In the parts I and II of this paper, the Author shows:1. how Leibniz’s arithmetico-intensional logical calculi of April 1679 can be completed and transformed in an intensional Boolean algebra (U, v, &:, -, e, -e) admitting, on the one hand, two different logical interpretations:li1: as a complete and consistent calculus of terms (properties) and syllogistic;li2: as a deontic first-order calculus and, on the other hand, two different arithmetical interpretations:ai1: as a numerical Boolean algebra (DM, lcm, ged, M/..., 1, M) (...) of all divisors of a natural number M;ai2: as a numerlcal Boolean algebra (BA, lcbc, gcbc, A-..., 0, A) of all binary components of a natural number A.Arithmetical representations of negation of a term x and of combination (intensional conjunction) and alternative (intensional disjunction) of two or more terms are respectively M/x, lcm (lower common multiple) and gcd (greatest common divisor) in the first Boolean algebra (ai1) and A-x, lcbc (lower common binary composite) and gcbc (greatest common binary component) in the second one (ai2).2. that, in this context, each possible world U of 2ⁿ elements (terms, acts) can be defined on the basis on n elements of U choosen as “saturated” (intensionally maximal, but possible) in U or, inversely, on the basis of n elements of U choosen as “primitive” (intensionally minimal, but not-universal) in U. In fact, each possible element of U can be defined now as an alternative of saturated elements of U, now as a combination of primitive (opposite to saturated) elements of U; all combinations of saturated elements of U being equivalent to the impossible element (-e, non-entity, resp. impossible act) of U and an alternatives of elements of U being equivalent to the universal element (e, entity, resp. possible act) of U.In the arithmetical representation of each possible world U, the maximal number M (for ai1) or A (for ai2) represents the impossible (non existent) element of U. Now, each possihle world defined by n saturated (resp. primitive) elements can be automatically enlarged (restricted) by the introduction (suppression) of m new (old) saturated (resp. primitive) elements, producing a new possible world U’ where m impossible elements (centaur, pegasus, syren, unicorn, etc.) of U become possible elements of U’ or inversely.After this first type of arithmetical representation of logical calculi, where the terms are -as in Leibniz’s 1679 calculi- represented by natural numbers and the propositions by equations (for universal resp. prescriptive) or inequations (for particular, resp. permissive), in the part III the Author presents a second type of arithmetical representation where propositions are represented by natural numbers and the valid (classical or deontic) syllogisms by true arithmetical relations between the numbers of premisses and the number of conclusion. Here the entire syllogistic adopts the form of a multiplication table wherea syllogism is valid if and only if the lcm (in ai1 ) or the lcbc (in ai2) of the characterlstic numbers of the premisses is a multiple (in ai1) or a binary composite (in ai2) of the characteristic number of the conclusion. (shrink)

Descrizione degli obbiettivi, le tecniche ed i primi risultati di una ricerca attualmente in corso all’Università del Paese Basco per applicare il metodo degli invarianti numerici delle classi di equivalenza -già applicato (si veda “Il programma Ars Judicandi”) sul piano di una legislazione nazionale- alla costruzione di un modello matematico per automatizzare le operazioni di analisi logica parallela, confronto e decisione su diverse legislazioni nazionaIi, simultaneamente iscritte nel quadro di una “rete deontica internazionale” che descrive per ciascuna di esse i (...) rapporti validi fra le condizioni, i casi e le soluzioni giuridiche. Il modelle comprende: a) la traduzione in un Iinguaggio aritmetico di base esadecimale delle componenti, le operazioni e le relazioni di una base multinazionale di dati legislativi; b) la formulazione, sotto forma di algoritmi aritmetici, delle regole di utilizzazione della base di dati per gli obbiettivi, classici ed attuali, di analisi e di decisione deI Diritto Comparato e dell’armonizzazione ed integrazione sopranazionale di legisIazioni diverse. (shrink)

An arithmetical language, whose words are natural numbers written in hexadecimal numeration system, is defined and its applications for the representation, analysis and decision of formulae of some logical and normative systems are described and illustrated.The formulae, operations and relations of the represented system are associated as follows respectively to the numbers and the arithmetical operations and relations of the proposed language:1. Each well-formed-formula of the system is associated to a number of a set of natural numbers between zero (associated (...) to all tautologies and theses) and the binary supremum Φ of the set (Φ is associated to all contradictions and antitheses and his value is 2n-1, n depending on system’s dimensions and structure).2. Negation of a formula and disjonction and conjonction of t wo other more formulae of the system are associated respectively to the binary complement Φ-N(f) of the number associated to the first and to the binary infimun and supremum of the numbers associated to the last.3. The logical relations “f1 implies f2” (f1 -> f2), “f1 is incompat.ible with f2” (f1|f2), “f1 is the contradictory opposite of f2” (f1wf2) and “f1 is alternative to f2” (f1vf2) are true in the represented system if and only if the associated arithmetical relations, respectively “N(f1) absorbs arithmetically N(f2)”, “binary supremum of N(f1) and N(f2) is equal to Φ”, “sum of of N(f1) and N(f2) is equal to Φ” and “binary infimum of N(f1) and N(f2) is equal to 0”, are true.The applications of the proposed arithmetical language as very rapid and simplified tool of analysis and decision method are shown for the following systems: propositional logic, syllogistic, some deontic and alethic modal systems and some normative systems of statutory law:1. In the propositional logic, after fixing the maximum of variables considered, the numbers associated to these variables and to the contradictions are calculated. On this permanent basis, the evaluation of a farmula (especially in the case of many occurrences of many variables) is performed by the calculation of the associated number faster as in the traditional way.2. In the syllogistic, after the calculation of the number associated to each type of premise or conclusion, an arithmetical table of syllogisms shows that a proposed syllogism is valid if and only if the binary supremum of the numbers associated to the premises absorbs arithmetically the number associated to the conclusion. It is well-known that the search of this sort of arithmetization of syllogistic and the study of its possibilitiy has been a recurrent topic in modern logic, from Leibniz to Łukasiewicz and to-day.3. In a deontic equivalent:ial system who includes Von Wright’s deontic system of 1982 far norms of the first order, the calculation of the numbers associated to an types of permission and obligation sentences allows the im mediate arithmetical verification of an the classical relations between those sentences. The same is shown for an alethic modal system including the modlities of the first order of Lewis’s S5.4. For normative systems of statutory law, the arithmetical verification of the logical and normative relations is shown in precedent author’s papers -especially in recent “The ‘Ars Judicandi.’ Programme”-, though not yet in hexadecimal numeration system but only in binary and decimal ones.In all the represented systems, the arithmetical verification of the metalogical properties of the system -consistency and completeness- is performed in a very easy and rapid manner, after the arithmetic representation of the axiomatic basis of the latter by a system of equations. (shrink)

Il s’agit d’une methode qui permet d’associer à chaque formule bien formée du système S5 de logique modale un nombre naturel invariant pour toutes les formules qui appartiennent a la même classe d’equivalence que la première.En particulier, étant donné que la méthode associe à toutes les tautologies du système le nombre o et a toutes les contradictions du système un certain nombre Φ, il suffit de calculer le nombre qui, en vertu des associations fondamentales, reste associé à n’importe quelle formule (...) pour décider si cette dernière est tautologique, contradictoire ou contingente.Les relations logiques reliant deux formules du systeme -par exemple, des implications, des incompatibilités, des oppositions contradictaires, etc.- sont révéciés par un simpIe examen oculaire des nombres assoc iés aux formules données et une rapide vérification manuelle ou informatique fondée sur la comparaison des chiffres du même rang de ces nombres, écrits en hexadécimal.Finalement, l’analyse de la composition binaire du nombre associé a une formule donnée permet d’obtenir l’expression de la première sous sa forme normale conjonctive.La méthode décrite constitue donc une nouvelle méthode arithmétique de décision pour le système modal indiqué. (shrink)

Obras escogidas/M. Sánchez-Mazas. - v.1.

The aim of this study is to try to make use of real numbers for representing an infinite analysis of individual notions in an infinity of possible worlds.As an introduction to the subject, the author shows, firstly, the possibility of representing Boole's lattice of universal notions by an associate Boole's lattice of rational numbers.But, in opposition to the universal notions, definable by a finite number of predicates, an individual notion, cannot admits this sort of definition, because each state of an (...) individual subject is characterized by the values taken by an infinite number of predicates, each of whom may appear or disappear in the next state.The notion of “degree of identification of an individual notion” is then introduced and arithmetized by a rational number.As an individual notion can be defined by a convergent succession of degrees of identification, the real characteristic number of such an individual notion can be defined by the corresponding convergent succession of rational numbers, satisfying Cauchy's conditions for the convergence of successions. (shrink)