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 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)

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)

© Springer International Publishing Switzerland 2016. We develop a systematic approach for dealing with informationally equivalent Aristotelian diagrams, based on the interaction between the logical properties of the visualized information and the geometrical properties of the concrete polygon/polyhedron. To illustrate the account’s fruitfulness, we apply it to all Aristotelian families of 4-formula fragments that are closed under negation and to all Aristotelian families of 6-formula fragments that are closed under negation.

This paper studies the logical context-sensitivity of Aristotelian diagrams. I propose a new account of measuring this type of context-sensitivity, and illustrate it by means of a small-scale example. Next, I turn toward a more large-scale case study, based on Aristotelian diagrams for the categorical statements with subject negation. On the practical side, I describe an interactive application that can help to explain and illustrate the phenomenon of context-sensitivity in this particular case study. On the theoretical side, I (...) show that applying the proposed measure of context-sensitivity leads to a number of precise yet highly intuitive results. (shrink)

We design a framework for assisted normative reasoning based on Aristotelian diagrams and algorithmic graph theory which can be employed to address heterogeneous tasks of deductive reasoning. Here we focus on two problems of normative determination: we show that the algorithms used to address these problems are computationally efficient and their operations are traceable by humans. Finally, we discuss an application of our framework to a scenario regulated by the GDPR.

© The Author 2018. Published by Oxford University Press. All rights reserved. Logical geometry provides a broad framework for systematically studying the logical properties of Aristotelian diagrams. The main aim of this paper is to present and illustrate the foundations of a computational approach to logical geometry. In particular, after briefly discussing some key notions from logical geometry, I describe a logical problem concerning Aristotelian diagrams that is of considerable theoretical importance, viz. the task of finding the maximal (...) Boolean complexity of a given family of Aristotelian diagrams, and I then present and discuss a simple algorithm for automatically solving this task. This algorithm is naturally implemented within the paradigm of logic programming. In order to illustrate the theoretical fruitfulness of this algorithm, I also show how it sheds new light on several well-known families of Aristotelian diagrams. (shrink)

© 2017 by the authors. Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron, the tetrakis (...) hexahedron, the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation between logical and geometrical distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design. (shrink)

The Region Connection Calculus (RCC) is a qualitative spatial reasoning formalism, developed in knowledge representation and geographical information systems. We argue that RCC can be viewed as a more fine-grained approach to the use of Euler diagrams to visualize categorical statements like ‘all A are B’. We present RCC using the syntax of first-order modal logic and a topological semantics. We compare the Gergonne relations (a well-known set of 5 jointly exhaustive and pairwise disjoint relations between two non-empty sets, visualized (...) using Euler-type diagrams) with RCC8 (a similar set of 8 relations for regions). (shrink)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of (...) another type of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

This book examines the transmission processes of the Aristotelian Mechanics. It does so to enable readers to appreciate the value of the treatise based on solid knowledge of the principles of the text. In addition, the book’s critical examination helps clear up many of the current misunderstandings about the transmission of the text and the diagrams. The first part of the book sets out the Greek manuscript tradition of the Mechanics, resulting in a newly established stemma codicum that illustrates (...) the affiliations of the manuscripts. This research has led to new insights into the transmission of the treatise, most importantly, it also demonstrates an urgent need for a new text. A first critical edition of the diagrams contained in the Greek manuscripts of the treatise is also presented. These diagrams are not only significant for a reconstruction of the text but can also be considered as a commentary on the text. Diagrams are thus revealed to be a powerful tool in studying processes of the transfer and transformation of knowledge. This becomes especially relevant when the manuscript diagrams are compared with those in the printed editions and in commentaries from the early modern period. The final part of the book shows that these early modern diagrams and images reflect the altered scope of the mechanical discipline in the sixteenth century. (shrink)

The paper examines Schopenhauer’s complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer’s partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) α -structures.

The paper examines Schopenhauer’s complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer’s partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) α -structures.

For an Aristotelian observer, the halo is a puzzling phenomenon since it is apparently sublunary, and yet perfectly circular. This paper studies Aristotle's explanation of the halo in Meteorology III 2-3 as an optical illusion, as opposed to a substantial thing (like a cloud), as was thought by his predecessors and even many successors. Aristotle's explanation follows the method of explanation of the Posterior Analytics for "subordinate" or "mixed" mathematical-physical sciences. The accompanying diagram described by Aristotle is one (...) of the earliest lettered geometrical diagrams, in particular of a terrestrial phenomenon, and versions of it can still be found in modern textbooks on meteorological optics. (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.

It is shown that Aristotelian dialectic can be analyzed as having two parts: a core formal model that has a formal dialogue structure and a set of ten definable supplementary characteristics that lie outside the core structure. Some current argumentation tools used in artificial intelligence and multi-agent systems are applied to the task of extending the core formal model to include the supplementary characteristics. Using these tools it is explained how the structure of a dialogue can be mapped into (...) an argument diagram that can be analyzed and evaluated using standard argumentation techniques such as argumentation schemes, types of dialogue, critical questions and a dialectical concept of burden of proof. Disputed issues on how the elenchus fits the standard dialogue typology are discussed, and it is concluded that it fits best into a type of dialogue called examination dialogue that is closely related to persuasion dialogue. (shrink)

Aristotelian imagination is widely understood as a psychological power by which retained perceptual states recur in consciousness. According to this view, imagination is decaying sense, a part of the psyche that is parasitic on perceptual acts for its content. This paper disputes this reading and provides an alternative account of Aristotle’s concept of imagination. I argue that Aristotelian imagination is a power of the psyche that is both productive like intellect, and presentational like perception. Unlike perception and intellect, (...) however, imagination does not correctly discriminate among beings, and thus cannot be relied upon to give one knowledge of the world. When one accepts this alternative conception of Aristotelian imagination, it becomes clear how it can take on the peculiar epistemic function of allowing a particular serve as the vehicle of a universal thought. This paper argues that Aristotle’s explanation of valid judgments in geometry depends on the imagination to allow the perception of a particular diagram to give rise to the intellectual grasp of a general proposition. (shrink)

This book constitutes the refereed proceedings of the 14th International Conference on the Theory and Application of Diagrams, Diagrams 2024, held in Münster, Germany, during September 27–October 1, 2024. -/- The 17 full papers, 19 short papers and 11 papers of other types included in this book were carefully reviewed and selected from 69 submissions. They were organized in topical sections as follows: Keynote Talks; Analysis of Diagrams; Euler and Venn Diagrams; Diagrams in Logic; Diagrams and Applications; Diagram Tools; (...) Historical Aspects of Diagrams; and Posters. (shrink)

© 2018, Springer International Publishing AG, part of Springer Nature. Aristotelian diagrams are used extensively in contemporary research in artificial intelligence. The present paper investigates the geometric and cognitive differences between two types of Aristotelian diagrams for the Boolean algebra B4. Within the class of 3D visualizations, the main geometric distinction is that between the cube-based diagrams and the tetrahedron-based diagrams. Geometric properties such as collinearity, central symmetry and distance are examined from a cognitive perspective, focusing on (...) class='Hi'>diagram design principles such as congruence/isomorphism and apprehension. The cognitive effectiveness of the different visualizations is compared for the representation of implication versus opposition relations, and for subdiagram embeddings. (shrink)

‘Byzantine logic diagrams’ have been used since at least late antiquity, and became popular in Europe in the 16th century. However, since the criticism of W. Hamilton and J. Venn in the 19th century, Byzantine diagrams have been largely dismissed as obsolete. This paper challenges this prevailing view. Initially, we provide a comprehensive overview of the current state of research pertaining to these diagrams, illustrating their applicability in analyzing assertoric syllogisms. Subsequently, we propose that the expressive capacity of these diagrams (...) extends far beyond their traditional use. In particular, we demonstrate that the expressivity of these diagrams is not limited to checking the validity of syllogisms. Rather, we show that there are various extensions of Byzantine diagrams, including opposition relations, modal logic, particular instances, n-term diagrams and logical connectives. These extensions are compositional and can therefore be combined to create very complex Byzantine diagrams that go far beyond what has been commonly believed since Hamilton and Venn. (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)

Around the turn of the 20th century, Keynes and Johnson extended the well-known square of opposition to an octagon of opposition, in order to account for subject negation (e.g., statements like ‘all non-S are P’). The main goal of this paper is to study the logical properties of the Keynes-Johnson (KJ) octagons of opposition. In particular, we will discuss three concrete examples of KJ octagons: the original one for subject-negation, a contemporary one from knowledge representation, and a third one (hitherto (...) not yet studied) from deontic logic. We show that these three KJ octagons are all Aristotelian isomorphic, but not all Boolean isomorphic to each other (the first two are representable by bitstrings of length 7, whereas the third one is representable by bitstrings of length 6). These results nicely fit within our ongoing research efforts toward setting up a systematic classification of squares, octagons, and other diagrams of opposition. Finally, obtaining a better theoretical understanding of the KJ octagons allows us to answer some open questions that have arisen in recent applications of these diagrams. (shrink)

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.

Venn diagrams, which are widely used in introductory logic courses, provide a convenient and illuminating way of presenting the various theories concerning the nature of law. When combined with the Aristotelian square of opposition, these diagrams show not only how the theories are related to one another, logically, which is essential to understanding them, but also which theories are compossible. One surprising result of this approach is that it shows the substantive compatibility of the theories of law set forth (...) by H. L. A. Hart and Ronald Dworkin, who are usually pitted against one another. I show, through quotation and visual representation, that there is no essential disagreement between these jurisprudential stalwarts concerning the relation of law and morality. (shrink)

In Baghdad in the mid twelfth century Abū al-Barakāt proposes a radical new procedure for finding the conclusions of premise-pairs in syllogistic logic, and for identifying those premise-pairs that have no conclusions. The procedure makes no use of features of the standard Aristotelian apparatus, such as conversions or syllogistic figures. In place of these al-Barakāt writes out pages of diagrams consisting of labelled horizontal lines. He gives no instructions and no proof that the procedure will yield correct results. So (...) the reader has to work out what his procedure is and whether it is correct. The procedure turns out to be insightful and entirely correct, but this paper may be the first study to give a full description of the procedure and a rigorous proof of its correctness. (shrink)

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.

This comment calls attention to the nature of the Aristotelian and classical logics, and the difficulty of representing their judgments and inferences by means of Venn diagrams. The meaning of ‘all’ in the different calculi produces problems. A second problem is that the specification of existence in Venn diagrams for statements and arguments cannot be restricted to a single class, overlooked by Wiebe. This problem is further complicated by his adoption of classical (Renaissance) syllogistic, which is inconsistent. Aristotle’s term (...) logic is consistent. So also is the medieval extension, though the inclusion of singular premisses renders it less perspicuous though more flexible. (shrink)

This paperback is a programed text designed for teaching introductory logic, either in conjunction with a standard text based upon traditional logic or as a do-it-yourself supplement for students taking courses stressing symbolic logic. The student learns logical theory by answering a variety of short answer, objective type exercises. The correct answer is given directly below each question or exercise, and the student is required to cover the answer while working the exercise; the purpose of this immediate access to the (...) answer is to enable the student to determine quickly whether or not he comprehends the material. The content of the book concentrates primarily upon categorical statements, presenting both the existential and hypothetical interpretations of the square of opposition, but unfortunately continues to promulgate the historically inaccurate terminology of "Aristotelian" and "Boolean." [Categorical statements were first expressed as existential statements by Franz Brentano in 1874.] Categorical statements not in standard form are standardized, e.g., "No roaches feel despair" is standardized by being rewritten as "No roaches are entities who feel despair." In addition to considering the logical relations of opposition with respect to A, E, I, O statements, these same relations are extended to non-categorical statements, e.g., "George Washington did not die in Europe" and "George Washington did not die in Asia" are subcontraries, while "There is philosophical activity on Venus" and "There is philosophical activity in the universe" are alterns. Determining validity of syllogisms is based upon the distribution of terms, and also by the use of Venn diagrams. The traditional figures and moods of the syllogism are ignored, as is the distinction between major and minor terms, with these latter two being lumped together as end terms. Immediate inferences and non-syllogistic arguments are treated by Venn diagrams. One-, two-, three- and four-term Venn diagrams are utilized throughout a substantial part of the book.--T. G. N. (shrink)

Aristotelian diagrams, such as the square of opposition and other, more complex diagrams, have a long history in philosophical logic. Alpha-structures and ladders are two specific kinds of Aristotelian diagrams, which are often studied together because of their close interactions. The present paper builds upon this research line, by reformulating and investigating alpha-structures and ladders in the contemporary setting of logical geometry, a mathematically sophisticated framework for studying Aristotelian diagrams. In particular, this framework allows us to formulate (...) well-defined functions that construct alpha-structures and ladders out of each other. In order to achieve this, we point out the crucial importance of imposing an ordering on the elements in the diagrams involved, and thus formulate all our results in terms of ordered versions of alpha-structures and ladders. These results shed interesting new light on the prospects of developing a systematic classification of Aristotelian diagrams, which is one of the main ongoing research efforts within logical geometry today. (shrink)

Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for (...) assigning bitstrings to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before. (shrink)

In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield (...) various asymmetric versions of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme. (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)

This paper compares two 3D logical diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron and the nested tetrahedron. Geometric properties such as collinearity and central symmetry are examined from a cognitive perspective, focussing on diagram design principles such as congruence/isomorphism and apprehension.

Burgess-Jackson has recently suggested that the debate between theism and atheism can be represented by means of a classical square of opposition. However, in light of the important role that the position of agnosticism plays in Burgess-Jackson’s analysis, it is quite surprising that this position is not represented in the proposed square of opposition. I therefore argue that the square of opposition should be extended to a slightly larger, more complex Aristotelian diagram, viz., a hexagon of opposition. Since (...) this hexagon does represent the position of agnosticism, it arguably yields a more helpful representation of the theism/atheism debate. It would be naïve to presume that Aristotelian diagrams can, by themselves, lead to a comprehensive solution of debates as intricate as that between theism and atheism. Nevertheless, this paper aims to show that these diagrams — especially if they are chosen carefully — have an important methodological role to play, by systematically organizing and clarifying the debate. (shrink)

The systematic account of deductive reasoning and the development of a formal logic to reveal the principles of such reasoning began with Aristotle's syllogistic. It was a term logic, a logic that dominated the field until the rise of modern predicate logic at the end of the Nineteenth century. That system quickly supplanted the old logic of terms. However, in the middle of the Twentieth century Fred Sommers took up the challenge to build a revised and strengthened term logic, one (...) fit to challenge the hegemony of predicate logic. The aim was to devise a formal logic that could better serve as the logic of natural language. -/- In recent years a new group of logicians have taken this version of term logic into many new directions. This book presents here for the first time some of the best examples of their work with new essays by philosophers, logicians, mathematicians, computational theorists, and historians of logic. The topics addressed include relative terms, logical copulation, Aristotelian diagrams, modality, truth, semantics, epistemic term logic, non-classical quantifiers, and more. (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 question of naturalness in logic is widely discussed in today’s research literature. On the one hand, naturalness in the systems of natural deduction is intensively discussed on the basis of Aristotelian syllogistics. On the other hand, research on “natural logic” is concerned with the implicitly existing logical laws of natural language, and is therefore also interested in the naturalness of syllogistics. In both research areas, the question arises what naturalness exactly means, in logic as well as in language. (...) We show, however, that this question is not entirely new: In his Berlin Lectures of the 1820s, Arthur Schopenhauer already discussed in depths what is natural and unnatural in logic. In particular, he anticipates two criteria for the naturalness of deduction that meet current trends in research: (1) Naturalness is what corresponds to the actual practice of argumentation in everyday language or scientific proof; (2) Naturalness of deduction is particularly evident in the form of Euler-type diagrams. (shrink)

The scholastic scientific diagram of the 13th century has a main component: the translations of Aristotle treatises. In this way Boethius’ works are highly significant both for his translations of Greek terms and for fixing a precise lexicology that allows us to interpret it. These records were enriched with meaningful translations and comments that began to spread in the 12th century and the following ones of the 13th century. However, Thomas Aquinas’ scientific view shows this tradition and enhances a (...) certain understanding of the Aristotelian texts. On this basis the following paper analyzes the notion of ‘dignitates’ on the thomistic opera omnia, because this is one of the terms which sets up the scientific scheme of Aristotelian root, with the objective to show its place, its meaning and its scope in the order of sciences. (shrink)

역설적⋅모순적 언어는 일상적으로는 접근 불가능한 심오한 종교적⋅철학적 통찰을 나타내기 위해 종종 사용되지만, 논리적으로 명료한 설명을 목적으로 하는 논서나 주석서에서는 그 사용을 억제하려는 경향이 강하다. 불교 인식논리학, 곧 인명(因明=hetuvidyā) 전통의 오류론 역시 비슷한 목적에서 출현하였다. 이러한 인명 전통에 대해서 서양의 분석철학 전통과의 비교 연구가 종종 행해지지만, 인명 관련 문헌에는 난해한 용어와 예시가 많이 등장하고, 그 문헌들의 한역과 주석 과정에서도 각종 난제들이 제기되었다. 본 논문은 『인명정리문론』과 『인명입정리론』에 소개된 ‘회토비월(懷兎非月=acandraḥ śaśī)’의 논증식을 분석하기 위한 예비적 단계로서 디그나가(Dignāga=陳那)가 고안한 인명학의 주요 개념에 대한 분석적 이해를 (...) 도모한다. 이를 위해 먼저 논증식의 일반적 구성 요소인 삼지(三支)와 인(因)의 삼상(三相) 개념은 약간의 수정을 가하면 현대 기호논리나 벤 다이어그램으로 나타낼 수 있음을 확인한다. 특히 삼지작법은 형식적으로 아리스토텔레스의 삼단논법과 유사해 보이지만, 근거나 예시를 경험 속에서 확인할 수 있어야 한다는 존재 함축을 지닌다는 점에서 연역 추리와 귀납 논리의 결합을 보여준다. 이는 또한 삼단논법의 시각적 이해를 위해 널리 활용되는 벤 다이어그램을 통해 논증식을 표시할 때 특정 구역이 공집합인지 아닌지를 통해 검토하여 그 논리적 타당성을 검증할 수 있음을 의미한다. ‘회토비월’ 논증식의 논리적 타당성 검토 또한 이에 기초해서 이루어져야 할 것이다. (shrink)

The founder of both American pragmatism and semiotics, Charles Sanders Peirce is widely regarded as an enormously important and pioneering theorist. In this book, scholars from around the world examine the nature and significance of Peirce’s work on perception, iconicity, and diagrammatic thinking. Abjuring any strict dichotomy between presentational and representational mental activity, Peirce’s theories transform the Aristotelian, Humean, and Kantian paradigms that continue to hold sway today and, in so doing, forge a new path for understanding the centrality (...) of visual thinking in science, education, art, and communication. The essays in this collection cover a wide range of issues related to Peirce’s theories, including the perception of generality; the legacy of ideas being copies of impressions; imagination and its contribution to knowledge; logical graphs, diagrams, and the question of whether their iconicity distinguishes them from other sorts of symbolic notation; how images and diagrams contribute to scientific discovery and make it possible to perceive formal relations; and the importance and danger of using diagrams to convey scientific ideas. This book is a key resource for scholars interested in Perice’s philosophy and its relation to contemporary issues in mathematics, philosophy of mind, philosophy of perception, semiotics, logic, visual thinking, and cognitive science. (shrink)

In this paper, I attempt a reconstruction of the theory of intersections in the geometry of Euclid. It has been well known, at least since the time of Pasch onward, that in the Elements there are no explicit principles governing the existence of the points of intersections between lines, so that in several propositions of Euclid the simple crossing of two lines (two circles, for instance) is regarded as the actual meeting of such lines, it being simply assumed that the (...) point of their intersection exists. Such assumptions are labelled, today, as implicit claims about the continuity of the lines, or about the continuity of the underlying space. Euclid’s proofs, therefore, would seem to have some demonstrative gaps that need to be filled by a set of continuity axioms (as we do, in fact, find in modern axiomatizations).I show that Euclid’s theory of intersections was not in fact based on any notion of continuity at all. This is not only because Greek concepts of continuity (such as the Aristotelian ones) were largely insufficient to ground a geometrical theory of intersections, but also, at a deeper level, because continuity was simply not regarded as a notion that had any role to play in the latter theory. Had Euclid been asked to explain why the points of intersections of lines and circles should exist, it would have never occurred to him to mention continuity in this connection. Ancient geometry was very different from ours, and it is only our modern views on continuity that tend to give rise to the expectation that this latter must be included in the foundations of elementary mathematics.We may hope to reconstruct Euclid’s views on intersections if we undertake a critical examination both of the extension and of the limits of ancient diagrammatic practices. This is tantamount to attempting to understand to what extent the existence of an intersection point may be inferred just from the inspection of a diagram, and in which cases we need rather to supplement such inference by propositional rules. This leads us to discuss Euclid’s definition of a point, and to work out the details of the complex interaction between diagrams and text in ancient geometry. We will see, then, that Euclid may have possessed a theory of intersections that was sufficiently rigorous to dispel the main objections that could be advanced in antiquity. (shrink)

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)

In this paper is constructed an analogue of the square of opposition for propositions about relations between two non-empty sets. Unlike the classical square of opposition, the proposed scheme uses all logically possible syllogistic constants, formulated in V.I. Markin’s universal language for traditional positive syllogistic theories. This scheme can be called “Logical lantern”. The basic constants of this language are representing the five basic relations between two non-empty sets: equity, strict inclusion, reversed strict inclusion, intersection and exclusion (considered are only (...) the relations between sets, but not the relations of the sets to the universe). The other relations between sets are combinations (disjunctions) of the basic ones. There are 32 constants and 32 respective propositions, and the proposed diagram expresses and/or allows to deduce the logical relations among those propositions. The constructed scheme, making it possible to visually see the logical relations among propositions about relations between two non-empty sets, allows us to suggest considering logical relations not discussed previously: many-place Aristotelian-like logical relations among propositions: exhaustive _n_-place contrariety and exhaustive _n_-place subcontrariety (_n_ is a natural number, \(n>2\) ). (shrink)

It has become an article of faith among historians of logic that the square of opposition diagram is due not to Aristotle, but to Apuleius. I examine three Aristotelian texts and argue that Prior Analytics I.46 contains a square of opposition, making Aristotle the discoverer of the diagram.

It is shown that a notion of natural place is possible within modern physics. For Aristotle, the elements—the primary components of the world—follow to their natural places in the absence of forces. On the other hand, in general relativity, the so-called Carter–Penrose diagrams offer a notion of end for objects along the geodesics. Then, the notion of natural place in Aristotelian physics has an analog in the notion of conformal infinities in general relativity.

First Logic is an introduction to the study of logic. Understanding the concepts of validity, invalidity, and acceptability, unacceptability of arguments is the primary focus of this book. The first chapter introduces the reader to some of the basic concepts, such as validity, soundness, and acceptability. Chapters two and three are devoted to Aristotelian logic, including the traditional square of opposition and Venn diagrams for sentences and arguments. Chapter four is a treatment of a number of important informal fallacies (...) of reasoning . Chapters five through eight are on various aspects of formal/symbolic logic: translating from natural language to the artificial language of logic; truth tables and truth trees; the method of natural deduction; predicate logic, including the logic of relations; and the concept and use of identity and its symbolization. (shrink)