Symbolic Knowledge Research Papers - Academia.edu (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... more

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.

C omo anunciamos en el primer número del volumen 46 de la Revista Latinoamericana de Filosofía, se publica aquí la segunda parte del "Dossier Leibniz", con trabajos que abordan los antecedentes escolásticos de la teodicea leibniziana y la... more

C omo anunciamos en el primer número del volumen 46 de la Revista Latinoamericana de Filosofía, se publica aquí la segunda parte del "Dossier Leibniz", con trabajos que abordan los antecedentes escolásticos de la teodicea leibniziana y la recepción de la que fueron objeto sus ideas en el pensamiento de Charles S. Peirce y Hermann Cohen. Al igual que los artículos de la primera parte, las versiones prelimi-nares de cada uno de ellos se discutieron en el simposio "Leibniz: ciencia, lógica y metafísica", celebrado el 12 y el 13 de diciembre de 2018 en la sede del Centro de Investigaciones Filosóficas (CIF). El primer artículo de esta segunda parte, "La metafísica de la creación de Duns Escoto: el entendi-miento divino como locus de los posibles", de Olga L. Larre, expone la doctrina escotista de la creación del mundo, con el objetivo de establecer las bases para un estudio comparativo con la teodicea leibniziana, que exhibe notables coincidencias con la metafísica de Duns Escoto. Por su parte, Javier Legris, en "La tradición del
http://rlfcif.org.ar/index.php/RLF/issue/current

En el sentido leibniziano, conocimiento simbólico (cogitatio caeca o symbolica) es un tipo de conocimiento que se obtiene mediante la utilización de algún tipo de estructura semiótica y se opone al conocimiento directo o “intuitivo”, como... more

En el sentido leibniziano, conocimiento simbólico (cogitatio caeca o symbolica) es un tipo de conocimiento que se obtiene mediante la utilización de algún tipo de estructura semiótica y se opone al conocimiento directo o “intuitivo”, como en ocasiones lo denomina Leibniz. De hecho, todo el conocimiento humano es simbólico en la medida en que que conocer presupone la construcción de estructuras simbólicas. Esta idea de conocimiento simbólico puede aplicarse al análisis de los aspectos cognoscitivos de la construcción de los sistemas formales y a algunas corrientes dentro de la inteligencia artificial. El presente trabajo se propone llevar a cabo un análisis del concepto de representación a partir de la noción de conocimiento simbólico concebida por Leibniz.

Este trabajo tiene por objetivo situar la representación diagramática en el contexto del “conocimiento simbólico”, empleando para ello conceptos tomados de la teoría del signo de Charles Sanders Peirce. En esta teoría los diagramas caen... more

Este trabajo tiene por objetivo situar la representación diagramática en el contexto del “conocimiento simbólico”, empleando para ello conceptos tomados de la teoría del signo de Charles Sanders Peirce. En esta teoría los diagramas caen dentro de la clase de los íconos, esto es, signos que representan por su semejanza estructural con el objeto representado. En el trabajo se mostrarán las limitaciones de una interpretación operacional de los íconos, es decir, como signos sobre los que se puede operar (componiendo o descomponiendo sus partes) y se discutirá una concepción amplia de ícono que destaque sus aspectos topológicos. Como ejemplo se analizará brevemente la interpretación del condicional material en el sistema lógico de los Gráficos Existenciales de Peirce.

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

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 de las concepciones rivales del siglo XIX.

The aim of this paper is to use the notion of symbolic knowledge, formulated originally by G.W. Leibniz. as a key concept to understand some aspects of Frege’s conceptual notation (Begriffsschrift) and its place in the origins of... more

The aim of this paper is to use the notion of symbolic knowledge, formulated originally by G.W. Leibniz. as a key concept to understand some aspects of Frege’s conceptual notation (Begriffsschrift) and its place in the origins of mathematical logic in the 19th Century. Furthermore, I hope to shed new light on the distinction between calculus and universal language, which played a decisive role in the historiography of mathematical logic. The paper will be focused on Frege´s book Begriffsschrift from 1879 and later articles, where Frege elucidates his conceptual notation and the logic built on it. Frege´s position seems to contradict some of the ideas underlying what I shall call the “tradition of symbolic knowledge, that were crucial in the development of mathematical logic.

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... more

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.

O livro apresenta, a partir de quatro questões centrais, um panorama geral do pensamento de Leibniz sobre a música. O capítulo 1 trata da tese de que a recepção estética do fenômeno musical pode ser entendida como um tipo de cálculo... more

O livro apresenta, a partir de quatro questões centrais, um panorama geral do pensamento de Leibniz sobre a música. O capítulo 1 trata da tese de que a recepção estética do fenômeno musical pode ser entendida como um tipo de cálculo aritmético oculto ou inconsciente. O capítulo 2 aborda o enfrentamento de Leibniz aos problemas da afinação, e sua resolução pelo desenvolvimento de um sistema de temperamento. No capítulo 3, são exploradas as aplicações, feitas pelo autor, de sua arte combinatória ao campo musical. Por fim, no capítulo 4, busca-se apresentar uma caracterização da notação musical tradicional em termos do conceito leibniziano de conhecimento cego ou simbólico. Encerram o livro, como anexos, as traduções para o português de 8 textos de Leibniz (em sua maioria, cartas) sobre a música e sua teoria.

D el 12 al 13 de diciembre de 2018, en la sede del Centro de Investigaciones Filo-sóficas (CIF), tuvo lugar el simposio "Leibniz: ciencia, lógica y metafísica". En él se expusieron y discutieron trabajos sobre la epistemología, la... more

D el 12 al 13 de diciembre de 2018, en la sede del Centro de Investigaciones Filo-sóficas (CIF), tuvo lugar el simposio "Leibniz: ciencia, lógica y metafísica". En él se expusieron y discutieron trabajos sobre la epistemología, la filosofía de la ma-temática y la filosofía natural de Leibniz y se presen-taron, además, contribuciones sobre los antecedentes de sus concepciones metafísicas en la escolástica, así como sobre la recepción de sus ideas en el desarrollo de la lógica y la filosofía de finales del siglo XIX y co-mienzos del XX. El resultado del encuentro se plasma en este dossier, que consta de dos partes. La primera de ellas, que es la que se publica aquí con el título "Creación, orden y mundo", incluye los textos que abordan temas estrictamente leibnizianos, mientras que la segunda, que aparecerá en el segundo número del volumen 46 de la Revista Latinoamericana de Filosofía, recogerá los textos que indagan en los antecedentes escolásticos y la recepción contemporánea de las ideas de Leibniz.

Con el fin de esclarecer la propuesta kantiana respecto al conocimiento metafísico, comenzaremos por examinar una de las caracterizaciones de la disciplina propuesta por Kant en Los progresos de la metafísica: La metafísica es, de acuerdo... more

Con el fin de esclarecer la propuesta kantiana respecto al conocimiento metafísico, comenzaremos por examinar una de las caracterizaciones de la disciplina propuesta por Kant en Los progresos de la metafísica: La metafísica es, de acuerdo a su fin, la ciencia del progreso de lo sensible a lo suprasensible. El ámbito de la metafísica no será entonces ni lo suprasensible –a lo cual no se puede acceder–, ni lo sensible –dado que es un conocimiento puro a priori–, sino el límite entre un ámbito y otro. Puesto que el límite, en tanto que demarca dos regiones, comparte algo de cada una de ellas, se nos presenta como un concepto clave para comprender el modo de conocimiento propio de la metafísica. Pondremos de manifiesto, además, la relación de la noción de analogía con el conocimiento simbólico tal como es presentada tanto en Prolegómenos como en Los progresos. Una vez esclarecidos estos tres conceptos fundamentales podremos dilucidar hasta qué punto se puede considerar que en el contexto de la filosofía trascendental kantiana el conocimiento metafísico es simbólico. Al mismo tiempo, este recorrido nos permitirá proponer que el conocimiento de la relación entre el ámbito nouménico y fenoménico también puede pensarse como un conocimiento de tipo simbólico.

Partindo de e encerrando com reflexões acerca de práticas de ensino de filosofia da matemática, proponho uma comparação entre os principais traços da noção leibniziana de conhecimento simbólico e algumas passagens tractarianas sobre a... more

Partindo de e encerrando com reflexões acerca de práticas de ensino de filosofia da matemática, proponho uma comparação entre os principais traços da noção leibniziana de conhecimento simbólico e algumas passagens tractarianas sobre a aritmética. Defendo que esta chave de leitura permite a um só tempo (i) projetar nova luz sobre as especificidades da definição tractariana de número em comparação com as de Frege e Russell; (ii) fazer despontar a compreensão da natureza do conhecimento matemático como conhecimento simbólico ou formal que Wittgenstein mobiliza em seu livro; (iii) elencar algumas razões para a alegação de que Wittgenstein pode ser considerado o filósofo da prática matemática avant la lettre. O trabalho se encerra com um apanhado, um retorno à reflexão inicial sobre os vínculos entre pesquisa e ensino, e uma defesa da chave de leitura aqui utilizada em termos de seu potencial para o desenvolvimento de estudos em filosofia da matemática.

In a famous paper from 1967, Jean van Heijenoort distinguished between logic as calculus and logic as language in order to describe two opposite trends in the earlier development of mathematical logic. The distinction was generalized by... more

In a famous paper from 1967, Jean van Heijenoort distinguished between logic as calculus and logic as language in order to describe two opposite trends in the earlier development of mathematical logic. The distinction was generalized by Jaakko Hintikka, who applied it to the interpretation of 20th century philosophy. According to the universalist conception of language, semantics cannot be defined in our only language without falling into a vicious circle. So, semantics cannot be expressible in the language. This fact motivated Hintikka to speak of the “ineffability of semantics”. This paper is an attempt to discuss these two notions in relation to the proof-theoretic semantics, as it was characterized and carried out by Michael Dummett, Dag Prawitz and Peter Schroeder-Heister, among many others. The case of proof-theoretic semantics is quite interesting not only because it is an alternative to model-theoretic semantics, but also because of its roots in mathematical intuitionism. This school had its own conception about the role of language, as ordinary as formalized, in foundational issues. For the intuitionists language was secondary in the construction and justification of mathematics. Arendt Heyting introduced formalization stricto sensu in intuitionism, and therefore paved the way for proof-theoretical semantics for intuitionistic logical constants. In the paper Heyting’s conception of formalization will be connected with the tradition of symbolic knowledge in formal sciences.
This is the final draft of a paper appeared in the volume: _Why is this a Proof? Festschrift for Luiz Carlos Pereira_. Ed. By Edward Hermann Haeusler, Wagner de Campos Sanz and Bruno Lopes. London, College Publications, 2015, pp. 74-86. ISBN: 978-1-84890-172-8.
http://www.collegepublications.co.uk/tributes/?00027

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... more

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.

Many of the difficulties related to Maimon’s philosophy arise from its antinomic nature, basically aimed at investigating the ways by which a finite understanding can approximate to the gnoseological perfection of an infinite... more

Many of the difficulties related to Maimon’s philosophy arise from its antinomic nature, basically aimed at investigating the ways by which a finite understanding can approximate to the gnoseological perfection of an infinite understanding. Beyond Maimon’s explicit confrontation with Spinoza, Leibniz, Hume and Kant, and his more or less direct influence on some crucial topics of the raising German idealism, the present contribution tries to clarify some internal passages of Maimon’s path. The aim is to show how the peculiar “differential” structure of thought, which in Maimon coincides with the structure of consciousness, can be illuminated by the analysis of notions that gravitate around it as its corollaries, namely, the concepts of truth and language, which are in turn strongly intertwined with each other.

The discussion concerning Kant’s knowledge of the Greek world has long been a subject of debate. Our contribution is intended to show that in the Dissertation of 1770 Kant is measured against some currents of Greek thought, and above all... more

The discussion concerning Kant’s knowledge of the Greek world has long been a subject of debate. Our contribution is intended to show that in the Dissertation of 1770 Kant is measured against some currents of Greek thought, and above all with Plato, on topics which will become very important in the articulated development of criticism in the 1770s. One aspect of our analysis deals with the texts that could have filtered Kant’s knowledge of ancient Greek tradition. We will then pore over some crucial features of the Dissertation, such as the distinction between sensible and intelligible knowledge and the ambiguous nature of the intellectualia, in order to assess how Kant’s understanding of certain issues of Greek classicism may have contributed to the outline of some still problematic theses in the text of 1770.

The existence of links between Peirce’s theory of signs and Leibniz’s arguments in favour of symbolic knowledge is beyond doubt. The aim of this paper is to place Peirce’s thought in the tradition of symbolic knowledge in formal sciences.... more

The existence of links between Peirce’s theory of signs and Leibniz’s arguments in favour of symbolic knowledge is beyond doubt. The aim of this paper is to place Peirce’s thought in the tradition of symbolic knowledge in formal sciences. The starting point is Peirce’s criticism to the Cartesian notion of intuition. Hereafter, the paper will focus on Peirce’s notion of iconicity according to the theory of signs, and its role in the characterization of formal sciences. This notion of iconicity will be related to the subrogative and ecthetic function of symbolic knowledge, but the essential analytic function of icons will be stressed too. On this basis, it will be argued that, even if Peirce’s thought is nourished by the tradition of symbolic knowledge, his iconic conception of formal sciences goes beyond this tradition.

Partindo de e encerrando com reflexões acerca de práticas de ensino de filosofia da matemática, proponho uma comparação entre os principais traços da noção leibniziana de conhecimento simbólico e algumas passagens tractaria-nas sobre a... more

Partindo de e encerrando com reflexões acerca de práticas de ensino de filosofia da matemática, proponho uma comparação entre os principais traços da noção leibniziana de conhecimento simbólico e algumas passagens tractaria-nas sobre a aritmética. Defendo que esta chave de leitura permite a um só tempo (i) projetar nova luz sobre as especificidades da definição tractariana de número em comparação com as de Frege e Russell; (ii) fazer despontar a compreensão da natureza do conhecimento matemático como conhecimento simbólico ou formal que Wittgenstein mobiliza em seu livro; (iii) elencar al-gumas razões para a alegação de que Wittgenstein pode ser considerado o filósofo da prática matemática avant la lettre. O trabalho se encerra com um apanhado, um retorno à reflexão inicial sobre os vínculos entre pesquisa e ensino, e uma defesa da chave de leitura aqui utilizada em termos de seu potencial para o desenvolvimento de estudos em filosofia da matemática.

We offer a reading of some passages from Wittgenstein’s Tractatus Logico-Philosophicus in which, dealing with the symbolic constructions of arithmetic, Wittgenstein puts in motion (more than explicitly assumes) the most outstanding... more

We offer a reading of some passages from Wittgenstein’s Tractatus Logico-Philosophicus in which, dealing with the symbolic constructions of arithmetic, Wittgenstein puts in motion (more than explicitly assumes) the most outstanding features of Leibniz’s concept of Symbolic Knowledge: the computational and the “ecthetic” functions of the notion of Symbolic Blind Thought. We begin with a brief presentation of some conceptual distinctions proposed by Oscar Miguel Esquisabel in his investigation about the Leibnizian origin of the tradition of Symbolic Knowledge. We then contrast these topics with the way Pasquale Frascolla interprets those same Tractarian passages, emphasizing the way in which he suspends any relevant sense in which we could speak about formal knowledge in Wittgenstein’s opera. To make our criticism better suited, our projection of Leibnizian lights into the Tractatus and into Frascolla’s interpretation of it is assisted with a concise exposition of one historical development in the Leibnizian tradition. Grounded in a study in which Javier Legris retraces the features of Symbolic Blind Though in Frege’s works, we then conclude proposing the advantages of our approach.