An Algebraic Approach to Latin Grammar
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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.