The paper explores the potential of Byzantine diagrams in syllogistic logic. Byzantine diagrams are originated by Byzantine scholars in the early modern period to use as tools for teaching and studying Aristotelian logic. This paper presents pioneering work on employing Byzantine diagrams for checking syllogistic validity through reduction.

In this paper, I explore general features of the “architecture” (relations of white space, diagram, and text on the page) of medieval manuscripts and early printed editions of Euclidean geometry. My focus is primarily on diagrams in the Arabic transmission, although I use some examples from both Byzantine Greek and medieval Latin manuscripts as a foil to throw light on distinctive features of the Arabic transmission. My investigations suggest that the “architecture” often takes shape against the backdrop of (...) an educational landscape. The constraints of the economic marketplace and cultural aesthetic ideals also appear to play a role in determining the “architecture” of both manuscripts and early printed editions. (shrink)

This paper analyses a hitherto unknown technique of using logic diagrams to create argument maps in eristic dialectics. The method was invented in the 1810s and -20s by Arthur Schopenhauer, who is considered the originator of modern eristic. This technique of Schopenhauer could be interesting for several branches of research in the field of argumentation: Firstly, for the field of argument mapping, since here a hitherto unknown diagrammatic technique is shown in order to visualise possible situations of arguments in (...) a dialogical controversy. Secondly, the art of controversy or eristic, since the diagrams do not analyse the truth of judgements and the validity of inferences, but the persuasiveness of arguments in a dialogue. (shrink)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of (...) most importance for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

The quotes in the sacro-profane florilegia have so far been neglected as documents for the 9th-century readership of the Greek novels. This article uses the quotes as intertextual links to the originals and reconstructs the excerption: mapped back onto the novels, the quotes highlight the excerptors’ points of interest and the patterns that connect them. Excerption is thus fully understood as reading practice. The quotes were collected not only because they could provide wisdom when decontextualised, but also because they played (...) a relevant role in the excerptors’ analysis and interpretation of the novels. The florilegia are therefore unique texts in revealing the experience of reading the novels in the 9thcentury. (shrink)

8 chapters are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

The way in which one understands information and concepts, and the way a student works to develop this, is an individual aspect of learning that cannot be universally defined as (at least manifested) the same for everyone. ‘Understanding’ is a broad term, and the way one achieves understanding is dependent on the way that material is presented. In this article, we argue that the philosophy of science can be important to nursing education—in particular, by showing that the way we imbue (...) understanding might depend on the meaning of ‘understanding’. Diagrams and concept maps are meant to guide newly formed knowledge and connections to develop proper thinking (e.g., the order in which nursing students must prioritize data) that a student requires in the field. We argue that whether or not an image/diagram/concept map confers understanding will depend on both what the object is and what we mean by ‘understanding’. (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 ontology of artefacts in Byzantine philosophy is still a terra incognita. One way of mapping this unexplored territory is to delve into Michael of Ephesus’ commentary on Aristotle’s Metaphysics. Written around 1100, this commentary provides a detailed interpretation of the most important source for Aristotle’s ontological account of artefacts. By highlighting Michael’s main metaphysical tenets and his interpretation of key-passages of the Aristotelian work, this study aims to reconstruct Michael’s ontology of artefacts and present it as one instance, (...) which is perhaps exemplary, of the Byzantine ontology of artefacts. In particular, the study shows that this commentary holds a definite position on the nature of artefacts, according to which they are neither substances nor hylomorphic compounds. Indeed, artefacts lack a form altogether and their forms exist only in thought. As a result, Michael’s commentary provides an ontological interpretation of artefacts as accidental beings, i.e., as matter which acquires a mere property as opposed to a substantial form. While such an interpretation shows originality when compared to the Aristotelian text, it also indicates adherence to the reading established by Alexander of Aphrodisias, despite important departures concerning the status of natural forms. (shrink)

There is substantial evidence from many domains that visual representations aid various forms of cognition. We aimed to determine whether visual representations of argument structure enhanced the acquisition and development of critical thinking skills within the context of an introductory philosophy course. We found a significant effect of the use of argument diagrams, and this effect was stable even when multiple plausible correlates were controlled for. These results suggest that natural⎯and relatively minor⎯modifications to standard critical thinking courses could provide (...) substantial increases in student learning and performance. (shrink)

It is well known that the historical texts composed under the Macedonian dynasty (Theophanes Continuatus, Genesios, but also Leo the Deacon, Manuel protospatharios, John Skylitzes or even Michael Psellos) display certain element, which can be seen as attempts to make a clean break with the past; the formalist style of unbroken historical narrative was largely rejected in favour of historical biography in which the influence of rhetorical methods is self-evident. It is not known which criteria tipped the balance in favour (...) of such a development, though we would do well to consider both the influence of Antiquity, and the close ties that bound biography to hagiography, which from the empire's very beginnings (Vita Constantini) also served as a uniquely Byzantine mode of promoting the image of the emperor. It may be legitimate to speak of a new type of historical writing which is specifically designed for the extremely beautified career of its subject and constitutes a contemporary movement which manifested itself in various cultural contexts. (shrink)

Saint Basil the Great wrote one of the most important and widely acknowledged Eucharistic texts in the Eastern Orthodox Church, a liturgical anaphora that bears his name. Before the dawn of the second millennium, this was the main Eucharistic text used in Constantinople and in the territories under its authority. In the modern time of digital media, the liturgical research methods have been improved by using the Thesaurus Linguae Graecae database. The emergence of patristic and liturgical texts in this novel (...) format was able to revolutionise the field of Comparative Liturgics, allowing to very quickly find the possible internal clues, which prove the basilian authorship of the anaphora bearing St. Basil's name. By the quick and accurate computer scanning of a part of the post-Sanctus prayer, a relatively complete picture of the relatedness of vocabulary, the author's theological and ascetical nomenclature, the recurring thought patterns, parallel passages and hapax legomena, rare terms and their frequency, is obtained. CONTRIBUTION: The present article aims to demonstrate that Theology has to be connected to modern research methods because the findings of such an academic approach are helpful for the development of ecumenical and interreligious dialogue, that is so well promoted by HTS Teologiese Studies/Theological Studies. (shrink)

Les travaux archéologiques sur l'apparition et le développement des édifices chrétiens en Jordanie sont déjà anciens et, dans ce pays longtemps préservé des destructions jusqu'à une époque récente, les vestiges, surtout connus par leurs belles mosaïques, offrent une exceptionnelle documentation à l'historien de l'antiquité tardive, de l'époque byzantine et de l'époque omayyade. Comme le dit N. Duval dans sa préface, la Jordanie est un terrain privilégié. Or, tandis que des répertoires et des études d'ensemble sont disponibles pour la Syrie-Palestine (...) et pour le Liban, on en manquait pour la Jordanie. L'ouvrage d'Anne Michel fait plus que combler ce manque : il nous offre une somme exhaustive des vestiges connus à ce jour. (shrink)

Die nach den Konventionen der Alten Geschichte gemeinhin mit dem Beginn der Regierungszeit des Kaisers Diokletian (284 n. Chr.) einsetzende Epoche der Spätantike (die Papyrologen sprechen seit diesem Zeitpunkt dagegen zumeist von der „Byzantinischen Zeit“) zeitigte eine Fülle von administrativen und fiskalischen Reformen, die sich auch in Änderungen der Zeitrechnung in den Provinzen des Reiches niederschlugen. Am besten studieren lassen sich diese Auswirkungen in der Provinz Ägypten, wo mit den Papyri eine Fülle von datierten administrativen Aufzeichnungen sowie öffentlichen und privaten (...) Urkunden zur Verfügung steht. Hier manifestiert sich in diesen Phänomenen zugleich die zunehmende Angleichung Ägyptens an die administrativen und fiskalischen Strukturen in den anderen Provinzen des Reiches, wenn man so will die „Romanisierung“ Ägyptens, deren Ergebnis das Verschwinden zahlreicher Besonderheiten war, durch welche sich in der frühen und hohen Kaiserzeit die Verhältnisse in Ägypten von denen in den übrigen Teilen des Reiches signifikant unterschieden haben, wenngleich dieser Umstand es nach der Auffassung des Rez. keinesfalls rechtfertigt, der Provinz als solcher einen Sonderstatus im römischen Reichsverband zuzuschreiben, wie dies eine traditionelle Richtung der althistorischen Forschung lange Zeit getan hat und teilweise immer noch tut. Die weltweit besten Kenner der diversen chronologischen Systeme des byzantinischen Ägypten und der mit ihnen zusammenhängenden, z. T. äußerst verwickelten Probleme sind zweifellos Roger S. Bagnall und Klaas A. Worp (B&W), die bereits vor 27 Jahren mit den „Chronological systems of Byzantine Egypt“ (CSBE. Studia Amstelodamensia, VIII. Zutphen 1978) eine Summe der aus ihrer Editionstätigkeit erwachsenden intensiven Beschäftigung mit den chronologischen Problemen byzantinischer Urkunden aus Ägypten vorgelegt und nicht nur dem Editor solcher Dokumente im besonderen, sondern auch der historischen und papyrologischen Forschung im allgemeinen ein unentbehrliches Hilfsmittel an die Hand gegeben haben. Schon ein Jahr später folgten mit den aus ihrer Feder stammenden „Regnal Formulas in Byzantine Egypt“ (RFBE. Bulletin of the American Society of Papyrologists, Supplement 2. Missoula/Montana 1979) und 1987 mit den zusammen mit A. Cameron und S. Schwartz verfaßten „Consuls of the Later Roman Empire“ (CLRE) zwei weitere solcher unverzichtbaren Hilfsmittel, die den auf dem Gebiet der epigraphischen und papyrologischen Hinterlassenschaft der Spätantike tätigen Forschern die Arbeit beträchtlich erleichtern. In Anbetracht des in der seit dem Erscheinen von CSBE vestrichenen Zeit enorm angewachsenen Quellenmaterials, welches nicht nur vielfältige neue chronologische Probleme aufwarf, sondern auch bereits bekannte in neuem Licht erscheinen ließ – einen Prozeß, den B&W mit zahlreichen Artikeln zur Chronologie der Papyrusdokumente des byzantinischen Ägypten (siehe die Bibliographie in CSBE) gewissermaßen kommentierend begleitet haben – bedarf diese Neuauflage der CSBE sicherlich keiner Rechtfertigung. Auch wenn mit der Volltextdatenbank der „Duke Data Bank of Documentary Papyri“ (DDBDP) sowie mit dem „Heidelberger Gesamtverzeichnis der griechischen Papyrusurkunden Ägyptens“ (HGV) nunmehr – zum Zeitpunkt der Erstauflage der CSBE noch nicht existente – äußerst nützliche digitale Ressourcen zur Verfügung stehen, die die Sammlung und chronologische Auswertung der Dokumente bedeutend erleichtern (worauf B&W in der Einleitung zu CSBE zu recht hinweisen [viii]), so ist dies zweifellos immer noch eine mühe- und entsagungsvolle Kärrnerarbeit, für die den beiden Autoren nicht hoch genug gedankt werden kann. Hinzu kommt, daß sich diese Arbeit nicht nur auf die Sammlung der Dokumente und der ggfs. seit ihrer Edition erfolgten Berichtigungen beschränkt, sondern B&W in zahlreichen Fällen auch selbst Lesungen überprüft und berichtigt haben. (shrink)

This article analyzes the organization and functioning of the Byzantine salt monopoly. Saltworks were owned by the state, but some were also owned by monasteries and laymen. State-owned saltworks were not run as state enterprises; rather, their operations were auctioned to private individuals by competitive bidding conducted by provincial governors. Auctions of saltworks could be combined with concessions of the salt sales tax or other rights to sources of local revenue (e. g. fisheries, wharfage). Concessionaires of saltworks often were (...) obligated to deliver salt free of charge and tax in perpetuity to influential monasteries because of long-standing local customs. The state had a fiscal, not a market monopoly on salt as the price of salt was determined by market forces. Still, topography and seasonality of production and trade could provide opportunities for exploitation of inter-spatial and intertemporal price differentials. Disaggregation of the salt market into definable producer–wholesaler, wholesaler–retailer, and retailer–consumer submarkets evinces the varying impact that market structure can have on the degree of competition and on the firms' pricing strategies. (shrink)

The monograph under review, written as a doctorate in 1999 and later revised and supplemented, offers a searching analysis of the form and function of rhetoric in the Pascal-Pentecost cycle of the 12th century homilist Cyril of Turov, together with an examination of the Byzantine sources. Lunde scrutinizes eight festal homilies for the period from Palm Sunday to the Sunday before Pentecost, using the edition by Igor' Erëmin [TODL 11–13,15 (1955–1958), reprinted as Literaturnoe nasledie Kirilla Turovskogo: Archeologičeskij obzor I (...) izdanie tekstov in 1989]. All Slavonic fragments are accompanied by English translations, predominantly drawn from Simon Franklin's Sermon and Rhetoric of Kievan Rus' [1991, reviewed by Francis Thomson in Slavica Gandensia 19], with minor alterations (e.g. p. 266, note 11) and critical remarks (e.g. p. 180, note 37). The study is positioned along the line that marks a shift in focus from philological and historical commentary towards the literary analysis of Cyril's sermons. Lunde's approach is clearly indebted to various developments in modern literary criticism, from structural linguistics to discourse analysis and communication theory. Questions of dating, attribution, and textual criticism are consciously left aside. (shrink)

This paper refers to habitation in the Byzantine Empire from the 7th to the 11th centuries and attempts a reappraisal of the patterns used to describe, evaluate and interpret the distribution of archaeological remains. Based on the study of a region in western Greek mainland, several contradictions between the historical and the archaeological evidence on settlement are being discussed; those reveal the prevalence of dispersed rather than nuclear patterns of habitation in the area. These patterns are further discussed within (...) the context of medieval settlement around the Mediterranean. The physical form of Middle Byzantine Episcopal sees is sought for while the term ‘κάστρον’ is examined against the “archaeological imaginary” of 6th–11th c. Byzantine fortified settlements. It is concluded that the contextualization of archaeological and historical evidence for Middle Byzantine settlements in Southern Epiros demonstrates instead a rather flexible organization of habitation; episcopal sees do not seem to have corresponded to any clear distinct type of settlement. It is therefore suggested that during periods of rapid social change expressed through a transformation of habitation, specific forms or sizes of settlements cannot be correlated to specific functions or other political or socioeconomic activities. There should rather be more of “hybrid” settlement formations of changing nature and mixed qualities. Given the dynamic development pattern of Byzantine habitats, there is room for such “open” analytical categories in the archaeology of settlements, in order to facilitate descriptions of find distributions which do not conform to the available theoretical patterns. These new categories could further lead to an emergence of new interpretation schemes bringing out the spatial differentiation and uniqueness which seems to have been very much the case in the evolution of Byzantine provincial settlement patterns. (shrink)

On September 24, 787, the works of the VII Ecumenical Synod were opened in the 'Saint Sophia' Church in Nicaea, after the first attempt, on August 7, 786, had failed. Although the nominal presidency was held by the legates of Pope Adrian I, the effective presidency was exercised by Patriarch Tarasios of Constantinople. A skilful church diplomat, with experience, gained as an imperial secretary and a remarkable theologian whose authority was imposed even during his election as a patriarch amongst the (...) laity, Tarasios meticulously and clairvoyantly prepared for the deployment of the Nicene synod. This is noticeable from the agreement made with the papal legates regarding the reading of the letters of Pope Adrian I whose content directly concerned the persona of the patriarch, agreeing to omit those compromising paragraphs, from the procedure of re-welcoming in the communion of the church of some former iconoclastic bishops, by correctly managing the resistance of the monks to whom he gave satisfaction regarding the patristic and traditional argumentation of the cult of the holy icons and by rejecting point-by-point the dogmatic decision of the iconoclastic synod of Hieria, a rejection of which the patriarch Tarasie is in all probability the author. Satisfied with the success of the synod, whose craftsman he indeed was, Patriarch Tarasios was able to communicate to both Pope Adrian I and the emperors and clergy of Constantinople that the unity of the Church residing in Christ had been restored and that the place in the church and due honour of the holy icons had been restored through the synodal decision of 302 participants. The success of the Seventh Ecumenical Council is unequivocally because of the tactful and competent preparation and management of Patriarch Tarasios. CONTRIBUTION: The perspective we promote on the events highlighted in the study, could contribute to unblocking the theological dialogue between Orthodox and Catholics on the issue of papal primacy, the study thus promotes HTS as an important forum for mediating interfaith dialogue. (shrink)

Dans la Delphes de l’époque protobyzantine, l’expression artistique et artisanale fait preuve d’une vive interaction entre les Delphiens et leurs homologues de villes proches ou lointaines. La céramique locale plus précisément, bien étudiée et circonscrite dans le temps, couvre un large éventail de produits de bonne qualité, déjà au ive s. de n. è., mais surtout aux vie et viie s. Quant à la céramique importée, amphores, lampes, sigillées et céramique peinte sont examinées ici pour la première fois en détail, (...) dans le souci de compléter l’image des objets en terre cuite utilisés dans cette ville moyenne de la Grèce Centrale. Les rapports commerciaux entre Delphes et certaines autres villes de la péninsule grecque et des îles y sont étudiés, en même temps que la présence dans toutes ces villes des grandes productions méditerranéennes. Cette étude nous permet d’inscrire Delphes dans un cadre commercial précis, la rapprochant des villes de la Grèce Centrale et du Péloponnèse où les produits africains l’emportent sur les importations micrasiatiques. (shrink)

Many systems of logic diagrams have been offered both historically and more recently. Each of them has clear limitations. An original alternative system is offered here. It is simpler, more natural, and more expressively and inferentially powerful. It can be used to analyze not only syllogisms but arguments involving relational terms and unanalyzed statement terms.

Diagrams are ubiquitous in mathematics. From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice. Long overlooked by philosophers focused on foundational and ontological issues, these roles have come to receive attention in the past two decades, a trend in line with the growing philosophical interest in actual mathematical practice.

This article presents diagrams developed from the insights of three middle school children with limited mobility about their experiences navigating social and spatial relations in their home, school and neighbourhoods. The paper explores the concept of assemblage as well as operationalising the Deleuzian idea of the diagram. The diagrams we produce are developed in connection with dominant idealisations of neighbourhood and home range that function in North America to choreograph children's progression from infancy through adolescence. We undertake this (...) diagramming in order to immediately and affectively illustrate the limits and possibilities of the challenges these young people face. (shrink)

Throughout the history of the philosophy of science, theories have been linked to formulas as a privileged representational format. In this paper, following, I defend a semantic-representational conception of theories, where theories are identified with sets of scientific re-presentations by virtue of their epistemic potential and independently of their format. To show the potential of this proposal, I analyze as a case study the use of phase diagrams in statistical mechanics to convey in a semantically consistent and syntactically correct (...) way theoretical principles such as Liouville’s theorem. I conclude by defending this philosophical position as a tool to show the enormous representational richness underlying scientific practices. (shrink)