Recent work in formal philosophy has concentrated over-whelmingly on the logical problems pertaining to epistemic shortfall - which is to say on the various ways in which partial and sometimes incorrect information may be stored and processed. A directly depicting language, in contrast, would reflect a condition of epistemic perfection. It would enable us to construct representations not of our knowledge but of the structures of reality itself, in much the way that chemical diagrams allow the representation (at a certain (...) level of abstractness) of the structures of molecules of different sorts. A diagram of such a language would be true if that which it sets out to depict exists in reality, i.e. if the structural relations between the names (and other bits and pieces in the diagram) map structural relations among the corresponding objects in the world. Otherwise it would be false. All of this should, of course, be perfectly familiar. (See, for example, Aristotle, Metaphysics, 1027 b 22, 1051 b 32ff.) The present paper seeks to go further than its predecessors, however, in offering a detailed account of the syntax of a working universal characteristic and of the ways in which it might be used. (shrink)

Leśniewski’s systems deviate greatly from standard logic in some basic features. The deviant aspects are rather well known, and often cited among the reasons why Leśniewski’s work enjoys little recognition. This paper is an attempt to explain why those aspects should be there at all. Leśniewski built his systems inspired by a dream close to Leibniz’s characteristica universalis: a perfect system of deductive theories encoding our knowledge of the world, based on a perfect language. My main claim is (...) that Leśniewski built his characteristica universalis following the conditions of de Jong and Betti’s Classical Model of Science (2008) to an astounding degree. While showing this I give an overview of the architecture of Leśniewski’s systems and of their fundamental characteristics. I suggest among others that the aesthetic constraints Leśniewski put on axioms and primitive terms have epistemological relevance. (shrink)

It is a commonplace that in the development of modern logic towards its actual shape at least two directions or traditions have to be distinguished. These traditions may be called, following the mo...

The task of this paper is to give a new, catholic interpretation of Leibniz’s concept of characteristica universalis. In § 2 we shall see that in different periods of his development, Leibniz defined this concept differently. He introduced it as “philosophical characteristic” in 1675, elaborated it further as characteristica universalis in 1679, and worked on it at least until 1690. Secondly, we shall see (in § 3) that in the last 130 years or so, different philosophers (...) have advanced projects similar to that of Leibniz, not always referring to him, at that. These very projects, we claim, threw some light on what Leibniz idea of characteristica universalis could be; or, in more positive sense, on how could it be reconstructed in a more workable way. Unfortunately, they failed to answer the question what exactly Leibniz’s philosophy was. Finally, despite the fact that Leibniz’s concept of characteristica universalis impressed generations of philosophers who tried to make sense of it, the result is that in the more than 300 years after it was introduced, it was never used in the scale its author dreamed of. This fact sets out the next objective which we are going to pursue in this paper: we shall try to find out (in §§ 4 and 5) how this concept can be interpreted in more practical form. (shrink)

In Descartes, the concept of a ‘universal science’ differs from that of a ‘mathesis universalis’, in that the latter is simply a general theory of quantities and proportions. Mathesis universalis is closely linked with mathematical analysis; the theorem to be proved is taken as given, and the analyst seeks to discover that from which the theorem follows. Though the analytic method is followed in the Meditations, Descartes is not concerned with a mathematisation of method; mathematics merely provides him (...) with examples. Leibniz, on the other hand, stressed the importance of a calculus as a way of representing and adding to what is known, and tried to construct a ‘universal calculus’ as part of his proposed universal symbolism, his ‘characteristica universalis’. The characteristica universalis was never completed—it proved impossible, for example, to list its basic terms, the ‘alphabet of human thoughts’—but parts of it did come to fruition, in the shape of Leibniz's infinitesimal calculus and his various logical calculi. By his construction of these calculi, Leibniz proved that it is possible to operate with concepts in a purely formal way. (shrink)

The discussions which follow rest on a distinction, first expounded by Husserl, between formal logic and formal ontology. The former concerns itself with (formal) meaning-structures; the latter with formal structures amongst objects and their parts. The paper attempts to show how, when formal ontological considerations are brought into play, contemporary extensionalist theories of part and whole, and above all the mereology of Leniewski, can be generalised to embrace not only relations between concrete objects and object-pieces, but also relations between what (...) we shall call dependent parts or moments. A two-dimensional formal language is canvassed for the resultant ontological theory, a language which owes more to the tradition of Euler, Boole and Venn than to the quantifier-centred languages which have predominated amongst analytic philosophers since the time of Frege and Russell. Analytic philosophical arguments against moments, and against the entire project of a formal ontology, are considered and rejected. The paper concludes with a brief account of some applications of the theory presented. (shrink)

Leibniz zamierzał stworzyć lingua characteristica universalis – język, w którym dałaby się zapisać wszelka wiedza, oraz calculus ratiocinator – metodę umożliwiającą rachunkowe określenie prawdziwości dowolnego zdania tego języka. Taka idea leży u podstaw współczesnej logiki. Taka idea w odniesieniu do poprawności sprzętu i programów leży u podstaw metod logicznych weryfikacji systemów informatycznych.Dziś widzimy, że dalszy rozwój informatyki zależy istotnie od postępu badań logicznych, a logika – ta starożytna dyscyplina – znalazła nowe bogate pole dociekań, zyskała nowe perspektywy badań. (...) Jej zastosowania mogą mieć tak znaczący wymiar praktyczny, o jakim nie śniło się filozofom. (shrink)

El punto de partida del calculus universalis leibniziano es la teoría aristotélica del silogismo, pero Leibniz se independiza de las ideas de Aristóteles para desarrollar su propio sistema lógico, mucho más general, aplicando el instrumento combinatorio a la silogística. Lo que propone es una importante modificación del modelo demostrativo axiomático, mediante la creación de cálculos lógico-simbólicos que no se limitan a los ámbitos tradicionales de la deducción, sino que admiten procedimientos discursivos más complejos que los de la lógica clásica, (...) ampliando con ello la noción de racionalidad. (shrink)

Eine Untersuchung der Präpositionen bei Leibniz kann aufgrund ihrer synkatego-rematischen Natur zeigen, in welchem Sinne die Sprache - als strukturiertes, bedeutendes Zeichensystem – das logische Verhältnis unter den Notionen ausdrü-cken kann, und damit der Zusammenhang zwischen Grammatik und Semantik einerseits, und Logik anderseits, erhellen. Meiner Ansicht nach bekommt auch Leibniz' Versuch des Aufbaus einer characteristica universalis dank dieser Per-spektive ein neues Forschungsinteresse. Um das Interesse für diese Redeteile zu wecken, werde ich zuvor in einem kurzen Exkurs die vorgängige (...) Tradition dar-stellen. Das Ziel ist, zu zeigen, was die Unterscheidung zwischen Kategoremen und Synkategoremen innerhalb der Tradition der sogenannten Mentalsprache im-pliziert; welche Fragen ihre Analyse erhoben hat und was diese Unterscheidung bei einem für Leibniz wichtigen Denker, wie Joachim Jungius, bedeutet. Beson-ders zeige ich, wie die Unterscheidung Kategoreme – Synkategoreme der zwi-schen Begriff und modi des Begreifens entspricht. Danach analysiere ich die viel-fältige Betrachtung dieser Redeteile bei Leibniz in Bezug auf seine Zurückfüh-rung der Präpositionen auf eine räumliche Grundbedeutung. Zum Schluss versu-che ich zu zeigen, warum diese Redeteile als cogitationes caecae bei Leibniz interpretiert werden können und welche Schlüsse man aus dieser Bestimmung im Rahmen von Leibniz‟ Sprachphilosophie ziehen darf. (shrink)

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be (...) fruitfully adopted in larger scale by philosophers. (shrink)

As recent scholarship has repeatedly shown, the history of Vienna Circle is to some extent rooted in the tradition of Austrian Philosophy which Neurath considered as not involved in the “Kantian interlude”. Nevertheless, it seems that the heritage of Leibniz and, in particular, of his “reform of logic” has been hitherto neglected. Indeed, Leibnizianism (along with Herbartianism) represents a main feature of this tradition stretching from Bolzano to quite forgotten figures as Exner and Zimmermann, and still influent on the Brentano (...) circle. This paper attempt to highlight the role played by Leibniz in framing of the “scientific world conception”, focusing in particular both on Carnap’s and Neurath’s stance towards the characteristica universalis underlying the project of the encyclopedia Leibniz aimed to realize. (shrink)

I will sustain in this article that Peirce can be seen as the last great representative of that inconspicuous but persistent tradition that, from the thirteenth to the seventeenth century, spent its energies on discovering a universal language. His project of Existential Graphs is in fact grounded on the isomorphism among a Sheet of Assertion, in which Graphs-signs are drawn, a Mind, with its thoughts-signs, and the Universe, with its facts-signs. In the same sense, Leibniz worked on his Characteristica (...) Universalis, seen as a general Encyclopedia or alphabet of human thoughts, and a pictum mundi amphitheatrum. (shrink)

Between the end of the nineteenth century and the first twenty years of the twentieth century, stimulated by the impetuous development of logical studies and taking inspiration from Leibniz's idea of a characteristica universalis, the three founding fathers of the analytic tradition in philosophy, i.e., Frege, Russell, and Wittgenstein, started to talk of a logically perfect language, as opposed to natural languages, all feeling that the latter were inadequate to their (different) philosophical purposes. In the second half of (...) the twentieth century, however, the very idea of a logically perfect language ceased for various reasons to seem attractive to analytic philosophers. Thus, it might appear that this idea could be classified together with the many other bizarre ideas that from time to time surface in the history of philosophy-an idea that perhaps had a beneficial impact on the development of twentieth century logic, but which can now be put to rest. In this brief note, I contend that this conclusion may be too hasty. Indeed, if a well-known empirical hypothesis advanced in 1975 by Jerry Fodor turns out to be true, then there is a logically perfect language, after all. More precisely, I argue that, if it exists, Fodor's language of thought possesses the main characteristics a logically perfect language is required to have. (shrink)

The disciplinary differentiation of sciences attracted Leibniz’s attention for a long period of time. From nowadays prospects it looks very well grounded as soon as in Leibniz’s manuscripts a modern scholar finds clue ideas of any research field which would tempt him to consider Leibniz as one of the founders of this particular discipline. We argue that this is possible only in retrospection and would significantly distort the essence of Leibniz’s epistemology. Our approach implies, in contrary, the investigation of the (...) Leibniz doctrine of signs on the background of the related philosophical problem, that of expression. The choice of semiotics is justified by the fact that it took a central place in his theoretical constructions, both those of natural sciences and of philosophy. In Leibniz system of knowledge the concept of notes (notae) and sings (signa) served a theoretical foundation of his most important and long-life aspiration to build up practical science of universal characteristics (characteristica universalis). In his eyes this practical science was the science of sciences (Scientia scientiarum), and we can consider it as the matrix for all possible scientific knowledge. (shrink)

In the first part of this paper the Leibnizian theory of relations is presented as the metaphysical background of his concept of space. The structure of metaphysical space is dominated by the monads while geometrical aspects are neglected. In the second part the paper deals with Leibniz' concept of space in his mathematical works and especially in the Analysis Situs . What emerges is that the attempt to transfer philosophical arguments into mathematical structures fails because of the incompatibility of notions (...) in different contexts. The same problem arises in his Characteristica universalis: Philosophy is not computable. (shrink)

Our present knowledge in the field of dynamical systems, information theory, probability theory and other similar domains indicates that the human brain is a complex dynamical system working in a strong chaotic regime in which random processes play important roles. In this environment our mental life develops. To choose a logically ordered sequence from a random or almost random stream of thoughts is a difficult and energy consuming task. The only domain in which we are able to do this with (...) a full success is mathematics. Leibniz’s life ambition was to extend this success, with the help of what he called _characteristica universalis_, to other areas of human activity. The belief that this is possible lies at the basis of Leibniz’s rationalist system. Reasoning within his system, Leibniz claimed that also fundamental laws of physics can be deduced from the “first principles”. Just as linguistic or conceptual units are at the basis of the _charactersistica universalis_, his monads are responsible for physical activity of material bodies. When this rationalistic strategy is applied to the philosophy of space and time, it leads to their radically relational conception. Leibniz’s rationalistic approach to philosophy and science arouses out sympathy but it was Newton’s mathematical-empirical method that turned out to be effective in human endeavour to understand the functioning of the physical world. Successes of the Newtonian method compel us to revise our concept of rationality. (shrink)

Our present knowledge in the field of dynamical systems, information theory, probability theory and other similar domains indicates that the human brain is a complex dynamical system working in a strong chaotic regime in which random processes play important roles. In this environment our mental life develops. To choose a logically ordered sequence from a random or almost random stream of thoughts is a difficult and energy consuming task. The only domain in which we are able to do this with (...) a full success is mathematics. Leibniz’s life ambition was to extend this success, with the help of what he called _characteristica universalis_, to other areas of human activity. The belief that this is possible lies at the basis of Leibniz’s rationalist system. Reasoning within his system, Leibniz claimed that also fundamental laws of physics can be deduced from the “first principles”. Just as linguistic or conceptual units are at the basis of the _charactersistica universalis_, his monads are responsible for physical activity of material bodies. When this rationalistic strategy is applied to the philosophy of space and time, it leads to their radically relational conception. Leibniz’s rationalistic approach to philosophy and science arouses out sympathy but it was Newton’s mathematical-empirical method that turned out to be effective in human endeavour to understand the functioning of the physical world. Successes of the Newtonian method compel us to revise our concept of rationality. (shrink)

Abstract:One of the models that Leibniz considered for his characteristica universalis was the Chinese script, which, after many oscillations, he ended up conceiving as a rational language. In order to explain what "rationality" means in this context, the role of symbolicity and metaphoricity in the characteristica is discussed here. Furthermore, it is argued that Leibniz' idea of a constitutive role of signs for thought led him to produce a concept of writing based not on representation but on (...) expression, in which written signs are structurally connected to the ideas they express. Taking inspiration from Rita Widmaier's metaphorological reading of Leibniz and in Kwan Tze-wan's philosophical theory of the Chinese script, it is shown how the procedures involved in the formation of sinographs shed light on the principles of the characteristica, and how, despite Leibniz' inaccurate understanding of Chinese writing as ideographic, his philosophy can still inspire contemporary research on this script. (shrink)

Although philosophising about given languages had been going on ever since the time of Plato's Kratylos , the idea of an artificial philosophical language or system of signs began to take shape in the seventeenth century. Both Descartes and Mersenne explored the ground for the foundations of a system of expressions which could meet all the requirements of logical thought; but the merit of presenting the first elaborate plans goes to the British authors George Dalgarno and John Wilkins. 1 Leibniz (...) followed soon after with his characteristica universalis. (shrink)

Es ist allgemein bekannt, daß Frege stark von Leibniz beeinflußt worden ist; jedoch behauptet Patzig (Frege, Leibniz u. d. sogenannte ,lingua characteristica universalis', in: Studia Leibnitiana, Suppl., vol. III no. 3 (1969), pp. 103-112), daß seine Leibniz-Kenntnisse nur auf indirekten Quellen fußten und im Grunde äußerst gering waren. Dieser Aufsatz soll zeigen, daß solche und ähnliche Behauptungen auf ungenügender Kenntnis von Freges Schriften beruhen.

Although philosophising about given languages had been going on ever since the time of Plato's Kratylos, the idea of an artificial philosophical language or system of signs began to take shape in the seventeenth century. Both Descartes and Mersenne explored the ground for the foundations of a system of expressions which could meet all the requirements of logical thought; but the merit of presenting the first elaborate plans goes to the British authors George Dalgarno and John Wilkins. 1 Leibniz followed (...) soon after with his characteristica universalis. (shrink)

The question of why Leibniz in his last letter to Nicolas Remond interrupted the explanation of his binary system is reducible to the question of which role it actually plays in this letter. This can be answered, when the mathematical meaning of the Dyadic as well as its analogical function (as symbol of the creation) are seen in the context of the so called Chinese Rites Controversy. In this context the role of the binary system changed in Leibniz`s thought. As (...) a mathematical decoding system of the mysterious characters of the oldest Chinese Philosopher Fuxi, it got a high status in epistemological as well as historical sciences. But as a pretended deeper knowledge of a creatio ex nihilo the dyadic attained immediately an enormous missiological meaning: It witnessed no more than the existence of an original monotheism and its theology of creation in China. However, when it became evident that the decisions in Rome would cause the end of the old Jesuit-mission in China, Leibniz within the frame of the natural theology (expected in pagan and old Chinese thinking) restricted the missiological meaning to its role as natural mathematical knowledge of China in primeval times. (shrink)

본고는 라이프니츠의 논리학과 직접적으로 연관되어 있는 증명을 위한 사고의 조합에 관해 연구한 논문이다. 그의 논리학 개념에는 다양한 측면들이 있으며, 그러한 측면들은 그의 여러 가 지 논리학적 구상들과 기획들에 반영되어 있다. 이 논문은 이러한 구상들과 기획들이 중요한 철 학적 함의들을 갖고 있음을, 그리고 그것들이 보편학으로서의 그의 논리학을 구성함을, 그리고 그의 논리학이 기호들의 조작과 사고의 조합에 근거한 증명법임을 보이고자 한다. 라이프니츠는 우선적으로 자신의 논리 체계를 정립하기 위해 다양한 수학적인 방법들을 도입했다. 그리고 그 는 인간 사고의 알파벳과 새로운 백과사전과 같은 예비적인 논리적 기획들을 (...) 감행했다. 여기서 전자는 복합 개념들을 분석하여 도달한 단순 개념들의 기호들의 목록이며, 후자는 모든 학문들 에 포함된 단순 개념들과 정의들을 산출한다. 이러한 기획들을 토대로, 그는 그의 논리 체계를 건설하고자 하였다. 이러한 논리 체계는 모든 학문적인 명제들을 증명하며 보편 기호법과 추리 계산법의 사용을 가능하게 한다. 비록 라이프니츠가 명확한 논리 체계를 건설하는 데 실패했다 할지라도, 그는 최초로 논리적 기호법을 고안하고자 노력했으며, 그의 논리학 개념은 부울과 프 레게에게 현대 논리학을 형성하도록 영감을 주었다. (shrink)

Leibniz hat eine große Anzahl von Fragmenten zur Logik verfaßt. Der umfangreichste dieser Texte sind die Generales inquisitiones (1982 als Band 338 der PhB erschienen). Leibniz hat jedoch verschiedene Systeme der Logik entworfen, von denen das in den Generales inquisitiones dargestellte nur eines ist, an dem er zudem später noch weitergearbeitet hat. Es lag daher nahe, weitere Texte zur Logik von Leibniz in einer ähnlichen Ausgabe zu bearbeiten, in denen einerseits diese Weiterentwicklungen exemplifiziert und andererseits weitere Systementwürfe vorgestellt werden. Diese (...) neue Ausgabe enthält zehn Texte aus den Jahren 1683-1690 zur Allgemeinen Wissenschaft (Scientia generalis), zur Allgemeinen Wissenschaftssprache (Characteristica universalis) und zum Logischen Kalkül (Calculus logicus). Im Bereich der Logikkalküle werden Texte zum sogenannten Identitätskalkül vorgestellt, die eine Weiterentwicklung der Generales inquisitiones darstellen, sowie Texte zum sogenannten Plus-Minus-Kalkül. Innerhalb beider Gruppen ist jeweils ein Text enthalten, der eher als Vorstufe angesehen werden kann, der also Leibniz "bei der Arbeit" zeigt. (shrink)

The standard account of the history of logic, in its crudest outline, runs as follows. Logic as a discipline was invented by Aristotle with his creation of syllogistic theory, which, despite the emergence of propositional logic in the work of the Stoics, refinements by mediæval logicians, and Leibniz's sketches of a characteristica universalis, dominated philosophy until the middle of the nineteenth century, when Boole's work on the "algebra of logic" and Frege's Begriffsschrift finally absorbed it into a larger (...) framework. Boolean and Fregean logic were rivals until the beginning of the twentieth century, when the superior power of the Fregean system, as developed by Peano and Russell, won the day, and modern logic as we know it began to exert its hegemony. However, as with any crude account, there are glaring omissions and anomalies that threaten its usefulness even as a first approximation, and Peirce's role in the story has perhaps been the most seriously underplayed of all. Whilst rooted in the Boolean tradition, Peirce also—admittedly slightly later, but nevertheless independently of Frege—invented a notation for quantifiers: a notation, indeed, that influenced modern notation to a far greater extent than Frege's own cumbersome two-dimensional script. Furthermore, Peirce's logic of relations was a significant development in its own right, and both this and the stirrings of the model-theoretic approach in his work was to feed into the mainstream tradition through the work of Schröder, Löwenheim, Skolem, and Tarski, in particular. (shrink)

In the first part of this paper a short review of the recently published 4th volume of Series 6 of the Akademie-Ausgabe of Leibniz’s Sämtliche Schriften und Briefe is given. This 3,000-page volume was edited by the Leibniz-Forschungsstelle in Münster, Germany. It contains unsurpassable, text-critical versions of more than 500 pieces which Leibniz composed between 1677 and 1690. One major topic dealt with in these essays is "Scientia Generalis, Characteristica, Calculus universalis". Here we find in particular various fragments (...) of a logical calculus that Leibniz developed around 1687. The main part of this paper presents a detailed reconstruction of this so-called "Plus-Minus-Calculus" which, by way of its somewhat unorthodox operators of "addition" and "subtraction", inclusion and identity, "communication", "commune" and "Nothing", provides an interesting alternative for the Boolean algebra of sets. (shrink)

Excerpt from Clavis Universalis Stuart, edited by Sir Wm. Hamilton, 1854, p. 349. 3 Robert Benson's Memoirs of the Life and Writings of the Rev. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection (...) in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

This paper discusses a nowadays completely forgotten 18th century attempt of constructing an artificial universal language in a Kantian framework. I give a brief sketch of this language and then address the continuing philosophical significance of the project, focusing in particular on the notions of predication and the copula and on the problem of psychologism.

Fondée sous les auspices du père de notre modernité philosophique Descartes, puis consolidée par des penseurs aussi importants que Leibniz, Bolzano ou Husserl, la mathesis universalis paraît représenter à elle seule l'ambitieux programme du « rationalisme classique ». Des philosophes tels que Husserl, Russell, Heidegger ou Cassirer ont pu s'accorder en ce point. Le développement de la « science moderne » aurait porté ce grand « rêve dogmatique » pour mener vers son terme le destin de la métaphysique occidentale. (...) Pourtant les recherches historiques récentes ont montré que l'idée de « mathématique universelle » existait bien avant Descartes, que ce dernier ne revendiquait d'ailleurs aucune rupture sur ce point et que sa réflexion se situait même assez clairement dans l'héritage des Anciens. Comment dès lors justifier que les Anciens, avec lesquels le programme des Classiques était censé rompre, aient pu déjà se préoccuper de « mathématique universelle »? Plus simplement encore, de quoi se préoccupaient donc ces philosophes sous ce concept? Le regain d'intérêt pour la mathesis universalis à la fin du XIXe siècle n'avait-il pas conduit paradoxalement à la perte de son sens comme problème? Cette étude a pour but de suivre ces questions jusqu'à leur origine et de montrer leur importance dans le dialogue entre mathématique et philosophie. Pages de début Remerciements Introduction La constitution de la « mathématique universelle » comme problème philosophique I. Aristote II. « Mathématique universelle » et théories mathématiques : Aristote, Euclide, Epinomis III. Le moment néo-platonicien Vers la science de l'ordre et de la mesure IV. La renaissance de la mathématique universelle V. La mathesis universalis cartésienne Conclusion Annexe I. La quaestio de scientia mathematica communi Annexe II. Essai bibliographique sur la mathesis universalis chez Descartes etLeibniz Bibliographie Index nominum Pages de fin. (shrink)