Algebre-De-Boole-Et-Portes-Logiques (original) (raw)
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Toute Theorie Est Algebrique et Topologique
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We want to explain how any mathematical theory is committed to a pulsation between algebra and topology. 1) Our main aim is to show on one hand how every theory is an algebraic one, according to various notions of an algebraic theory, through sketches and up to figurative algebras, that is to say a question of equations between laws of composition of figures; and on the other hand how every theory is topological or toposical (toposique), that is to say an expression of facts of continuity and a geometrical organisation of these facts (but this approach comes an ambiguity). So we get insights into a possible construction of an Algebraic Theory similar to the Algebraic Geometry of Grothendieck. It could be underlined also that on the way we prove two new facts which are essential to our analysis, one of a general nature, and the other which is a rather peculiar observation: 2.1) We lay stress on figurative algebras, and as a by-product we get that : every category of models is the ful...
Dans d'autres textes que l'on peut trouver concernant la MCL, les étapes sont moins nombreuses, il n'y en a parfois que 6 parce que l'on a réuni quelques-unes des étapes ci-dessus, comme par exemple celle de la planification des activités et celle des ressources. L'idée de base de la méthode est toutefois toujours la même. Dans ce texte l'Asdi a voulu préciser très clairement toutes les étapes et a donc donné à chacune d'elles un titre individuel.
Babbage et Boole : les lois du calcul symbolique
Intellectica. Revue de l'Association pour la Recherche Cognitive, 2004
Dans la première moitié du 19 ème siècle, Charles Babbage (1791-1871) conçoit les plans de sa "machine analytique", aujourd'hui catégorisée comme calculatrice automatique et mécanique à programme externe, avant que George Boole (1815-65) ne produise la première formulation mathématique d'une logique attachée depuis Aristote à l'analyse du langage. Le propos de cet article est de montrer que ces auteurs, qui sont en général convoqués indépendamment l'un de l'autre, appartiennent à un même réseau d'algébristes anglais, et de resituer leur entreprise parmi les effets, conceptuels et institutionnels, de la Révolution Industrielle. Ces algébristes, nourris de l'empirisme et de la philosophie du langage de Locke, n'interviennent pourtant pas dans une perspective constructiviste. Réformateurs anglicans, ils attribuent la nécessité des mathématiques à un calcul symbolique radicalement premier, explicitant les opérations de l'esprit indépendamment de toute contingence interprétative. Ils inaugurent une séparation radicale, mais cependant hiérarchique, entre opérativité et signification du calcul symbolique.
Algèbres graphiques (suites : Les bidules)
1982
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Op'erateurs Bool'eens S'equentiels Et Logique Multivalu'ee
Since the seminal paper by J. McCarthy [2], three-valued logic is the standard tool to study the behaviour of the sequential boolean operators: a special value ? denotes the result of undefined computations. In this paper we propose an more precise approach based on a multi-valued logic, in order to distinguish between several types of discernible errors. We present two results about the equational axiomatizability of boolean op'erators in multivalued logics: one is about the signature if Gamma then Gamma else; true; false (the three-valued case was presented by J. McCarthy in 1963), and the other one about the classical signature and; or; not; true; false (similar to the result obtained by Guzm`an and Squier in 1990). R'esum'e Depuis l'article fondateur de J. McCarthy [2] l"etude des comportement des op'erateurs bool'eens s'equentiels a 'et'e men'ee grace `a la logique trivalu'ee, dans laquelle une valeur sp'eciale ? repr'ese...
Discrete Mathematics, 1985
D&li6 a E. Corominas An analysis of Pierce's work on compact zero-dimensional spaces of finite type and of Hanfs work on primitive Boolean algebras shows that it is possible to obtain a description of the semiring generated by all primitive Boolean algebras in term of simple quasi-ordered systems. All Boolean algebras considered here are assumed to be denumerable. If B is (such) a Boolean algebra and a e B, let B(a) denote the Boolean algebra {x ~ B Ix ~a}. The algebra B is pseudo-indecomposable (abbreviated p.i.) if for all a~B, either B-~B(a) or B~B(a¢), where a c denotes the complement of a. The element aeB is p.i. if B(a) is p.i. A Boolean algebra B is primitive if it is p.i. and each dement is the sup of a finite family of disjoint p.i. elements, it is quasi-primitive if it is a finite direct product of primitive Boolean algebras. By a result of Williams, the free product of quasi-primitive Boolean algebras is again quasi-primitive and we denote by ~g the semiring of all (isomorphism classes of) quasi-primitive Boolean algebras with product and free product. To have a more concrete version of s~, we need to recall Pierce's concept of quasi-ordered system (Q.O. system). A Q.O. system is merely a set equipped with a binary transitive relation-which we shall always denote by R. A morphism between Q.O. systems Q and Q' is a map h:Q ~ Q' such that hR(q)= Rh(q), where R(q)={p[pRq}. A Q.O. system is simple if any morphism from Q is iniective. Q.O. systems make two complementary appearances in the theory of quasi-primitive Boolean algebras. (1) Let B be a primitive Boolean algebra. Define S(B)= {Ix]ix p.i. in B} where [x] is the isomorphism type of B(x); and let [x]R[y] whenever B(x)xB(y)=B(y). Then Hard and Williams proved that S(B) is a simple Q.O. system that characterizes B (up to isomorphism). Note that it is possible in some cases (following Pierce) to have an utterly different description of S(B). (2) A Q.O. semigroup is a structure D = (D;., R) where (D, .) is a semigroup and (D, R) is a compatible O.O. system (i.e. xRy implies xzRyz and zxRzy). Let Pi(~) denote the Q.O. semigroup of all (isomorphism classes of) primitive Boolean algebras, with free product and relation R defined by BRB' if B x B' ~ B'. Then it is a consequence of a theorem of Pierce that Pi(~) determines ~, up to isomorphism. Moreover, there exists a universal simple Q.O. system V, which contains a unique representative of each isomorphism class of countable simple Q.O. systems (take the direct limit of all countable simple O.O. systems). This system V can be uniquely endowed with a multiplication which makes it into a O.O. semigroup isomorphic with Pi(~). Thus the remaining problems to elucidate the structure of s~ are of the following kind: describe V (that is, give necessary and sufficient conditions for a countable Q.O. system to be simple), and characterize the Q.O. semigroup V. These problems are solved in the case of well-founded Q.O. systems.