Algebra of Logic Research Papers (original) (raw)

The following are the slides of my presentation at the meeting "GBMS Theme 2: From Boole's Algebra of Logic to Boolean Algebra, and Beyond" within the BOOLE CONFERENCES, Cork, Ireland, 27 - 28 Aug 2015. The aim of the presentation was to trace the notion of symbolic knowledge that can be found implicitly in George Boole’s Algebra of Logic. G. W. Leibniz introduced around 1684 the idea of symbolic knowledge (cogitatio caeca or cogitatio symbolica) in order to draw a fundamental distinction between forms of cognitive representations. It can be described as knowledge obtained by means of a sign system. Even if in Boole’s work the notion of symbolic knowledge is not explicitly mentioned, it was presupposed in the methodology of British algebraists at the beginning of the 19th Century that constituted the base for Boole’s algebraic approach to logic. In the paper the presence of different features of symbolic knowledge in Boole´s work will be shown. Boole conceived two basic ways of producing symbolic knowledge: (a) by manipulation of signs according to rules (this way would be knowledge by formal calculus in the stricter sense), (b) by application of the sign system to a new domain, so that new properties of the domain can be known. In the last case knowledge of formal structures is involved and it could be possible to speak of ‘structural knowledge’ as a form of symbolic knowledge. The following conclusions will be also drawn:
(1) The algebraic formulation of logic played in Boole a more pragmatic than semantic role, namely the solution of logical problems.
(2) Boole solved logical problems through ‘computations’ in algebra. However, compared with the preceding attempts at logical calculi in the 18th Century, Boole devised an algebraic structure for logic, so that a whole new perspective for the analysis of logic opened up.

- by
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- History of Logic, Symbolic Knowledge, George Boole, Algebra of Logic

One of the most interesting topics in the modern part of the history of logic is concerned with the essence of the relationship between logic and mathematics. Putting aside the questions such as what has caused this relationship and which stages it has passed through, according to some the mathematical logic (ML) is mathematics in fact, some insists that it is a new form of the traditional logic (TL), some thinks it is neither TL nor mathematics even though it depends on the common grounds of mathematics and logic, and yet according to some it is nothing. In this study, it will be attempted to outline the views of the English philosopher George Boole (1815-1864) who has a significant place in the contemporary history of logic and mathematics as well. His views are known as an endeavor of synthesis between TL and mathematics, i.e. the algebra of logic (AL). I hope that the study will give light on what is the AL which forms a very significant stage in the progress of the ML, and on the methodical variations between TL, AL, and algebra as well.

- by sadık türker
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- Logic, Algebra of Logic

Gottlob Frege (1848-1925) et Ernst Schröder (1844-1902) participèrent de façon remarquable à l'émergence et au développement de la logique symbolique. Leur contribution à cette discipline est incontestée aujourd’hui (principalement celle de Frege) et ils y font figures de « pères fondateurs ». Tous deux se sont aussi consacrés - bien qu’avec des optiques différentes- aux problèmes de fondements des mathématiques qui faisaient l'objet de controverses dans le dernier quart du XIXe siècle. Par la suite, Frege influencera la philosophie des mathématiques, alors que le travail de Schröder sera plus lié à la praxis mathématique de son temps. Quoi qu'il en soit, les systèmes qu'ils proposèrent eurent un curieux destin. Le système des Grundgesetze de Frege, ainsi qu'on le sait, se révéla inconsistant et Schröder ne parvint pas à développer complètement son programme de fondement des mathématiques par ailleurs excessivement ambitieux. Je ne m'occuperai pas tant des aspects techniques de leurs propositions respectives que de leurs caractéristiques programatiques. En particulier, ces propositions seront analysées comme exemples de la relation entre la logique symbolique et les mathématiques à la fin du XIXe siècle. Dans l'histoire de la logique symbolique, la tendance a été de placer les deux auteurs dans des positions opposées, et il y a de bonnes raisons à cela. Cependant, on peut aussi relever, dans leurs objectifs, d’importantes similitudes. Mon opinion est que l'on doit parvenir à un juste équilibre entre ressemblances et différences. L'idée commune à la base de leurs programmes respectifs, est celle d'un langage scientifique universel ; dans la discussion je recourerai à cette idée de manière récurrente.

- by JAVIER LEGRIS
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- Philosophy Of Mathematics, Gottlob Frege, Algebra of Logic

Analiza la relación entre el álgebra de la lógica de Venn y su método diagramático, relación que es decisiva para esclarecer la concepción de Venn sobre las relaciones entre lenguaje, lo simbólico y lo gráfico, al tiempo que la distingue... more

- by Bruno Ramos Mendonça
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- History of Logic, Diagrammatic Reasoning, Venn diagrams, Syllogistic

The algebra of logic, as an explicit algebraic system showing the underlying mathematical structure of logic, was introduced by George Boole (1815-1864) in his book The Mathematical Analysis of Logic (1847). The methodology initiated by Boole was successfully continued in the 19th century in the work of William Stanley Jevons (1835-1882), Charles Sanders Peirce (1839-1914), Ernst Schröder (1841-1902), among many others, thereby establishing a tradition in (mathematical) logic. Furthermore, this tradition motivated the investigations of Leopold Löwenheim (1878-1957) that eventually gave rise to model theory. The tradition of the algebra of logic played a key role in the notion of Logic as Calculus as opposed to the notion of Logic as Universal Language. This entry is divided into 10 sections:
0. Introduction
1. 1847—The Beginnings of the Modern Versions of the Algebra of
Logic.
2. 1854—Boole's Final Presentation of his Algebra of Logic.
3. Jevons: An Algebra of Logic Based on Total Operations.
4. Peirce: Basing the Algebra of Logic on Subsumption.
5. De Morgan and Peirce: Relations and Quantifiers in the Algebra
of Logic.
6. Schröder’s systematization of the Algebra of Logic.
7. Huntington: Axiomatic Investigations of the Algebra of Logic.
8. Stone: Models for the Algebra of Logic.
9. Skolem: Quantifier Elimination and Decidability.

- by JAVIER LEGRIS
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- History of Mathematics, History of Logic, Philosophy of Logic, Symbolic Knowledge

This paper deals with conceptions of formality underlying 19th Century symbolic logic, where notations and manipulation of signs played an important role. It is devoted specifically to the case of Ernst Schröder's " formal algebra " , which extended with the algebra of relatives (as developed by C. S. Peirce) constituted the basis for a Pasigraphy as a universal notation system. The discussion will begin with the well-known distinction devised by Gottlob Frege between two sorts of formal theories. In the paper both conceptions of formality will be connected with the corresponding attempts of constructing universal scientific notations (Schröder's Pasigraphy and Frege's Begriffsschrift). It will be shown that the Pasigraphy was an interpretation of that formal algebra. As a further conclusion it will be suggested that each of the two conceptions of formality places logic in different levels and determines different conceptions of universality.

- by JAVIER LEGRIS
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- History of Logic, Universal Logic, Formality, Algebra of Logic

The aim of this chapter is to apply the notion of symbolic knowledge, conceived by G. W. Leibniz, to the understanding of some problems in the origins of mathematical (symbolic) logic in the 19th Century. In this sense, it can be regarded as a collection of notes for the study of the origins of mathematical logic with the notion of symbolic knowledge as Leitfaden. With its introduction I also hope to contribute to the current discussion in historiography of mathematical logic, where several distinctions, like the distinction between calculus and universal language, played an important role.

- by JAVIER LEGRIS
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- Philosophy of Logic, Gottlob Frege, History of Symbolic Logic, Symbolic Knowledge

Le couple < 0, 1 > de Boole a deux lectures dont la différence, pour être vraiment mesurée, demande un apologue.

- by Jean-Claude Dumoncel
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- Boolean Algebra, Algebra of Logic, C.S. Peirce, Transcendental terms