Logic diagrams have been increasingly studied and applied for a few decades, not only in logic, but also in many other fields of science. The history of logic diagrams is an important subject, as many current systems and applications of logic diagrams are based on historical predecessors. While traditional histories of logic diagrams cite pioneers such as Leibniz, Euler, Venn, and Peirce, it is not widely known that Kant and the early Kantians in Germany and England played a crucial (...) role in popularising Euler(-type) diagrams. In this paper, the role of the Kantians in the late eighteenth and early nineteenth centuries will be analysed in more detail. It shows that diagrams (or intuition in general) were a highly contentious topic that depend on the philosophical attitude and went beyond logic to touch on issues of physics, metaphysics, linguistics and, above all, mathematics. (shrink)

This paper describes methods for generating interactive Euler diagrams. User interaction is needed to improve the aesthetic quality of the drawing without writing tedious formal specifications. More precisely, the user can modify the diagram’s layout on the fly by mouse control. We prove that the satisfiability problem is in \ and we provide two syntactic fragments such that the corresponding restricted satisfiability problem is already \-hard. We describe an improved local search based approach, a method inspired from the (...) gradient method and a hybrid method mixing both and. A software tool was implemented and its implementation is described. We also experimentally compare the different methods. We first see that the improved local search and the hybrid method outperforms the local search from the literature and the gradient method for generating a diagram. Concerning interaction, the local search approach is not suitable but hybrid method and gradient method give both good results in terms of quality of drawings and stability. Specifications are written using region connection calculus ), radius constraints and disjunctions. Euler diagrams are described as set of circles. (shrink)

One of the traditional applications of Euler diagrams is as a representation or counterpart of the usual set-theoretical models of given sentences. However, Euler diagrams have recently been investigated as the counterparts of logical formulas, which constitute formal proofs. Euler diagrams are rigorously defined as syntactic objects, and their inference systems, which are equivalent to some symbolic logical systems, are formalized. Based on this observation, we investigate both counter-model construction and proof-construction in the framework of Euler (...) diagrams. We introduce the notion of “counter-diagrammatic proof”, which shows the invalidity of a given inference, and which is defined as a syntactic manipulation of diagrams of the same sort as inference rules to construct proofs. Thus, in our Euler diagrammatic framework, the completeness theorem can be formalized in terms of the existence of a diagrammatic proof or a counter-diagrammatic proof. (shrink)

Mario Savio is widely known as the first spokesman for the Free Speech Movement. Having spent the summer of 1964 as a civil rights worker in segregationist Mississippi, Savio returned to the University of California at a time when students throughout the country were beginning to mobilize in support of racial justice and against the deepening American involvement in Vietnam. His moral clairty, his eloquence, and his democratic style of leadership inspired thousands of fellow Berkeley students to protest university regulations (...) that had severely limited political speech and activity on campus. The nonviolent campaign culminated in the largest mass arrest in American history, drew widespread faculty support, and resulted in a revision of university rules to permit political speech and organizing. This significant advance for student freedom rapidly spread to countless other colleges and universities across the country. Mario Savio went on to become a college teacher of physics, logic, and philosophy, to speak and organize in favor of immigrant rights and affirmative action and against U.S. intervention in Central America. He died on November 6, 1996, in the middle of a struggle against California State University fee hikes that hurt working-class students. Savio had submitted this article to the Notre Dame Journal of Formal Logic before he died. Final revisions were made by Philip Clayton with the assistance of Mario's colleagues at Sonoma State University. As reader for the Journal, George Englebretsen not only provided an extensive commentary on the article--much of which has been incorporated here--but also assisted in the difficult task of making revisions without changing the substance of Mario's style or thought. It is fitting that this, Savio's final publication, would be pedagogical in orientation. For him, moral considerations were no less pertinent in logic than in philosophy's less abstract fields. The usual student confusion with Venn diagrams led him to develop the new pictorial device presented in the following pages, which he believed was more sensitive to user psychology. It is hard to miss the political overtones in Savio's closing worry that in Venn diagrams "information of real significance may occasionally appear hidden and distorted." The decision by the Notre Dame Journal of Formal Logic to publish this piece posthumously is a testimony that logic, no less than other fields of philosophy, can be a tool of free speech and political change--as powerful in its way as the rhetorical brilliance of that young man standing on top of a police car who launched a worldwide movement with the words, "There is a time when the operation of the machine becomes so odious, makes you so sick at heart, you can't take part.". (shrink)

Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables (...) us to formalize and analyze free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs. (shrink)

How can Euler diagrams support non-consequence inferences? Although an inference to non-consequence, in which people are asked to judge whether no valid conclusion can be drawn from the given premises, is one of the two sides of logical inference, it has received remarkably little attention in research on human diagrammatic reasoning; how diagrams are really manipulated for such inferences remains unclear. We hypothesized that people naturally make these inferences by enumerating possible diagrams, based on the logical notion of self-consistency, (...) in which every Euler diagram is true in a set-theoretical interpretation. The work is divided into three parts, each exploring a particular condition or scenario. In condition 1, we asked participants to directly manipulate diagrams with size-fixed circles as they solved syllogistic tasks, with the result that more reasoners used the enumeration strategy. In condition 2, another type of size-fixed diagram was used. The diagram layout change interfered with accurate task performances and with the use of the enumeration strategy; however, the enumeration strategy was still dominant for those who could correctly perform the tasks. In condition 3, we used size-scalable diagrams, which reduced the interfering effect of diagram layout and enhanced participants’ selection of the enumeration strategy. These results provide evidence that non-consequence inferences can be achieved by diagram enumeration, exploiting the self-consistency of Euler diagrams. An alternate strategy based on counter-example construction with Euler diagrams, as well as effects of diagram layout in inferential processes, are also discussed. (shrink)

Euler famously used diagrams to illustrate syllogisms in his Lettres à une princesse d’Allemagne [1]. His diagrams are usually seen as suffering from a fatal “ambiguity problem” [11]: as soon as they involve intersecting circles, which are required for the representation of existential statements, it becomes unclear what exactly may be read off from them, and as Hammer & Shin conclusively showed, any set of reading conventions can lead to erroneous conclusions. I claim that Euler diagrams can, however, (...) be used rigorously, if they are read in conjunction with the premises they are supposed to illustrate. More precisely, I give rigorous “heterogeneous” inference rules (in the sense of Barwise and Etchemendy) – rules whose premises are a sentence and a diagram and whose conclusion is a sentence – which allow to use them safely. I conclude that one should abandon the preconception that diagrams can only be used rigorously if they can be given a context-independent semantics. Finally, I suggest that context-dependence is a widespread feature of diagrams: for instance, Mumma [12] noticed that what may be read off from a Euclidean diagram depends not only on the diagram’s appearance, but also on the way it was constructed. (shrink)

Logicians have often suggested that the use of Euler-type diagrams has influenced the idea of the quantification of the predicate. This is mainly due to the fact that Euler-type diagrams display more information than is required in traditional syllogistics. The paper supports this argument and extends it by a further step: Euler-type diagrams not only illustrate the quantification of the predicate, but also solve problems of traditional proof theory, which prevented an overall quantification of the predicate. Thus, (...) Euler-type diagrams can be called the natural basis of syllogistic reasoning and can even go beyond. In the paper, these arguments are presented in connection with the book Nucleus Logicae Weisaniae by Johann Christian Lange from 1712. (shrink)

Logicians commonly speak in a relatively undifferentiated way about pre-euler diagrams. The thesis of this paper, however, is that there were three periods in the early modern era in which euler-type diagrams (line diagrams as well as circle diagrams) were expansively used. Expansive periods are characterized by continuity, and regressive periods by discontinuity: While on the one hand an ongoing awareness of the use of euler-type diagrams occurred within an expansive period, after a subsequent phase of regression (...) the entire knowledge about the systematic application and the history of euler-type diagrams was lost. I will argue that the first expansive period lasted from Vives (1531) to Alsted (1614). The second period began around 1660 with Weigel and ended in 1712 with lange. The third period of expansion started around 1760 with the works of Ploucquet, euler and lambert. Finally, it is shown that euler-type diagrams became popular in the debate about intuition which took place in the 1790s between leibnizians and Kantians. The article is thus limited to the historical periodization between 1530 and 1800. (shrink)

This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic (...) and introduce an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences. (shrink)

The evolution of Euler diagrams is examined from Euler's original system through the modifications made by Venn and Peirce. It is shown that these modifications were motivated by an attempt to increase the expressivity of the diagrams, but that a side effect of these modifications was a loss of the visual clarity of Euler's original system. Euler's original system is reconstructed from a modern, logical point of view. Formal semantics and rules of inference are provided for (...) this reconstruction of Euler's system, and basic logical properties are proved. (shrink)

Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions (...) as in the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem. (shrink)

In this paper we argue that there were several currents, ideas and problems in 19th-century logic that motivated John Venn to develop his famous logic diagrams. To this end, we first examine the problem of uncertainty or over-specification in syllogistic that became obvious in Euler diagrams. In the 19th century, numerous logicians tried to solve this problem. The most famous was the attempt to introduce dashed circles into Euler diagrams. The solution that John Venn developed for this problem, (...) however, came from a completely different area of logic: instead of orienting to syllogistic like Euler diagrams, Venn applied Boolean algebra to improve visual reasoning. Venn’s contribution to solving the problem of elimination also played an important role. The result of this development is still known today as the ‘Venn Diagram’. (shrink)

Although a number of authors have used diagrams extensively in their studies of Navya-Nyāya, they have done so to explain and illustrate concepts, not with the goal of reasoning with the diagrams themselves. Adherents of diagrammatic reasoning have made claims for its potential by pointing to key structural correspondences between diagrams and logical concepts, arguably lacking in sentential representations, and describing these relations using concepts such as “well matchedness” and “iconicity”. A canonical example of this iconicity is the use of (...) Euler diagrams to depict categorical syllogisms. Since the meaning of expressions in Indian logic differs in so many important ways from logic in the Western tradition, the use or adaptation of diagrams developed in the latter would seem to preclude iconicity. Thus, the development of diagrams which reflect the nature of inference in Navya-Nyāya, which centres on the anumāna inference schema, is motivated. In this paper we extend Ganeri’s method of depicting the Vaiśeṣika ontology with graphs to include syntax intended to expose the nature of anumāna. The diagrams are given a formal basis: i.e. abstract syntax, inference rules defined abstractly and a graph-theoretic semantics. These are the first formalised logical diagrams that aim to reflect the nature of the anumāna inference. This paper lays the way for further work in extending the formalism to cover more of Navya-Nyāya, and in exploring a dialogue between properties of the formalism and of Navya-Nyāya. (shrink)

In this paper, we analyze and discuss Schopenhauer’s n-term diagrams for eristic dialectics from a graph-theoretical perspective. Unlike logic, eristic dialectics does not examine the validity of an isolated argument, but the progression and persuasiveness of an argument in the context of a dialogue or even controversy. To represent these dialogue situations, Schopenhauer created large maps with concepts and Euler-type diagrams, which from today’s perspective are a specific form of graphs. We first present the original method with Euler-type (...) diagrams, then give the most important graph-theoretical definitions, then discuss Schopenhauer’s diagrams graph-theoretically and finally give an example of how the graphs or diagrams can be used to analyze dialogues. (shrink)

It is of common use in modern Venn diagrams to mark a compartment with a cross to express its non-emptiness. Modern scholars seem to derive this convention from Charles S. Peirce, with the assumption that it was unknown to John Venn. This paper demonstrates that Venn actually introduced several methods to represent existentials but felt uneasy with them. The resistance to formalize existentials was not limited to diagrammatic systems, as George Boole and his followers also failed to provide a satisfactory (...) symbolic representation for them. This difficulty points out issues that are inherent to the very nature of existentials. This paper assesses the various methods designed for the representation of existential statements with Venn diagrams. First, Venn’s own attempts are discussed and compared with other solutions proposed by his contemporaries and successors, notably Lewis Carroll and Peirce. Since disjunctives hold an important role in an effective representation of existentials, their representation is also discussed. Finally, recent methods for the diagrammatic representation of existing individuals, rather than mere existence, are surveyed. (shrink)

What is the primacy of logic? Concepts, judgments, or inferences? Whereas representationalists traditionally argue for a primacy of the conceptual, rationalists, referring to the context principle and the use theory of meaning, consider judgments and inferences to be primary. This dispute also seems to be applicable to logic diagrams: Whereas “Euler-type diagrams” are actually only for judgments and inferences, “concept diagrams” represent ontologies by using concepts. With reference to Schopenhauer, the paper develops a position called “rational representationalism.” According to (...) this point of view, the question of primacy is decided by analyzing the functions of the logic principles: For the explanation of logic and language, concepts are primary, but for understanding it is judgments. The mediation between intuitive representation and logical rationality is ensured by concept diagrams. (shrink)

Logic, the discipline that explores valid reasoning, does not need to be limited to a specific form of representation but should include any form as long as it allows us to draw sound conclusions from given information. The use of diagrams has a long but unequal history in logic: The golden age of diagrammatic logic of the 19th century thanks to Euler and Venn diagrams was followed by the early 20th century's symbolization of modern logic by Frege and Russell. (...) Recently, we have been witnessing a revival of interest in diagrams from various disciplines - mathematics, logic, philosophy, cognitive science, and computer science. This book aims to provide a space for this newly debated topic - the logical status of diagrams - in order to advance the goal of universal logic by exploring common and/or unique features of visual reasoning. (shrink)

It is little known that Schopenhauer (1788–1860) made thorough use of Euler diagrams in his works. One specific diagram depicts a high number of concepts in relation to Good and Evil. It is, hence, uncharacteristic as logicians of that time seldom used diagrams for more than three terms (the number demanded by syllogisms). The objective of this paper is to make sense of this diagram by explaining its function and inquiring whether it could be viewed as an (...) early serious attempt to construct complex diagrams. (shrink)

Spider diagrams are based on Euler and Venn/Peirce diagrams, forming a system which is as expressive as monadic first order logic with equality. Rather than being primarily intended for logicians, spider diagrams were developed at the end of the 1990s in the context of visual modelling and software specification. We examine the original goals of the designers, the ways in which the notation has evolved and its connection with the philosophical origins of the logical diagrams of Euler, Venn (...) and Peirce on which spider diagrams are based. Using Peirce’s concepts and classification of signs, we analyse the ways in which different sign types are exploited in the notation. Our hope is that this analysis may be of interest beyond those readers particularly interested in spider diagrams, and act as a case study in deconstructing a simple visual logic. Along the way, we discuss the need for a deeper semiotic engagement in visual modelling. (shrink)

Investigamos a lógica de Leonhard Euler com ênfase no papel dos “diagramas de Euler”. Concluímos que os referidos diagramas constituem um instrumento intuitivo, embora não sistemático, para determinar validade na silogística tradicional, isto é, a silogística munida de implicações conversacionais. Nisto, contrastam-se com os diagramas de Venn que constituem um instrumento sistemático, porém menos intuitivo, para determinar validade numa silogística mais voltada para os fundamentos da matemática moderna.: We investigate the logic of Leonhard Euler, giving emphasis to (...) the role of “Euler diagrams”. We conclude that these diagrams are an intuitive, non-systematic instrument for determining validity in the traditional syllogistic, that is, the syllogistic furnished with conversational implications. In this regard, they contrast with Venn diagrams which are a systematic, less intuitive instrument for determining validity in a syllogistic more appropriate for the foundation of modern mathematics. Keywords: History of Logic. Leonhard Euler. Euler Diagrams. Venn Diagrams. (shrink)

From the beginning of the 16th century to the end of the 18th century, there were not less than ten philosophers who focused extensively on Venn’s ostensible analytical diagrams, as noted by modern historians of logic (Venn, Gardner, Baron, Coumet et al.). But what was the reason for early modern philosophers to use logic or analytical diagrams? Among modern historians of logic one can find two theses which are closely connected to each other: M. Gardner states that since the Middle (...) Ages certain logic diagrams were used just in order to teach “dull-witted students”. Therefore, logic diagrams were just a means to an end. According to P. Bernhard, the appreciation of logic diagrams had not started prior to the 1960s, therefore the fact that logic diagrams become an end the point of research arose very late. The paper will focus on the question whether logic resp. analytical diagrams were just means in the history of (early) modern logic or not. In contrast to Gardner, I will argue that logic diagrams were not only used as a tool for “dull-witted students”, but rather as a tool used by didactic reformers in early modern logic. In predating Bernhard’s thesis, I will argue that in the 1820s logic diagrams had already become a value in themselves in Arthur Schopenhauer’s lectures on logic, especially in proof theory. (shrink)

Der Kardinal Nikolaus von Kues bzw. Nicolaus Cusanus (1401 - 1464) war der bedeutendste Philosoph des 15. Jahrhunderts und zugleich einer der einflussreichsten Kirchenpolitiker seiner Zeit. Neben seiner rastlosen Tätigkeit als Advokat von Fürsten und Päpsten, als Fürstbischof von Brixen und Kurienkardinal verfasste er philosophisch-theologische Schriften von historischem Rang. Ihn interessierten die großen Köpfe der europäischen Geistesgeschichte und die Intellektuellen seiner Zeit genauso wie die Alltagsprobleme der Gläubigen seines Tiroler Bistums. Besondere Beachtung verdient zudem die Tatsache, dass er einen Großteil (...) seiner Schriften in Dialogform verfasste. Die 14 Beiträge dieses Bandes zeigen in exemplarischer Weise, dass Nikolaus von Kues sowohl ein Denker im Dialog als auch ein dialogischer Denker war. (shrink)

Proceedings of an international conference held at the Herzog August Bibliothek Wolfenbèuttel in 2009.

Name der Zeitschrift: Das Mittelalter Jahrgang: 19 Heft: 1 Seiten: 72-85.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

1,857 adults rated the grandparental solicitude they received in childhood. Through a simple model based on the evolutionary concepts of ontogenetically differentiated reproductive strategy and paternity confidence, an ordered discriminative pattern of grandparental caregiving was predicted and confirmed by solid main effects, based on 603 complete cases. The maternal grandmother was the most caring. Unlike prevalent gender stereotypes, she was followed by the maternal grandfather, the paternal grandmother, and the paternal grandfather. The preferential grandparental solicitude was not influenced by residential (...) distance, grandparent age, and availability of other grandparents. A predicted higher correlation for male than for female progenitors between solicitude and phenotypic resemblance could be confirmed. (shrink)

Name der Zeitschrift: Kant-Studien Jahrgang: 110 Heft: 1 Seiten: 172-177.

Core sustainability issues concerning the governance of seeds revolve around knowledge aspects, such as intellectual property rights over genetic information or the role of traditional knowledge in plant breeding, seed production and seed use. While the importance of knowledge management for efficient and equitable seed governance has been emphasized in the scientific discourse on Seed Commons, knowledge aspects have not yet been comprehensively studied. With this paper, we aim to (i) to analyze the governance of knowledge aspects in both global (...) and local/regional Seed Commons, (ii) to highlight disconnections in knowledge governance between the local Seed Commons and global governance of plant genetic resources, and (iii) to investigate the contribution of knowledge commoning to environmentally sustainable and culturally adapted food systems. For this purpose, we will analyze knowledge governance by the International Seed Treaty (ITPGRFA) and by two local Seed Commons, a Philippine farmer-led network and a German organic-breeding association. We take the analytical lens of commoning, focusing on social practices rather than specific resources. The main challenges include finding institutional arrangements, which fruitfully integrate aspects from both traditional and scientific knowledge systems, taking into account the complex interrelation between knowledge-related, material and cultural aspects of seeds. (shrink)

This article has two purposes. First, it aims to present a detailed analysis of the argument of “recognition” or even of the “fight for recognition”, which Hegel uses in his fragments from Jena treating of the system of philosophy, especially of the philosophy of spirit. It will be necessary to determine precedently by means of an exact interpretation the content of that expressions, in order to criticize and to compare, his original significance in Hegel with the theoretical application made by (...) Honneth. The special aim hereby is to correct and to explain the function of the “fight for recognition”, even in its sharpening significance as a “fight for life and death”, with relation to the genesis of the state of right. Second, the figure of “recognition” is related to Hegel’s Philosophy of right in the version of 1820. In this work, recognition participates in the theory of “institutions” which is analyzed by Honneth as well. Apart from the merits gained through the endeavour to introduce a classic philosophical theory into a conception of justice in society, we feel obliged to state an error in Honneth‘s interpretation in so far as he understands an institution as a simple social community of communication. For that reason, a conclusion based on a critical commentary can only include a negative recommendation, that means, not to follow those traces of interpretation but rather to defend and to consult the texts originally composed by Hegel. (shrink)