Metalogicon (1992) V, 1 Algebra and Grammar: A Peanian Analysis of Everyday English Michele Malatesta* 1. Peano's Latino sine flexione. It is well known that, by following Leibniz's intuition, Giuseppe Peano constructed the Latino sine flexione, i.e. Latin without inflexion. The rules of LSF (Latino sine flexione) are the following: (1) grammatical cases are superfluous since they can be expressed by prepositions; (2) grammatical genders are superfluous as in Chinese; (3) grammatical numbers are superfluous as in Chinese (N.B. Grammatical numbers not arithmetic numbers are superfluous); (4) conjugations of verbs are superfluous: it is sufficient to express the personal pronouns I, You, He, She, It, etc. as in Chinese; (5) verb tenses are superfluous since the time is expressible by means of adverbs as in Chinese. Note that Peano did not axiomatize LSF.1 * Presented at 3rd International Symposium on Systems Research Informatics and Cybernetics, Baden-Baden (Germany), August 12-18, 1991, in the Plenary Session on Logic and Mathematics of day August 16. 1 An attempt to axiomatize LSF is found in FREGUGLIA [1977] 305-317 and [1978] 72-75. In both works, the author given five general axioms and three specific axioms, proves two theorems. “Assiomi generali A.6.1: Siano , , ecc. simboli per elementi grammaticali, allora se = + 21 Metalogicon (1992) V, 1 si ha : = - = - A.6.2: Indichi 0 l'elemento grammaticale «de valore nullo» allora se = si ha - = 0 A.6.3: ± 0 = 0 + = A.6.4: - = - + = 0 A.6.5: ± ( ± ) = ( ± ) ± C Assiomi specifici A'.6.1: V = es + A = habe + N dove es e habe, imperativi latini dei verbi esse e habere, vengono usati secondo l'interlingua per indicare genericamente i verbi essere e avere. A'.6.2: A = que + V = cum + N = V + -nte A'.6.3: N = V + -ore. Teoremi TEOREMA 6.1: L'elemento grammaticale cum ha la stessa funzione grammaticale dell'elemento que habe (nel senso che possono sostituirsi scambievolmente in una generica frase che contenga o l'uno o l'altro). Dimostrazione: Dall'assioma A3.6.2 si ha: A = cum + N applicando ora lo A.6.1 si ha cum = A - N applicando lo A.6.2 " " cum = A + 0 - N " " A.6.4 " " cum = A - V + V - N " " A.6.5 " " cum = (A - V) + (V - N) (i) Sempre dall'assioma A'.6.2 si ha: A = que + V da cui per A.6.1 si ottiene (ij) que = A - V. A sua volta dall'assioma A'.6.1 si ha : V = habe + N da cui per A.6.1 discende (iij) habe = V - N. Sostituendo (ij) e (iij) in (i) si ha cum = que habe, c.v.d. TEOREMA 6.2: Gli elementi grammaticali que es (colui che è), es -nte, habe -ore hanno valore nullo (nel senso che grammaticalmente non rappresentano niente, cioè sono «eliminabili»). Dimostrazione: Dall'assioma A'.6.1 per A.6.1 si ha es = V - A " " A'.6.2 " " " que = A - V " " A'.6.2 " " " -nte = A - V allora: que es = (A - V) + (V - A) = applicando A.6.5 = A - V + V - A = " A.6.4 = A + 0 - A = " A.6.3 = A - A = " A.6.4 22 Metalogicon (1992) V, 1 2. English language trend to Chinese. The English Language is placed at the periphery of the Indo- European Group and, among the Western languages, expresses the maximum trend towards the Chinese. Let we examine the following table: Classical German Classical Italian English Greek Latin Spanish French C ( ) ( ) ( ) indef ( ) + a Art def + + ( ) ( ) ( ) s Subst + + + ( ) ( ) e Adj + + + ( ) ( ) s — attrib + + ( ) ( ) predic G ( ) ( ) + — indef + e Art def + + ( ) + — n Subst NR NR NR NR NR d Adj + + + — + e attrib + — + + — r predic s G n ( ) ( ) + + indef + r u Art def + + ( ) + — a m Subst + + + + + m b Adj + + + — + m e attrib + — + + — a r predic t s = 0 così anche: es -nte = (V - A) + (A - V ) = 0 Dall'assioma A'.6.1 per A.6.1 si ha: habe = V - N " A'.6.3 " " " -ore = N - V per cui analogamente a sopra si ha: habe -ore = (V - N) + (N - V) = 0 c.v.d.” 23 Metalogicon (1992) V, 1 T Past + + + + — e Present + + + + + n Future + + + + — s e s 14/18 14/18 11.18 12.18 3/18 77.77% 77.77% 61.11% 66.66% 16.66% This is a synoptic table of some grammatical categories. It considers only the following languages: Classical Greek, German, Classical Latin, Italian, Spanish, French, English. The sign ‘( )’ shows that a given grammatical part does not exist in the corresponding language; the sign ‘+’ indicates that a given grammatical part belongs to the corresponding language and has inflections; the sign ‘ — ’ denotes that a given grammatical part belongs to the corresponding language but is without inflexions; the signe 'NR’ shows that a given grammatical part belongs to the corresponding language but is not remarkable from the inflexion standpoint. Classical Greek has cases, and therefore inflexions, in the definite articles, in substantives, in attributive and predicative adjectives. German has cases, and therefore inflexions, in the indefinite and definite articles, in substantives and in attributive adjectives. Classical Latin has cases, and therefore inflexions, in substantives and in attributive and predicative adjectives. Italian, Spanish, French and English do not have cases. Classical Greek has genders in the definite articles, in substantives, in attributive and predicative adjectives. German has genders in the indefinite and definite articles, in substantives, in attributive adjectives. Classical Latin has genders in substantives, in attributive and predicative adjectives. Italian, Spanish and French have genders in the indefinite and definite articles, in substantives, in attributive and predicative adjectives. English has genders in substantive but they are not relevant 24 Metalogicon (1992) V, 1 from the inflexion standpoint. Classical Greek has grammatical numbers in the definite articles, in substantives, in attributive and predicative adjectives. German has grammatical numbers in the indefinite and definite articles, in substantives, in attributive adjectives. Classical Latin has grammatical genders in substantives, in attributive and predicative adjectives. Italian, Spanish and French have grammatical numbers in the indefinite and definite articles, in substantives, in attributive and predicative adjectives. English has grammatical numbers only in indefinte articles and in substantives. Classical Greek, German, Classical Latin, Italian, Spanish and French have tense inflexions in the past, in the present and in the future. The English language, apart from the verb to be and the indicative present of to have, has tense inflexions only in the third singular person of the indicative present. With regard to examined grammatical elements and leaving aside the substantive gender which is not remarkable for its own inflexions, note the following: - in Classical Greek the inflexions are 14/18 equal to 77,77 %; if we consider that this language has no indefinite articles, then we obtain the following inflexion number 14/15 equal to 93,33 %; - in German the inflexions are 14/18 equal to 77,77 %; - in Classical Latin the inflexions are 11/18 equal to 61,11 %; if we consider that this language has no indefinite and definite articles, then we obtain the following inflexion number 11/12 equal to 91,66 %; - in Italian, Spanish, French the inflexions are 12/18 equal to 66,66 %; if we consider that these languages have no cases, then we obtain the following inflexion number 12/13 equal to 92,30 %; - in English the inflexions are 3/18 equal to 16,66 %; now, if we consider that this language has no cases, then we obtain the following inflexion number 3/13 equal to 23,07 %. Therefore, whatever statistical criterion we choose when we analyze West languages, English is the one with the lowerst number of 25 Metalogicon (1992) V, 1 inflexions, and, consequently, the one nearest to Chinese. At this point we see that a Peanian analysis of English is very easy. 3. English without inflexion. Now we construct an artificial English language by reducing every English verb to its theme, by selecting among relative pronouns the only pronoun that, and by eliminating every inflexion from the verbs to be and to have. We study EWI (English without inflexion) by means of algebraic equations analogous to those in Peano's De derivatione and Algebra de grammatica. We shall give two distinct axiomatic patterns: in the first one we take as axioms four laws of standard algebra without proving their independence, as is usual in abstract algebra handbooks;2 in the second one, the most rigorous, we put as axioms three linguistic equations whose independence we prove. 4. First axiomatic version. 3.1 Syntax Let ‘A ’, ‘B ’ be symbols of grammatical elements. Let ‘A’ (Adjective), ‘V’ (Verb), ‘S’ (Abstract Substantive), ‘Es’ (Ending of Abstract Substantive) be the names of grammatical elements. The signs ‘0’, ‘+’, ‘-’, ‘=’ belong to the language. RULES OF FORMATION - 0 is a grammatical element; 2 Amongst others see P. J. HIGGINS [1975] 7 sgg. Freguglia also does not give any proof of independence of the algebraic axioms of Peano's Algebra de grammatica. See FREGUGLIA [1977] and [1978] cit. 26 Metalogicon (1992) V, 1 - all the adjectives, verbs and abstract substantives belonging to the English language are grammatical elements and the words ‘that ’, ‘with ’, ‘to’ are grammatical elements; the sign ‘Es’ i.e. “Ending of Abstract Substantive” is a grammatical element; - the names of grammatical elements are grammatical elements; - if A is a grammatical element, then +A and -A are grammatical elements; - if A and B are grammatical elements, then A +B and A -B are grammatical elements; - if A and B are grammatical elements, then A =B is a linguistic equation; - if A =B is a linguistical equation, and A = 0 or B= 0, then A =B is a satisfied linguistic equation; - there are no other grammatical elements nor linguistic equations nor satisfied linguistic equations.