A common misunderstanding is that there is something logically amiss with the classical square of opposition, and that the problem is related to Aristotle’s and medieval philosophers’ rejection of empty terms. But [Parsons 2004] convincingly shows that most of these philosophers did not in fact reject empty terms, and that, when properly understood, there are no logical problems with the classical square. Instead, the classical square, compared to its modern version, raises the issue of the existential import of words (...) like all; a semantic issue. I argue that the modern square is more interesting than Parsons allows, because it presents, in contrast with the classical square, notions of negation that are ubiquitous in natural languages. This is an indirect logical argument against interpreting all with existential import. I also discuss some linguistic matters bearing on the latter issue. (shrink)

© Springer International Publishing AG, part of Springer Nature 2018. Nearly all squares of opposition found in the literature represent both the Aristotelian relations and the duality relations, and exhibit a very close correspondence between both types of logical relations. This paper investigates the interplay between Aristotelian and duality relations in diagrams beyond the square. In particular, we study a Buridan octagon, a Lenzen octagon, a Keynes-Johnson octagon and a Moretti octagon. Each of (...) these octagons is a natural extension of the square, both from an Aristotelian perspective and from a duality perspective. The results of our comparative analysis turn out to be highly nuanced. (shrink)

In the recent debate on future contingents and the nature of the future, authors such as G. A. Boyd, W. L. Craig, and E. Hess have made use of various logical notions, such as the Aristotelian relations of contradiction and contrariety, and the ‘open future square of opposition.’ My aim in this paper is not to enter into this philosophical debate itself, but rather to highlight, at a more abstract methodological level, the important role that Aristotelian (...) diagrams can play in organizing and clarifying the debate. After providing a brief survey of the specific ways in which Boyd and Hess make use of Aristotelian relations and diagrams in the debate on the nature of the future, I argue that the position of open theism is best represented by means of a hexagon of opposition. Next, I show that on the classical theist account, this hexagon of opposition ‘collapses’ into a single pair of contradictory statements. This collapse from a hexagon into a pair has several aspects, which can all be seen as different manifestations of a single underlying change. (shrink)

The aim of this paper is to elucidate the relationship between Aristotelian conceptual oppositions, commutative diagrams of relational structures, and Galois connections.This is done by investigating in detail some examples of Aristotelian conceptual oppositions arising from topological spaces and similarity structures. The main technical device for this endeavor is the notion of Galois connections of order structures.

The square of opposition is a diagram related to a theory of oppositions that goes back to Aristotle. Both the diagram and the theory have been discussed throughout the history of logic. Initially, the diagram was employed to present the Aristotelian theory of quantification, but extensions and criticisms of this theory have resulted in various other diagrams. The strength of the theory is that it is at the same time fairly simple and quite rich. The theory of oppositions (...) has recently become a topic of intense interest due to the development of a general geometry of opposition (polygons and polyhedra) with many applications. A congress on the square with an interdisciplinary character has been organized on a regular basis (Montreux 2007, Corsica 2010, Beirut 2012, Vatican 2014, Rapa Nui 2016). The volume at hand is a sequel to two successful books: The Square of Opposition - A General Framework of Cognition, ed. by J.-Y. Béziau & G. Payette, as well as Around and beyond the Square of Opposition, ed. by J.-Y. Béziau & D. Jacquette, and, like those, a collection of selected peer-reviewed papers. The idea of this new volume is to maintain a good equilibrium between history, technical developments and applications. The volume is likely to attract a wide spectrum of readers, mathematicians, philosophers, linguists, psychologists and computer scientists, who may range from undergraduate students to advanced researchers. (shrink)

A general theory of logical oppositions is proposed by abstracting these from the Aristotelian background of quantified sentences. Opposition is a relation that goes beyond incompatibility (not being true together), and a question-answer semantics is devised to investigate the features of oppositions and opposites within a functional calculus. Finally, several theoretical problems about its applicability are considered.

The traditional square of opposition is generalized and extended to a cube of opposition covering and conveniently visualizing inter-sentential oppositions in relational syllogistic logic with the usual syllogistic logic sentences obtained as special cases. The cube comes about by considering Frege–Russell’s quantifier predicate logic with one relation comprising categorical syllogistic sentence forms. The relationships to Buridan’s octagon, to Aristotelian modal logic, and to Klein’s 4-group are discussed.GraphicThe photo shows a prototype sculpture for the cube.

In Aristotelian logic, categorical propositions are divided in Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative. Possible relations between two of the mentioned type of propositions are encoded in the square of opposition. The square expresses the essential properties of monadic first order quantification which, in an algebraic approach, may be represented taking into account monadic Boolean algebras. More precisely, quantifiers are considered as modal operators acting on a Boolean algebra and the square of opposition (...) is represented by relations between certain terms of the language in which the algebraic structure is formulated. This representation is sometimes called the modal square of opposition. Several generalizations of the monadic first order logic can be obtained by changing the underlying Boolean structure by another one giving rise to new possible interpretations of the square. (shrink)

The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the (...) class='Hi'>Aristotelian relations in Boolean terms and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)

The argument of this paper is that the Aristotelian analysis of justice and related concepts provides the best framework for understanding the structure and importance of justice in Dante’s Commedia. After giving a synopsis of the principle features of Aristotle’s account of justice in Book 5 of the Nicomachean Ethics, I consider a few scenes from the Inferno, the Purgatorio, and the Paradiso, showing how the punishments and rewards Dante describes are based on the Aristotelian analysis of justice. (...) Finally, I show that Dante, following Aristotle’s views on the metaphysics of the human person, recognizes that argument alone is inadequate to the task of educating the reader on the proper care of the soul, and that Dante’s effective use of images, makes the abstract Aristotelian concept of justice vivid and attractive, and its opposite repulsive. (shrink)

The argument of this paper is that the Aristotelian analysis of justice and related concepts provides the best framework for understanding the structure and importance of justice in Dante’s Commedia. After giving a synopsis of the principle features of Aristotle’s account of justice in Book 5 of the Nicomachean Ethics, I consider a few scenes from the Inferno, the Purgatorio, and the Paradiso, showing how the punishments and rewards Dante describes are based on the Aristotelian analysis of justice. (...) Finally, I show that Dante, following Aristotle’s views on the metaphysics of the human person, recognizes that argument alone is inadequate to the task of educating the reader on the proper care of the soul, and that Dante’s effective use of images, makes the abstract Aristotelian concept of justice vivid and attractive, and its opposite repulsive. (shrink)

Pendant l’ete de 1928 Heidegger a offert un seminaire sur le troisieme livre de la Physique d’Aristote et donc sur l’explication aristotelicienne de la nature du mouvement. La derniere seance de ce cours, qui eut lieu le 25 juillet, est d’une grande importance parce que c’est a cette occasion que Heidegger va au livre neuf de la Metaphysique pour essayer de comprendre la notion ontologique qui est a la base de l’interpretation aristotelicienne du mouvement : l’energeia. Mais dans les protocoles (...) de ce seminaire qui se trouvent parmi les papiers de Heidegger et qui ont ete publies recemment dans le volume 83 de la Gesamtausgabe, la seance du 25 juillet se trouve absente. Ce fait a conduit l’editeur a conclure que le seminaire avait pris fin le 23 juillet, sans s’apercevoir donc que la conclusion du seminaire manquait. Il existe heureusement une transcription preservee parmi les papiers de l’etudiante de Heidegger, Helene Weiss, et accessible aujourd’hui dans les archives de l’universite de Stanford. Cette transcription montre que la derniere session eut bien lieu le 25 juillet et nous offre la lecture heideggerienne de Metaphysique IX qui ne se trouve pas dans la version de la Gesamtausgabe. C’est dans le contexte de cette lecture que Heidegger fait la declaration etonnante qui nous concerne ici : ≪Dans la derniere instance, la Metaphysique Θ revient a Platon ; la priorite de l’energeia est fondamentalement la meme chose que l’epekeina des Idees. ≫La première tâche que je me propose ici sera d’expliquer cette déclaration qui suggère une relation tres etroite, ou meme une identite, entre la notion aristotelicienne de l’energeia comme ayant une priorite vis‑a‑vis de la dunamis et la notion platonicienne de l’Idee du Bien comme etant epekeina de l’ousia. Pour cette explication je ferai appel non seulement au contexte du seminaire de 1928, mais aussi aux textes plus tardifs comme les Beitrage et les cours sur Nietzsche dans lesquels Heidegger semble presupposer et développer sa déclaration de 1928. Ma seconde tâche sera de comparer cette thèse heideggerienne a la tentative de Gadamer de surmonter l’opposition traditionnelle entre les ontologies de Platon et d’Aristote en faisant appel a l’idee du bien chez les deux. Cette tentative se trouve dans le texte Die Idee des Guten zwischen Plato und Aristoteles. La comparaison que j’entreprends ici va montrer certaines affinites entre les interpretations de l’Idee du Bien chez Heidegger et Gadamer, mais aussi de profondes differences qui vont determiner leurs differents projets philosophiques. (shrink)

This paper studies John Buridan's octagons of opposition for the de re modal propositions and the propositions of unusual construction. Both Buridan himself and the secondary literature have emphasized the strong similarities between these two octagons (as well as a third one, for propositions with oblique terms). In this paper, I argue that the interconnection between both octagons is more subtle than has previously been thought: if we move beyond the Aristotelian relations, and also take Boolean considerations (...) into account, then the strong analogy between Buridan's octagons starts to break down. These differences in Boolean structure can already be discerned within the octagons themselves; on a more abstract level, they lead to these two octagons having different degrees of Boolean complexity (i.e. Boolean closures of different sizes). These results are obtained by means of bitstring analysis, which is one of the key tools from contemporary logical geometry. Finally, I argue that this historical investigation is directly relevant for the theoretical framework of logical geometry, and discuss how it helps us to address certain open questions in this framework. (shrink)

Peters and Westerståhl (Quantifiers in Language and Logic, 2006), and Westerståhl (New Perspectives on the Square of Opposition, 2011) draw a crucial distinction between the “classical” Aristotelian squares of opposition and the “modern” Duality squares of opposition. The classical square involves four opposition relations, whereas the modern one only involves three of them: the two horizontal connections are fundamentally distinct in the Aristotelian case (contrariety, CR vs. subcontrariety, SCR) but express the same Duality (...) relation of internal negation (SNEG). Furthermore, the vertical relations in the classical square are unidirectional, whereas in the modern square they are bidirectional. The present paper argues that these differences become even bigger when two more operators are added, namely the U ( ${{\equiv} {\rm A}\,{\vee} \,{\rm E} }$ , all or no) and Y ( ${\equiv{\rm I} \,{\wedge} \,{\rm O}}$ , some but not all) of Blanché (Structures Intellectuelles, 1969). In the resulting Aristotelian hexagon the two extra nodes are perfectly integrated, yielding two interlocking triangles of CR and SCR. In the duality hexagon by contrast, they do not enter into any relation with the original square, but constitute a independent pair of their own, since they are their own SNEGs. Hence, they not only stand in a relation of external NEG, but also in one of duality. This reflexive nature of the SNEG will be shown to result in defective monotonicity configurations for the pair, namely the absence of right-monotonicity (on the predicate argument). In the second half of the paper, we present an overview of those hexagonal structures which are both Aristotelian and Duality configurations, and those which are only Aristotelian. (shrink)

The relationship between fear and courage has been discussed in terms of opposite though mutually involving notions. However, their link has not been inquired extensively. Recently, new light has been shed on the topic thanks to recent empirical evidence within emotion theories that stress the role played by perception and/or cognition in the experience of fear, as well as the role played by the “emotional virtue” of courage in fear regulation. Questions arise whether fear has a fundamentally perceptual structure or (...) is a biologically-grounded natural kind, and whether such an emotion-related virtue as courage is intrinsically or extrinsically related to fear. This paper considers the last problem first, then enlarges the picture to fear modeling, finally drives some conclusions which aim at deepening the relationship between fear and courage. As a result, it emerges that the emotion of fear has a conceptual, emotional, situational and subjective dimension. Assuming fear as a possible emotional center within the subject’s cognitive experience, the virtue of courage appears to balance the excess and lack of fear, and is consequently related to rational thoughts and consistent behavior, laying the foundations for a new Aristotelian-Thomistic account for it. (shrink)

To translate the Aristotelian square of opposition into Chinese requires restructuring the Aristotelian system of genus-species into the Chinese way of classification and understanding of the focus-field relationship. The feature of the former is on a tree model, while that of the later is on the focusfield model. Difficulties arise when one tries to show contraries betweenA- type and E-type propositions in the Aristotelian square of opposition in Chinese, because there is no clear distinction between (...) universal and particular in a focus-field structure of thinking. If there could be a chance to discuss the analytic identity between the two logical systems, then it might be only constituted during a face to face conversation in the present, or, in other words, in the translation of particular propositions (singular subjective,I-type, andO-type propositions) in a particular case. The best hope for a translator is that in the actual temporally situated practice,now he or she might find a temporary way to map the concepts of one to the other with relatively little loss of structure. (shrink)

According to a rule of traditional logic concerning the relation between general (or universal) concepts, if a given concept is more general than a second one, then the opposition (or contradictory) of the first concept is more specific than the opposition (or contradictory) of the second one. Kātibī, one of the Muslim logicians in the 13th century, has raised a question against this rule and, by giving some counterexamples, claims that it results in contradiction. Some Muslim logicians have (...) replied to Kātibī, and in this paper I have examined their replies. Also, by using rules of modern logic, we may easily show that either Kātibī's argumentation is fallacious or it does not result in contradiction; however, it seems that if modern logic rules had been represented to Muslim logicians, some of those rules would have been rejected by them. (shrink)

This dissertation is a comparative study of Aristotelian and Chinese logic. I briefly overview the reports of difficulties in understanding that derives from cultural differences. I claim that these difficulties not only result from the fact that concepts in each language fail to match properly, but also from the fact that the logical spaces themselves are structured differently. Aristotelian logic is based on the structure of a classificatory system---a hierarchical structure of names for kinds of things organized into (...) genera/species. Chinese logic is based on a Chinese binary system---a yin-yang structure of associative relations. ;I discuss the difficulties of translating the first three chapters of Matthew Lipman's Harry Stottlemeier's Discovery---one of a series of novels used in the Philosophy for Children Project---into Chinese. These chapters discuss the logical relations constituting the Aristotelian Square of Opposition. I find that most translation problems result from the fact that the logical structures of these two languages fail to match. ;I report my research on the responses of American and Chinese school children to reading chapter one of Harry Stottlemeier's Discovery , one in the original English version and the other in Chinese. The study of discussions among older children offers evidence for my claim that logical spaces of Western and Chinese thinking are structured differently, but the study of discussions among younger children gives a hope that there is a chance for there to be communication between the two very different ways of thinking. ;The conclusion is that Aristotelian logic and Chinese logic have two very different objectives, and as a consequence, different patterns for constructing logical relations. These two patterns represent two thinking models. The former is like a tree. It represents a finite number of things according to a presumed order. The latter is like a net; from any point on the net, one can link to any number of related points in an ever-changing world. The possibility of a mutually beneficial rapprochement between Chinese and Aristotelian logic depends on whether we can lay out a way for each side to understand the other in a common discourse. (shrink)

In the past four or five years I have been especially dependent on Aristotle's writings as I have initiated a series of experiments that can legitimately be called empirical efforts to prove Aristotelian conceptions to be true. In actuality, of course, I am trying to prove my own theory to be true—that is, worthy of consideration because it is consistent with observed human actions. However, by extension, I am surely seeking evidence for Aristotle's image of human cognition. There are (...) two Aristotlelian conceptions that underwrite my theoretical and empirical efforts: predication and opposition. When we speak of a "predication" in cognition we refer to a process that is fundamentally creative. Predications deal in meanings; and the way in which we align meanings, lending the meaning of one concept to another is what predication is all about. Predication is the act of affirming, denying, or qualifying certain patterns of meaning in relation to other patterns of meaning. The second Aristotelian conception that I have been employing theoretically and investigating empirically is "opposition." Although association through frequent contact was recognized by Aristotle , he also appreciated that there is often an intrinsic tie of opposite meanings to be seen in human reason. In my own interpretation of Aristotle, I believe that we can see the ultimate necessity of predication stemming from the opositional ties of meanings like this. If I am cognizing within a congerie of interlacing meanings which relate to and, indeed, delimit the definitions of their opposite meanings, then it is up to me as a cognizer to continually "take a position" on just what I am going to affirm—or not! 2012 APA, all rights reserved). (shrink)

In the past four or five years I have been especially dependent on Aristotle's writings as I have initiated a series of experiments that can legitimately be called empirical efforts to prove Aristotelian conceptions to be true. In actuality, of course, I am trying to prove my own theory to be true—that is, worthy of consideration because it is consistent with observed human actions. However, by extension, I am surely seeking evidence for Aristotle's image of human cognition. There are (...) two Aristotlelian conceptions that underwrite my theoretical and empirical efforts: predication and opposition. When we speak of a "predication" in cognition we refer to a process that is fundamentally creative. Predications deal in meanings; and the way in which we align meanings, lending the meaning of one concept to another is what predication is all about. Predication is the act of affirming, denying, or qualifying certain patterns of meaning in relation to other patterns of meaning. The second Aristotelian conception that I have been employing theoretically and investigating empirically is "opposition." Although association through frequent contact was recognized by Aristotle, he also appreciated that there is often an intrinsic tie of opposite meanings to be seen in human reason. In my own interpretation of Aristotle, I believe that we can see the ultimate necessity of predication stemming from the opositional ties of meanings like this. If I am cognizing within a congerie of interlacing meanings which relate to and, indeed, delimit the definitions of their opposite meanings, then it is up to me as a cognizer to continually "take a position" on just what I am going to affirm—or not! 2012 APA, all rights reserved). (shrink)

This pioneering study by Sister Prudence Allen traces the concept of woman in relation to man in more than seventy philosophers from ancient and medieval traditions. The fruit of ten years' work, this study uncovers four general categories of questions asked by philosophers for two thousand years. These are the categories of opposites, of generation, of wisdom, and of virtue. Sister Prudence Allen traces several recurring strands of sexual and gender identity within this period. Ultimately, she shows the paradoxical influence (...) of Aristotle on the question of woman and on a philosophical understanding of sexual coomplemenarity. Supplemented throughout with helpful charts, diagrams, and illustrations, this volume will be an important resource for scholars and students in the fields of women's studies, philosophy, history, theology, literary studies, and political science. (shrink)

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square (...) of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation. (shrink)

In this work, we provide an extensive analysis of Hohfeld’s theory of normative relations, focusing in particular on diagrammatic structures. Our contribution is threefold. First, we specify an extensional formal language to represent the main notions in the two families of normative relations identified by Hohfeld (i.e. the deontic and the potestative family). Our primary focus is on the part of the theory concerning potestative relations. In this regard, we assign a key role to the concept of (...) ability, which is treated as a primitive notion and used to formulate three fine-grained definitions of power (outcome-centered, change-centered and force-centered). Second, on the basis of these definitions we build Aristotelian diagrams of opposition for deontic and potestative relations, improving, extending and systematizing previous proposals formulated in the literature. Third, we present a model-theoretic interpretation and a logic programming (ASP) implementation of the proposed framework, elaborating on the procedural dimension of normative reasoning. (shrink)

The paper proposes two logical analyses of (the norms of) justification. In a first, realist-minded case, truth is logically independent from justification and leads to a pragmatic logic LP including two epistemic and pragmatic operators, namely, assertion and hypothesis. In a second, antirealist-minded case, truth is not logically independent from justification and results in two logical systems of information and justification: AR4 and AR4¢, respectively, provided with a question-answer semantics. The latter proposes many more epistemic agents, each corresponding to a (...) wide variety of epistemic norms. After comparing the different norms of justification involved in these logical systems, two hexagons expressing Aristotelian relations of opposition will be gathered in order to clarify how (a fragment of) pragmatic formulas can be interpreted in a fuzzy-based question-answer semantics. (shrink)

In this paper evidence will be provided that Wittgenstein’s intuition about the logic of colour relations is to be taken near-literally. Starting from the Aristotelian oppositions between propositions as represented in the logical square of oppositions on the one hand and oppositions between primary and secondary colors as represented in an octahedron on the other, it will be shown algebraically how definitions for the former carry over to the realm of colour categories and describe very precisely the (...) class='Hi'>relations obtaining between the known primary and secondary colours. Linguistic evidence for the reality of the resulting isomorphism will be provided. For example, the vertices that resist natural single-item lexicalization in logic (such as the O-corner, for which there is no natural lexicalization *nall (=not all)) are not naturally lexicalized in the realm of colour terms either. From the perspective of the architecture of cognition, the isomorphism suggests that the foundations of logical oppositions and negation may well be much more deeply rooted in the physiological structure of human cognition than is standardly assumed. (shrink)

The paper uses the theory of generalized quantifiers to discuss existential import and its implications for Aristotelian logic, namely the square of opposition, conversions and the assertoric syllogistic, as well as for more recent generalizations to intermediate quantifiers like “most”. While this is a systematic discussion of the semantic background one should assume in order to obtain the inferences and oppositions Aristotle proposed, it also sheds some light on the interpretation of his writings. Moreover by applying tools from (...) modern formal semantics to the investigation of classical Aristotelian logic and its extensions, we combine different approaches to the logic of quantification. We will present variants of quantifiers that are associated to the four corners of the square of opposition with and without existential import and discuss their role for the logical square, conversions and the syllogistic. It will turn out that there is no way to ascribe existential import that validates all inferences and relations which one is willing to hold in Aristotelian logic. Two options, however, provide reasonable results. Existential import should either be ascribed only to affirmative statements or only be ascribed to universal quantification. The former option is preferable for a mere reconstruction of the classical Aristotelian logic while the latter option is more attractive if Aristotelian logic is generalized to intermediate quantifiers. (shrink)

This book presents a historical and scientific analysis as historical epistemology of the science of weights and mechanics in the sixteenth century, particularly as developed by Tartaglia in his Quesiti et inventioni diverse, Book VII and Book VIII (1546; 1554). In the early 16th century mechanics was concerned mainly with what is now called statics and was referred to as the Scientia de ponderibus, generally pursued by two very different approaches. The first was usually referred to as Aristotelian, where (...) the equilibrium of bodies was set as a balance of opposite tendencies to motion. The second, usually referred to as Archimedean, identified statics with centrobarica, the theory of centres of gravity based on symmetry considerations. In between the two traditions the Italian scholar Niccolò Fontana, better known as Tartaglia (1500?-1557), wrote the treatise Quesiti et inventioni diverse (1546). This volume consists of three main parts. In the first, a historical excursus regarding Tartaglia's lifetime, his scientific production and the Scientia de ponderibus in the Arabic-Islamic culture, and from the Middle Ages to the Renaissance, is presented. Secondly, all the propositions of Books VII and VIII, by relating them with the Problemata mechanica by the Aristotelian school and Iordani opvsculvm de ponderositate by Jordanus de Nemore are examined within the history and historical epistemology of science. The last part is relative to the original texts and critical transcriptions into Italian and Latin and an English translation. This work gathers and re-evaluates the current thinking on this subject. It brings together contributions from two distinguished experts in the history and historical epistemology of science, within the fields of physics, mathematics and engineering. It also gives much-needed insight into the subject from historical and scientific points of view. The volume composition makes for absorbing reading for historians, epistemologists, philosophers and scientists. (shrink)

This paper first investigates logical characterizations of different structures of opposition that extend the square of opposition in a way or in another. Blanché’s hexagon of opposition is based on three disjoint sets. There are at least two meaningful cubes of opposition, proposed respectively by two of the authors and by Moretti, and pioneered by philosophers such as J. N. Keynes, W. E. Johnson, for the former, and H. Reichenbach for the latter. These cubes exhibit four (...) and six squares of opposition respectively. We clarify the differences between these two cubes, and discuss their gradual extensions, as well as the one of the hexagon when vertices are no longer two-valued. The second part of the paper is dedicated to the use of these structures of opposition for discussing the comparison of two items. Comparing two items usually involves a set of relevant attributes whose values are compared, and may be expressed in terms of different modalities such as identity, similarity, difference, opposition, analogy. Recently, J.-Y. Béziau has proposed an “analogical hexagon” that organizes the relations linking these modalities. Elementary comparisons may be a matter of degree, attributes may not have the same importance. The paper studies in which ways the structure of the hexagon may be preserved in such gradual extensions. As another illustration of the graded hexagon, we start with the hexagon of equality and inequality due to R. Blanché and extend it with fuzzy equality and fuzzy inequality. Besides, the cube induced by a tetra-partition can account for the comparison of two items in terms of preference, reversed preference, indifference and non-comparability even if these notions are a matter of degree. The other cube, which organizes the relations between the different weighted qualitative aggregation modes, is more relevant for the attribute-based comparison of items in terms of similarity. (shrink)

In logical geometry, Aristotelian diagrams are studied in a precise and systematic way. Although there has recently been a good amount of progress in logical geometry, it is still unknown which underlying mathematical framework is best suited for formalizing the study of these diagrams. Hence, in this paper, the main aim is to formulate such a framework, using the powerful language of category theory. We build multiple categories, which all have Aristotelian diagrams as their objects, while having different (...) kinds of morphisms between these diagrams. The categories developed here are assessed according to their ability to generalize previous work from logical geometry as well as their interesting category-theoretical properties. According to these evaluations, the most promising category has as its morphisms those functions on fragments that increase in informativity on both the opposition and implication relations. Focusing on this category can significantly increase the effectiveness of further research in logical geometry. (shrink)

The recognition and reconciliation of ‘opposites’ lies at the heart of our most personal and global problems. These problems are ‘wicked’ in the sense that they are difficult or impossible to solve and arise at the interface of interdependent polarities. By exploring the human tendency to divide the world into two parts, _Wicked Problems & the Reconciliation of Opposites_ argues that our relationship with such pairings and polarities is profoundly important to the way we recognise and resolve wicked problems. Using (...) an original theory of bi-relational development, Adam proposes that our everyday ways of knowing and being can be powerfully located and understood in terms of the creation, emergence, opposition, convergence, collapse and trans-positioning of dyadic constituents. Chapters use this concept to frame key debates in and across domains of knowledge, including education, psychology, law and theology, as well as to offer a new perspective on the most profound problems of the twenty-first century: globalisation, sustainability and secularisation. This book is a comprehensive study of dyadic structrures and relationships and provides a multidisciplinary and original approach to human development. It will be of great interest to students and academics of social psychology, psychosocial development, psychoanalysis and transpersonal psychology, but it is also relevant within the fields of education, human development and theology. (shrink)

In spite of the fact that Henry of Gent had a major and lasting influence on the developments at the University of Paris after the condemnation of the errores philosophorum in 1277, the Gandavistae – pupils of Henry of Gent – are hardly known by their proper names in the history of philosophy. As a member of the theological and philosophical faculty, Henry broke with the predominant Averroistic approach to Aristotle’s conception of science and concentrated, instead, on the Aristotelian (...) tradition. He defended a revised version of the Aristotelian doctrine of the categories along the lines of the pseudo-Boethian Liber de sex principiis and ascribed fundamental eminence to relatio, a category which was considered ontologically “debilissimus” among the Aristotelians. John de Polliaco was a pupil of Albert the Great, Thomas Aquinas, Henry of Gent and Godfrey of Fontaines during the seventies of the 13th century. In his Quodlibet I, q. 7, dating from 1307, he gave a controversial account of the notion of relation that was favoured by Henry’s adherents: “Does the relation, expressed as a respect, differ from the respects expressed by the six principles?” In this discussion he attacks the intentional and modal interpretation given by the Gandavistae and calls them non-reales. Is this accusation already an indication of the rise of 14th century nominalism? (shrink)

Various semantics for studying the square of opposition have been proposed recently. So far, only [14] studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and show how to construct probabilistic versions of (...) the square of opposition by forming suitable tripartitions. Finally, as an application, we present a new square involving generalized quantifiers. (shrink)

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

The Vienna Circle’s relations to Polish Semantics comprise a rather large spectrum that reaches from Camap’s whole-hearted reception, Gödel’s partial anticipation that Neurath would completely reject semantics. According to Neurath, semantics was of no help for the empiricist program of the Vienna Circle. Quite the contrary, Neurath seems to have regarded Tarski as a sort of evil genius who led astray his closest philosophical friend Carnap, seducing him to leave the path of empiricist virtue and to become addicted to (...) the vice of Aristotelian metaphysics. (shrink)

The present paper expounds the logic notation proposed by Augustus De Morgan in 1850 from within the original context of De Morgan’s account of syllogistic logic and his approach to quantification. The notational system of 1850 is shown to be a flexible tool to state inferences, to prove their validity and to derive formulæ of the respective system by ‘blind’ application of transformation rules. These pertain to the swapping of operator signs, which are of inverse ‘character’ in a two-fold sense: (...) The signs both represent inverse operations and exemplify oppositeness due to their very shapes. Thus, while De Morgan’s initial choice of symbol shapes is due to certain diagrammatic features, his notational system of 1850 leads to an account of logical operations as effected by manipulation of symbol tokens. It is concluded that as an instantiation of what De Morgan terms a ‘calculus of opposite relations’, his notation of 1850 may also be conceived of as a ‘graphicism’ affording for the working of a ‘symbolic machine’ on ‘de-semantification’. (shrink)

. Each predicate of the Aristotelian square of opposition includes the word “is”. Through a twofold interpretation of this word the square includes both classical logic and non-classical logic. All theses embodied by the square of opposition are preserved by the new interpretation, except for contradictories, which are substituted by incommensurabilities. Indeed, the new interpretation of the square of opposition concerns the relationships among entire theories, each represented by means of a characteristic predicate. A generalization of (...) the square of opposition is achieved by not adjoining, according to two Leibniz’ suggestions about human mind, one more choice about the kind of infinity; i.e., a choice which was unknown by Greek’s culture, but which played a decisive role for the birth and then the development of modern science. This essential innovation of modern scientific culture explains why in modern times the Aristotelian square of opposition was disregarded. (shrink)

Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of opposition by forming (...) suitable tripartitions of the set of all coherent assessments. Finally, as an application, we present new versions of the square and of the hexagon involving generalized quantifiers. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. (...) We then introduce two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

A novel theoretical formulation of Categorical Logic based on two properties of categorical propositions and three simple axioms has been introduced recently. This formulation allowed for the suppression of the distinction between immediate and mediate inferences, and also provided a theoretical framework to study opposition relations, thus restoring the theoretical unity of traditional Aristotelian logic. By using this approach, it has been reported that a total of 3072 conclusive syllogistic moods can be found when including indefinite terms (...) in classical syllogistic, but this result has yet to be proven. This paper presents an overview of the recently proposed theoretical formulation of Categorical Logic, along with the derivation of the 48 canonical syllogistic moods that are capable of generating the 3072 conclusive moods previously reported. (shrink)

The article makes a detailed analysis of the understanding of women in the philosophical works of Corpus Aristotelicum. It is established that the specificity of the view of this ancient thinker on the problem of research is due to the fact that he considers it in the whole body of his philosophical works, reflecting on it in logical, ethical-aesthetic and socio-philosophical aspects. It has been found that the key issue around which Stagirite reflects on women is the concept of «domination». (...) In the context of reflections on domination, he emphasizes that in the ontological context, women are not inferior to men, but in marital and family relations, a woman must obey a man. At the same time, it is proved that Aristotle speaks only briefly about the possible need for education and upbringing of women, and mentions it not for the benefit and self-development of women, but only to avoid all sorts of chaos and disorder in a particular society. However, he notes that a woman can hold a high position if she has a good reputation and authority among citizens who pursue politics. It is argued that the ancient thinker emphasizes that there are a number of issues, namely those concerning children, in which women clearly understand each other better, and emphasizes that they should deal with them and resolve disputes under their guidance. It is interesting that in the context of this issue, Aristotle mentions there is an image of a Delphic knife [Polit.1252b1–7], because he used in various aspects - military, domestic and sacred. As we can see, this ancient philosopher continues Plato's opinion that it is impossible to «spray» on certain goals and objectives, because, mainly, it leads to mistakes, tragedies, etc. Everyone, including women, must do their job, which is given to them by nature. This does not diminish this or that personality, but on the contrary makes it unique and inimitable among all creations. (shrink)

The square of opposition has never been drawn by classical Arabic logicians, such as al-Fārābī and Avicenna. However, in some later writings, we do find squares, which their authors call rather ‘tables’ (sing. _lawḥ_). These authors are Shihāb al-Dīn al-Suhrawardī and Muhammed b. Yūsuf al-Sanūsī. They do not pertain to the same geographic area, but they both provide squares of opposition. The aim of this paper is to analyse these two squares, to compare them with each other and (...) with the traditional square of opposition drawn by Apuleius and Boethius, with regard to their particular structure and their vertices and to evaluate their validity. We will show that both squares are different from each other and from the traditional square both with regard to their vertices, and to the arrangements of the oppositional relations. In addition, while al-Suhrawardī uses the notion of matter modalities to define the oppositions, al-Sanūsī relies rather on al-Rāzī’s and al-Khūnajī’s distinctions. Both authors agree, however, on the fact that affirmatives have an import while negatives do not. For this reason, their squares are valid from a logical viewpoint. As diagrams, these squares are nevertheless entirely original, due to the particular arrangements of their oppositional relations. Our analysis shows also that the later author does not seem to know the square drawn by the earlier one and that both authors do not seem to know Apuleius’ or Boethius’ squares. (shrink)

We show that we in ways related to the classical Square of Opposition may define a Cube of Opposition for some useful statements, and we as a by-product isolate a distinct directive of being inviolable which deserves attention; a second central purpose is to show that we may extend our construction to isolate hypercubes of opposition of any finite cardinality when given enough independent modalities. The cube of opposition for obligations was first introduced publically in a (...) lecture for the Square of Opposition Conference in the Vatican in May 2014. -/- A COLORFUL ERRATUM: The article Cubes and Hypercubes of Opposition, with Ethical Ruminations on Inviolability, suffered a couple of mishaps when it was published in Logica Universalis, ISSN 1661-8297, 10(2-3), p. 373–376. The worst of them was that the colorful cube was printed in shades of grey, instead of decorating the sides with two transparent sides, and then a red, a yellow, a blue and a green side. The text of the article is not quite understandable because of this mishap, so I avail a colorfully updated version of the original article here. -/- . (shrink)

. In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples.

In this essay Suzanne Rice examines Aristotle's ideas about virtue, character, and education as elements in an Aristotelian conception of good listening. Rice begins by surveying of several different contexts in which listening typically occurs, using this information to introduce the argument that what should count as “good listening” must be determined in relation to the situation in which listening actually occurs. On this view, Rice concludes, there are no “essential” listening virtues, but rather ways of listening that may (...) be regarded as virtuous in the context of particular concrete circumstances. (shrink)

In this paper, I discuss the neo-Aristotelian approaches, which usually reinterpret Aristotle’s ideas on form and/or borrow the notion of formal cause without engaging with the broader implications of Aristotle’s metaphysics. In opposition to these approaches, I claim that biosemiotics can propose an alternative view on life’s form. Specifically, I examine the proposals to replace the formal cause with gene-centrism, functionalism, and structuralism. After critically addressing these approaches, I discuss the problems of reconciling Aristotelianism with the modern view (...) of life’s organization. I claim that the notion of the final cause proposes a cosmological hierarchy and that this is the main problem with applying the formal cause to biology. An alternative categorization of conceptualizing life’s form involves the processual identity, relational property clusters, and context-dependent transmission of representational units. The third category points to a semiotic basis of the form of life. In this context, I offer to readjust the focus of the problem of matter-form duality by pointing out that form is primarily an issue of the subject-object relation. Biosemiotics helps to understand the constructive role of symbolic representation in living systems, which is crucial to extend the analysis of the form from cognitive representations to external phenomena. Emergence of subjectivity and perspectivity of interactions are key elements to bridge the form and actual processes within a non-hylomorphic account. To demonstrate transitions from the physical influence of shapes to the organic recognition of forms, I address the biological studies on the synchronization of coincidental inputs and enzyme specificity. (shrink)

The traditional Square of Opposition consists of four sentence types. Two are universal and two particular; two are affirmative and two negative. Examples, where ‘S’ and ‘P’ designate the subject and the predicate, are: ‘every S is P’, ‘no S is P’, ‘some S is P’ and ‘some S is not P’. Taking the usual sentences of the square of opposition, quantifying over their predicates exhibits non-standard sentence forms. These sentences may be combined into non-standard Squares of (...) class='Hi'>Opposition , and they reveal a new relationship not found in the usual Square. Medieval logicians termed ‘disparatae’ pairs of sentences like ‘every S is some P’ and ‘some S is every P’, which are neither subaltern nor contrary, neither contradictory nor subcontrary. Walter Redmond has designed a special language L to express the logical form of these sentences in a precise way. I will use this language to show how Squares of Opposition, standard and non-standard, form a complex network of relations which bring to .. (shrink)