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)

John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus (...) from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations. (shrink)

Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler in 1730, when he was trying to find the general term for the factorial function by means of an algebraic expression. Nevertheless, 70 years before, Pietro Mengoli (1626–1686) had computed such integrals for natural and half-integer exponents in his Geometriae Speciosae Elementa (1659) and Circolo(1672) and displayed the results in triangular tables. In particular, his new arithmetic–algebraic method allowed him to compute the quadrature of the (...) circle. The aim of this article is to show how Mengoli calculated the values of these integrals as well as how he analysed the relation between these values and the exponents inside the integrals. This analysis provides new insights into Mengoli’s view of his algorithmic computation of quadratures. (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)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during the (...) transitional period of his thought (1929-33). (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)

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)

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)

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)

Interpersonal sensitivity, particularly threat of potential exclusion, is a critical condition in borderline personality disorder (BPD) which impairs patients’ social adjustment. Current evidence-based treatments include group components, such as mentalization-based group therapy (MBT-G), in order to improve interpersonal functioning. These treatments additionally focus on the therapeutic alliance since it was discovered to be a robust predictor of treatment outcome. However, alliance is a multidimensional factor of group therapy, which includes the fellow patients, and may thus be negatively affected by the (...) exclusion-proneness of BPD patients. The aim of this pilot study was to examine the predictive value of threat of social exclusion for the therapeutic alliance in MBT-G. In the first part of the study, social exclusion was experimentally induced in 23 BPD inpatients and 28 healthy subjects using the Cyberball paradigm, a virtual ball tossing game. The evoked level of threat was measured with the Need-Threat Scale (NTS) which captures four dimensions of fundamental human needs, i. e. the need for belongingness, for self-esteem, for control, and for a meaningful existence. In the second part of the study, therapeutic alliance was measured on three dimensions, the therapists, the fellow patients and the group as a whole, using the Group-Questionnaire (GQ-D). BPD patients scored higher in their level of threat according to the NTS in both, the inclusion and the exclusion condition. The level of threat after exclusion predicted impairments of the therapeutic alliance in MBT-G. It was associated with more negative relationships, lower positive bonding and a lower positive working alliance with the fellow patients and lower positive bonding to the group as a whole whilst no negative prediction of the alliance to the therapists was found. Consequently, our translational study design has shown that Cyberball is an appropriate tool to use as an approach for clinical questions. We further conclude that exclusion-proneness in BPD is a critical feature with respect to alliance in group treatments. In order to neutralize BPD patients’ exclusion bias, therapists may be advised to provide an “inclusive stance”, especially in initial sessions. It is also recommendable to strengthen patient to patient relations. (shrink)

In order to evaluate the impact of contemporary grandparenting, the view should not be restricted to developmental achievement effects in grandchildren. Both child happiness and grandparent happiness are high-ranking goals with implications for public policy. The beneficial impact of grandparenting for risk families appears unequivocal, and modern life still encounters all kinds of unpredictable risks. Contemporary grandparental care is no evolutionary mismatch.

Theodor W. Adorno: Kant's Critique of Pure Reason . Hrsg. von Rolf Tiedemann. Übers. von Rodney Livingstone. Cambridge: Polity Press 2001 . xi/300 S. ISBN 0-80474-426-2. Lara Denis : Moral Self-Regard. Duties to Oneself in Kant's Moral Theory. New York & London: Garland Publishing, Inc. 2001. xvi/245 S. . 0-815-33967-4.

Le sujet de l’article est la situation actuelle de la recherche internationale sur le Cusain et ses perspectives pour l’avenir. Au cours du xxe siècle, la recherche internationale sur Nicolas de Cues a connu une expansion énorme. L’édition historique-critique de son œuvre est achevée. Aujourd’hui, il existe des centres sur le Cusain pas seulement en Allemagne, au Japon et aux États-Unis, mais aussi aux Pays-Bas, en Italie, en Argentine, en Russie et en France. à l’échelle mondiale, beaucoup de thèses de (...) doctorat sont écrites sur Nicolas de Cues, sans parler des nombreux articles le concernant. à l’occasion du 25e anniversaire de l’American Cusanus Society en mai 2008 à Kalamazoo, le philosophe américain Clyde Lee Miller a étudié la recherche sur Nicolas de Cues sous quatre angles différents que l’article reprend pour esquisser les futures perspectives de la recherche sur le Cusain. (shrink)

The article compares for the first time Luther‘s reflections on Islam with Cusanus‘s. Both thinkers didn‘t engage in Islam on their own initiative, but because they were prompted by political developments. Luther‘s writings on Islam are mostly authored in German. He addresses the public in the empire and tries to encourage Christians challenged in their Christians faith, especially those who are in Turkish captivity. Nicholas of Cusa addresses also Islamic receivers in his Cribratio Alkorani. Luther stresses the contrast between the (...) gospel of Jesus Christ and the message of Muhammad, whereas Cusanus tries to build theological bridges between Christianity and Islam. (shrink)