + 3 (n-1). Thus we have the forms Pgpc . For example, P2 =
"
ego tu ea
me te 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 a r → 1 → ar a
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 s πp oc
for for for for
sentences in the t-th tense-mood sentences when the tense does not matter the subject in the p-th person 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 π p , 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 π p , 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 π p , 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 π p , 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 π p , 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 π p , 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
C3tp (morior ) : π rp st Catp (amo)
: or4 π rp st
Catp (do)
: or4 or3 π rp st
”
Catp (appello) : or4 or4 π rp st
”
C3tp (amo)
(a = 1, 2)
: or5 π rp 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. If C1tp (V) has type . . . or4 π rp st , then C2tp (V) has the same type, but C3tp (V) has type . . . or5 π rp st , provided or5 occurs only once. The proviso is required for appello, since C3tp (appello) : or5 or1 π rp st .
Contents
First
Last
Prev
Next
J
(1) (the horse dies)
eqˇ uus moriatur π 3 (π r3 s1 ) → s1
Contents
First
Last
Prev
Next
J
(1) (the horse dies)
eqˇ uus moriatur π 3 (π r3 s1 ) → s1
(2) (the boy had loved the girl)
puer puellam amaverat π 3 o4 (or4 π r3 s3 ) → s3
Contents
First
Last
Prev
Next
J
(3) (the girl will give the boy an apple)
puella puero m¯ alum dabit π3 o3 o4 (or4 or3 π r3 s5 ) → s5
Contents
First
Last
Prev
Next
J
(3) (the girl will give the boy an apple)
puella puero m¯ alum dabit π3 o3 o4 (or4 or3 π r3 s5 ) → s5
(4) (the boys have named the girl queen)
pueri puellam reginam appellaverunt π6 o4 o4 (or4 or4 π r6 s1 ) → s1
Contents
First
Last
Prev
Next
J
(5) (the girl was loved by the boy)
puella puero amabatur π3 o5 (or5 π r3 s3 ) → s3
Contents
First
Last
Prev
Next
J
(5) (the girl was loved by the boy)
puella puero amabatur π3 o5 (or5 π r3 s3 ) → s3
(6) (the girl is named queen by the boys)
puella regina pueris appellatur π3 o1 o5 (or5 or1 π r3 s1 ) → s1
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. If the verb form Catp (V) has been assigned type xr1 . . . xrn st , then (1) any xri can be omitted , (2) xri and xrj can be permuted , (3) any xri on the left of st can be moved to the right as x`i .
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 : or4 π r3 s1 π r3 or4 s1 s1 π `3 o`4 s1 o`4 π `3 π r3 s1 o`4 or4 s1 π `3 or4 s1 , π r3 s1 , s1 o`4 , s1 π `3 , s1
, , , , , ,
(puer puellam amat) (puellam puer amat) (amat puellam puer) (amat puer puellam) (puer amat puellam) (puellam amat puer) (puellam amat) (puer amat) (amat puellam) (amat puer) (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 Dgnc (N) : n gnc , Dgnc (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 → π 3 , n g21 → π 6 , 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 : Pgpc : π gpc
Contents
First
Last
Prev
Next
J
Personal pronouns may dispaly gender, person, and case : Pgpc : π gpc we postulate π gp1 → π p , π gpc → oc . n g1c → π g3c , n g2c → π g6c .
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 n g11 n g2c n g1c n g2c
→ π g31 → π g6c → π g3c → π g6c
→ π3 → π6 → oc → oc
Contents
, , , .
First
Last
Prev
Next
J
(7) (he loved the girl)
is puellam amabat π 131 n214 (or4 π r3 s2 ) → s2
Contents
First
Last
Prev
Next
J
(7) (he loved the girl)
is puellam amabat π 131 n214 (or4 π r3 s2 ) → s2
because n214 → o 4 and π 131 → π 3 ;
Contents
First
Last
Prev
Next
J
(7) (he loved the girl)
is puellam amabat π 131 n214 (or4 π r3 s2 ) → s2
because n214 → o 4 and π 131 → π 3 ; (8)
puellae te amabunt (girls will love you) r r n221 π 124 (o4 π 6 s5 ) → s5
Contents
First
Last
Prev
Next
J
(7) (he loved the girl)
is puellam amabat π 131 n214 (or4 π r3 s2 ) → s2
because n214 → o 4 and π 131 → π 3 ; (8)
puellae te amabunt (girls will love you) r r n221 π 124 (o4 π 6 s5 ) → s5
because π 124 → o 4 and n221 → π 6 .
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 (nrgnc ngnc ) and (ngnc n`gnc ) .
Contents
First
Last
Prev
Next
J
(9)
puellam bonam amo r n214 (n214 n214 ) (or4 s1 ) → s1
Here Metarule 2 is used to delete π ri from the type of amo, and Metarule 3 is invoked to assign the new type (nr214 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 Catp (sum) : or1 π rp st , argn1 π rgp1 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 Catp (sum) : or1 π rp st , argn1 π rgp1 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 Catp (sum) : or1 π rp st , argn1 π rgp1 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 Catp (sum) : or1 π rp st , argn1 π rgp1 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 π p is omitted.
Contents
First
Last
Prev
Next
J
(10)
consul o1
fui (I have been consul) r (o1 s1 ) → s1
(11)
puer consul erat (the boy was consul) n111 o1 (or1 π r3 s3 ) → s3
(12)
puella n211
consul erat (the girl was consul) r r o1 (o1 π 3 s3 ) → s3
Contents
First
Last
Prev
Next
J
(13)
puella n211
bona est (the girl is good) r r a211 (a211 π 231 s1 ) → s1
Contents
First
Last
Prev
Next
J
(13)
puella n211
bona est (the girl is good) r r a211 (a211 π 231 s1 ) → s1
(14)
puer persona bona erat (the boy was a good person) n111 n211 (nr211 n211 ) (or1 π r3 s3 ) → s3
Contents
First
Last
Prev
Next
J
Here we have an example of feature conflict (15) *
puella n211
bonus erat (the girl was good[masculine]) r r a111 (agn1 π gp1 s1 ) → 6 s1 ? ?
Contents
First
Last
Prev
Next
J
Here we have an example of feature conflict (15) *
puella n211
bonus erat (the girl was good[masculine]) r r a111 (agn1 π gp1 s1 ) → 6 s1 ? ?
To cancel a111 argn1 → 1 we require g = 1 and n = 1,
Contents
First
Last
Prev
Next
J
Here we have an example of feature conflict (15) *
puella n211
bonus erat (the girl was good[masculine]) r r a111 (agn1 π gp1 s1 ) → 6 s1 ? ?
To cancel a111 argn1 → 1 we require g = 1 and n = 1, but to cancel n211 π rgp1 → 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 when C3tp (V) has type or5 π rp s1 , say amatur in (16)
puella n211
puero amatur (the girl is loved by the boy) r r n115 (o5 π 3 s1 ) → s1
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 n211 n115 (o5 a211 ) (a211 π r231 s1 ) → s1
Here amata has type or5 a211 . In general the passive participle amatus is like an adjective, yet it inherits a possible complement from the passive verb amor, hence has type or5 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 n311 n213 n115 (or5 or3 π r3 s3 ) → s3
Contents
First
Last
Prev
Next
J
(18) (an apple was given to the girl by the boy)
malum puellae puero dabatur n311 n213 n115 (or5 or3 π r3 s3 ) → s3
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 n311 n213 n115 (o5 o3 a311 )(a311 π r331 s3 ) → s3
Contents
First
Last
Prev
Next
J
10.
Relative clauses
The relative pronoun Dgnc (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: Dgnc0 (qui ) : nrgnc ngnc s` ngnc0 , ngnc s` ngnc0
Contents
First
Last
Prev
Next
J
(20) (to the boy who loves the girl)
puero qui amat puellam r ` n113 (n113 n113 s n111 ) (π r3 s1 o`4 ) n214 → n113
Contents
First
Last
Prev
Next
J
(20) (to the boy who loves the girl)
puero qui amat puellam r ` n113 (n113 n113 s n111 ) (π r3 s1 o`4 ) n214 → n113
(21) (to the boy whom the girl loves)
puero quem amat puella r ` r n113 (n113 n113 s n114 ) (o4 s1 π `3 ) n211 → n113
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 o4 ) n214 (n214 n214 s n213 ) (o3 s1 o`4 ) n314 → s1
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 o4 ) n214 (n214 n214 s n213 ) (o3 s1 o`4 ) n314 → s1
(23) (I love whom I have given the apple)
amo cui dedi malum ` ` r (s1 o4 ) (n214 s n213 ) (o3 s1 o`4 ) n314 → s1
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 n211 (n211 n211 s n213 ) (o3 s1 o`4 ) n314 (π r3 s3 o`4 ) n114 → s3
Contents
First
Last
Prev
Next
J
11.
Concluding remarks
We have investigated a small part of Latin grammar, in the expectation 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 Romance 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 Ctp (V) =
a = ASP ECT
t = T EN SE
p = P ERSON
Basic : 1 P erf ect : 2 P assive : 3
pres. ind. : 1 pres. subj. : 2 imperf. ind. : 3 imperf. subj. : 4 f uture : 5
f irst : 1 second : 2 third : 3 f ourth : 4 f if th : 5 sixth : 6
Contents
First
Last
Prev
Next
J
g Dnc (A) =
g Dnc (N) =
g = GEN DER n = N U M BER c = CASE M asculine : 1 F eminine : 2 N euter : 3
singular : 1 plural : 2
nominative : 1 genitive : 2 dative : 3 accusative : 4 ablative : 5
g = GEN DER n = N U M BER c = CASE M asculine : 1 F eminine : 2 N euter : 3
singular : 1 plural : 2
Contents
nominative : 1 genitive : 2 dative : 3 accusative : 4 ablative : 5
First
Last
Prev
Next
J
Metarule 1. If C1tp (V) has type . . . or4 π rp st , then C2tp (V) has the same type, but C3tp (V) has type . . . or5 π rp st , provided or5 occurs only once. Metarule 2. If the verb form Catp (V) has been assigned type xr1 . . . xrn st , then (1) any xri can be omitted , (2) xri and xrj can be permuted , (3) any xri on the left of st can be moved to the right as x`i . Metarule 3. An adjective of type agnc , also has types (nrgnc ngnc ) and (ngnc n`gnc ) .
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 sentence structure, in P. de Groote, G. Morrill and C. Retor´e (eds.), Logical Aspects of Computational Linguistics, 62-78, SpringerVerlag, Berlin, 2001. [4] Caesar, J., The Gallic War, Loeb Classical Library, Harvard University 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.), Logical Aspects of Computational Linguistics, 110–124, Springer-Verlag, Berlin, 2001. [10] Casadio, C. and J. Lambek, A tale of four grammars, Studia Logica, 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 Headlink Plc, London, 2002 . [13] Lambek, J., A mathematician looks at Latin conjugation, Theoretical Linguistics, 6 (1979), 221-234. [14] Lambek, J., Type grammar meets German word order, Theoretical 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
