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

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

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

Transformed RAVAL NOTATION solves Syllogism problems very quickly and accurately. This method solves any categorical syllogism problem with same ease and is as simple as ABC… In Transformed RAVAL NOTATION, each premise and conclusion is written in abbreviated form, and then conclusion is reached simply by connecting abbreviated premises.NOTATION: Statements (both premises and conclusions) are represented as follows: Statement Notation a) All S are P, SS-P b) Some S are P, S-P c) Some S are not P, S / PP (...) d) No S is P, SS / PP (- implies are and / implies are not) All is represented by double letters; Some is represented by single letter. No S is P implies No P is S so its notation contains double letters on both sides. -/- RULES: (1) Conclusions are reached by connecting Notations. Two notations can be linked only through common linking terms. When the common linking term multiplies (becomes double from single), divides (becomes single from double) or remains double then conclusion is arrived between terminal terms. (Aristotle’s rule: the middle term must be distributed at least once) -/- (2)If both statements linked are having – signs, resulting conclusion carries – sign (Aristotle’s rule: two affirmatives imply an affirmative) -/- (3) Whenever statements having – and / signs are linked, resulting conclusion carries / sign. (Aristotle’s rule: if one premise is negative, then the conclusion must be negative) -/- (4)Statement having / sign cannot be linked with another statement having / sign to derive any conclusion. (Aristotle’s rule: Two negative premises imply no valid conclusion) Syllogism conclusion by Tranformed Raval’s Notation is in accordance with Aristotle’s rules for the same. It is visually very transparent and conclusions can be deduced at a glance, moreover it solves syllogism problems with any number of statements and it is quickest of all available methods. By new Raval method for solving categorical syllogism, solving categorical syllogism is as simple as pronouncing ABC and it is just continuance of Aristotle work on categorical syllogism. It’s believed that Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, it’s claimed that Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle". Above conclusion is reached at a glance with Raval's Notations (Symbolic Aristotle’s syllogism rules). Premise: "No (square that is a quadrangle) is a (rhombus that is a rectangle)" Raval's Representations: S – Q, S – Q / Rh – Re, Rh – Re Premise: "No (rhombus that is a rectangle) is a (square that is a quadrangle)". Raval's Representations: Rh – Re, Rh – Re / S – Q, S - Q Conclusion: "No (quadrangle that is a square) is a (rectangle that is a rhombus)" Raval’s Representations: Q – S, Q – S / Re – Rh, Re – Rh As “ Q – S” follows from “S – Q” and “Re – Rh” from “Rh – Re”. Given conclusion follows from the given premises. Author disregards existential fallacy, as subset of a null set has to be a null set. -/- . (shrink)

Resumen: En la actualidad uno de los libros más usados para dar lógica elemental es el de Irving Copi y Carl Cohen (Introducción a la lógica, 2001), allí se presentan unas reglas para decidir la validez de los silogismos categóricos de forma estándar. Pero en tal texto ni en ninguno que nosotros conozcamos se ofrece una fundamentación de las mismas. Es decir, una demostración de que ellas son realmente una condición necesaria y suficiente de la validez de un silogismo categórico (...) de forma estándar, lo único que hemos leído es su motivación como condición necesaria. El objetivo de este trabajo es ofrecer una fundamentación de las mismas usando la noción de conjunto (o clase) y la técnica de diagramas de Venn, un concepto y una técnica que están expuestos en los principales textos de lógica elemental, incluyendo el referido anteriormente. -/- Abstract: Currently, one of the most widely used textbooks in elementary logic is the one by Irving Copi and Carl Cohen (Introduction to Logic, 2001). They offer rules for deciding the validity of standard categorical syllogisms. But neither that textbook, nor any other we know of, provides the foundations of these rules; that is to say, the demonstration that they are a necessary and sufficient condition for the validity of a standard categorical syllogism. We have only found motivations for the necessary condition. The main goal of this work is to provide a foundation of these rules using the notion of set as well as Venn diagrams, a concept and a technique exposed in the main textbooks of logic, including the aforementioned. (shrink)

Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers, even of those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that commonly called "Eulerian circles," has met with any general acceptance. A variety of others indeed have been proposed by ingenious and celebrated logicians, several of which would (...) claim notice in a historical treatment of the subject; but they mostly do not seem to me to differ in any essential respect from that of Euler. They rest upon the same leading principle, and are subject all alike to the same restrictions and defects. We must therefore cast about for some new scheme of diagrammatic representation which shall be competent to indicate imperfect knowledge on our part; for this will at once enable us to appeal to it step by step in the process of working out our conclusions. I have never seen any hint at such a scheme, though the want seems so evident that one would suppose that something of the kind must have been proposed before. (shrink)

In this paper we argue that the different positions taken by Dyson and Feynman on Feynman diagrams’ representational role depend on different styles of scientific thinking. We begin by criticizing the idea that Feynman Diagrams can be considered to be pictures or depictions of actual physical processes. We then show that the best interpretation of the role they play in quantum field theory and quantum electrodynamics is captured by Hughes' Denotation, Deduction and Interpretation theory of models (DDI), where “models” are (...) to be interpreted as inferential, non-representational devices constructed in given social contexts by the community of physicists. (shrink)

This article synthesizes the current theoretical attempts to understand human-machine interactions and introduces seven premises to understand our emerging dynamics with increasingly competent, pervasive, and instantly accessible algorithms. The hope that these seven premises can build toward a social theory of human-AI cocreation. The focus on human-AI cocreation is intended to emphasize two factors. First, is the fact that our machine learning systems are socialized. Second, is the coevolving nature of human mind and AI systems as smart devices form an (...) indispensable part of our cognitive scaffolds, thus, shaping how we perceive the world and ourselves. The seven premises include: primacy of social structures; human’s desire for freedom and autonomy; AI systems will form inseparable parts of our cognitive/affective scaffolds and can change our self-understanding; philosophy and humanistic foundations on human flourishing as a guide to human-AI interaction; mindsponge information-filtering process; acculturative process of how values change and emerge from human-AI interaction; overlapping Venn diagram of the human-human, human-nature, and human-machine interactions. This article concludes with a discussion on human agency in our entanglements with socialized machines and the illusoriness of the Cartesian agent view of the mind. (shrink)

The following technique has proven effective in helping beginning logic students locate the sections of a three-circled Venn Diagram in which they are to represent a categorical sentence. Very often theses students are unable to identify the parts of the diagram they are to shade or bar.

In a time which it is not amiss to term “the Dark Ages of logic”, Karl Christian Friedrich Krause stayed not only true to logic but actually did something for its advancement. Besides making systematic use of Venn-diagrams long before Venn, Krause — once more taking his inspiration from Leibniz — propounded what appears to be the first completely symbolic systematic representation of logical forms, strongly suggestive of the powerful symbolic languages that have become the mainstay of logic since the (...) beginning of the 20th century. However, Krause’s limits in logic are also clearly visible: Krause’s method in logic is, in the main, not axiomatic; it is combinatorial. More importantly, Krause remained entirely within the confines of traditional syllogistics, neglecting propositional logic and, of course, first-order relational terms. (shrink)

When interpreting the meanings of visual features in information visualizations, observers have expectations about how visual features map onto concepts (inferred mappings.) In this study, we examined whether aspects of inferred mappings that have been previously identified for colormap data visualizations generalize to a different type of visualization, Venn diagrams. Venn diagrams offer an interesting test case because empirical evidence about the nature of inferred mappings for colormaps suggests that established conventions for Venn diagrams are counterintuitive. Venn diagrams represent classes (...) using overlapping circles and express logical relationships between those classes by shading out regions to encode the concept of non-existence, or none. We propose that people do not simply expect shading to signify non-existence, but rather they expect regions that appear as holes to signify non-existence (the hole hypothesis.) The appearance of a hole depends on perceptual properties in the diagram in relation to its background. Across three experiments, results supported the hole hypothesis, underscoring the importance of configural processing for interpreting the meanings of visual features in information visualizations. (shrink)

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...) diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. (shrink)

This long essay was published in Vital Beauty, a collection including Wendy Steiner and Tim Ingold, which investigates the possibility of new ways toward beauty. This is my first encounter with Hartshorne’s Diagram of Aesthetic Values, a mandala-like structure explaining the relations between aesthetic experiences. The essay looks into the awkward history of the diagram in Hartshorne’s philosophy, its connection to Max Dessoir’s work, to Whitehead’s chapter on beauty in Adventures of Ideas and the notion of creativity in (...) Schelling. (shrink)

In his 1903 Syllabus, Charles S. Peirce makes a distinction between icons and iconic signs, or hypoicons, and briefly introduces a division of the latter into images, diagrams, and metaphors. Peirce scholars have tried to make better sense of those concepts by understanding iconic signs in the context of the ten classes of signs described in the same Syllabus. We will argue, however, that the three kinds of hypoicons can better be understood in the context of Peirce's sixty-six classes of (...) signs. We analyze examples of hypoicons taken from the field of information design, describing them in the framework of the sixty-six classes, and discuss the consequences of those descriptions to the debate about the order of determination of the 10 trichotomies that form those classes. (shrink)

In his article “Insight: Genesis and Ongoing Context,” Fred Crowe calls out Lonergan’s line “the diagram is more important than…is ordinarily believed” as the “philosophical understatement of the century.” Sixteen pages later he identifies elaborating an invariant cognitional theory to underlie generalized emergent probability and thus “the immanent order of the universe of proportionate being,” as “our challenge,” “but given the difficulty” he does not “see any prospect for an immediate answer.” Could this have something to do with the (...) lack of a comprehensive diagram of cognitional theory? Appendix A of The Boston College Lectures on Mathematical Logic and Existentialism offers diagrams of the dynamics of knowing and doing perhaps copied from Lonergan’s own blackboard work, but they do not distinguish explanatory and descriptive insights, let alone statistical insights, and do not illustrate the pull upwards or the fusing of routinized insights. Before we can effectively relate cognitional theory to generalized emergent probability, we must have an adequately rigorous and precise cognitional theory. I firmly believe in the truth of Lonergan’s fundamental insights, but in order to rigorously undergird generalized emergent probability there are many pertinent questions about his cognitional theory which must be asked and answered. In this paper I (1) review some of Lonergan’s attempts to diagram cognitional theory and discuss what insights they do and do not express, (2) elaborate and defend principles for making our formulations of insight into insight rigorous and clear, and (3) attempt to build up a diagram which makes full use of those principles in a maximally expressive way. (shrink)

This article explores aspects of Rosenzweig’s Star of Redemption from the perspective of systems theory. Mosès, Pollock, and others have noted the systematic character of the Star. While “systematic” does not mean “systems theoretic,” the philosophical theology of the Star encompasses ideas that are salient in systems theory. The Magen David star to which the title refers, and which deeply structures Rosenzweig’s thought, fits the classic definition of “system” – a set of elements (God, World, Human) and relations between the (...) elements (Creation, Revelation, Redemption). The Yes and No of the elements and their reversals illustrate the bridging of element and relation with the third category of “attribute,” a notion also central to the definition of “system.” In the diachronics of “the All,” the relations actualize what is only potential in the elements in their primordial state and thus remedy the incompleteness of these elements, fusing them into an integrated whole. Incompleteness is a major theme of systems theory, which also explicitly examines the relations between wholes and parts and offers a formal framework for expressing such fusions. In this article, the systems character of Parts I & II of the Star is explored through extensive use of diagrams; a systems exploration of Part III is left for future work. Remarkably, given its highly architectonic character, diagrams are absent in Rosenzweig’s book, except for the triangle of elements, the triangle of relations, and the hexadic star, which are presented on the opening page of each part of the book. While structures can be explicated entirely in words, diagrams are a visual medium of communication that supplements words and supports a nonverbal understanding that structures both thought and experience. (shrink)

This document diagrams the forms OOA, OOE, OOI, and OOO, including all four figures. Each form and figure has the following information: (1) Premises as stated: Venn diagram showing what the premises say; (2) Purported conclusion: diagram showing what the premises claim to say; (3) Relation of premises to conclusion: intended to describe how the premises and conclusion relate to each other, such as validity or contradiction. Used in only a few examples; (4) Distribution: intended to create a (...) system in which each syllogism has a unique code. In each premise and conclusion, the terms are each assigned a one or a zero, based on whether the term is distributed; (5) Rules: lists the rules of the syllogism and shows whether that particular syllogism follows, violates, or is unaffected by, each rule. (shrink)

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

Using as case studies two early diagrams that represent mechanisms of the cell division cycle, we aim to extend prior philosophical analyses of the roles of diagrams in scientific reasoning, and specifically their role in biological reasoning. The diagrams we discuss are, in practice, integral and indispensible elements of reasoning from experimental data about the cell division cycle to mathematical models of the cycle’s molecular mechanisms. In accordance with prior analyses, the diagrams provide functional explanations of the cell cycle and (...) facilitate the construction of mathematical models of the cell cycle. But, extending beyond those analyses, we show how diagrams facilitate the construction of mathematical models, and we argue that the diagrams permit nomological explanations of the cell cycle. We further argue that what makes diagrams integral and indispensible for explanation and model construction is their nature as locality aids: they group together information that is to be used together in a way that sentential representations do not. (shrink)

Robert Brandom’s expressivism argues that not all semantic content may be made fully explicit. This view connects in interesting ways with recent movements in philosophy of mathematics and logic (e.g. Brown, Shin, Giaquinto) to take diagrams seriously - as more than a mere “heuristic aid” to proof, but either proofs themselves, or irreducible components of such. However what exactly is a diagram in logic? Does this constitute a semiotic natural kind? The paper will argue that such a natural kind (...) does exist in Charles Peirce’s conception of iconic signs, but that fully understood, logical diagrams involve a structured array of normative reasoning practices, as well as just a “picture on a page”. (shrink)

Reism or concretism are the labels for a position in ontology and semantics that is represented by various philosophers. As Kazimierz Ajdukiewicz and Jan Woleński have shown, there are two dimensions with which the abstract expression of reism can be made concrete: The ontological dimension of reism says that only things exist; the semantic dimension of reism says that all concepts must be reduced to concrete terms in order to be meaningful. In this paper we argue for the following two (...) theses: (1) Arthur Schopenhauer has advocated a reistic philosophy of language which says that all concepts must ultimately be based on concrete intuition in order to be meaningful. (2) In his semantics, Schopenhauer developed a theory of logic diagrams that can be interpreted by modern means in order to concretize the abstract position of reism. Thus we are not only enhancing Jan Woleński’s list of well-known reists, but we are also adding a diagrammatic dimension to concretism, represented by Schopenhauer. (shrink)

The recent wave of data on exoplanets lends support to METI ventures (Messaging to Extra-Terrestrial Intelligence), insofar as the more exoplanets we find, the more likely it is that “exominds” await our messages. Yet, despite these astronomical advances, there are presently no well-confirmed tests against which to check the design of interstellar messages. In the meantime, the best we can do is distance ourselves from terracentric assumptions. There is no reason, for example, to assume that all inferential abilities are language-like. (...) With that in mind, I argue that logical reasoning does not have to be couched in symbolic notation. In diagrammatic reasoning, inferences are underwritten, not by rules, but by transformations of self-same qualitative signs. I use the Existential Graphs of C. S. Peirce to show this. Since diagrams are less dependent on convention and might even be generalized to cover non-visual senses, I argue that METI researchers should add some form of diagrammatic representations to their repertoire. Doing so can shed light, not just on alien minds, but on the deepest structures of reasoning itself. (shrink)

The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagrams in a visual half field task design. In the first part, (...) we tested whether the processing of metric vs topological relations yielded the same hemispheric specialization as the processing of coordinate vs categorical relations. In the second part, we investigated the specific performance patterns for the processing of five pairs of exact/co-exact relations, where stimuli for the co-exact relations were divided into three categories depending on their distance from the exact case. Regarding the processing of metric vs topological relations, hemispheric differences were found for only a few of the stimuli used, which may indicate that other processing mechanisms might be at play. Regarding the processing of exact vs co-exact relations, results show that the level of agreement among participants in judging co-exact relations decreases with the distance from the exact case, and this for the five pairs of exact/co-exact relations tested. The philosophical implications of these empirical findings for the epistemological analysis of Euclid's diagram-based geometric practice are spelled out and discussed. (shrink)

This paper describes how bodily positions and gestures were used to teach argument diagramming to a student who cannot see. After listening to short argumentative passages with a screen reader, the student had to state the conclusion while touching his belly button. When stating a premise, he had to touch one of his shoulders. Premises lending independent support to a conclusion were thus diagrammed by a V-shaped gesture, each shoulder proposition going straight to the conclusion. Premises lending dependent support were (...) diagrammed by a T-shaped gesture, the shoulder premises meeting at the collar bone before moving down to the belly button. Arguments involving two pairs of entailments were diagrammed by an I-shaped gesture, going from the collar bone to a mid-way conclusion above the abdomen before travelling to the final conclusion at the belly button. The student’s strong performance suggests that placing propositions at different locations on the body and uniting them with gestures can help one discern correct argumentative structures. (shrink)

After determining one set of skills that we hoped our students were learning in the introductory philosophy class at Carnegie Mellon University, we performed an experiment twice over the course of two semesters to test whether they were actually learning these skills. In addition, there were four different lectures of this course in the first semester, and five in the second; in each semester students in some lectures were taught the material using argument diagrams as a tool to aid understanding (...) and critical evaluation, while the other students were taught using more traditional methods. In each lecture, the students were given a pre-test at the beginning of the semester, and a structurally identical post-test at the end. We determined that the students did develop the skills in which we were interested over the course of the semester. We also determined that the students who were taught argument diagramming gained significantly more than the students who were not. We conclude that learning how to construct argument diagrams significantly improves a student’s ability to analyze arguments. (shrink)

This article presents some results of a research on computational strategies for the visualization of sign classification structures and sign processes. The focus of this research is the various classifications of signs described by Peirce. Two models are presented. One of them concerns specifically the 10-fold classification as described in the 1903 Syllabus (MS 540, EP 2: 289–299), while the other deals with the deep structure of Peirce’s various trichotomic classifications. The first is 10cubes, an interactive 3-D model of Peirce’s (...) 10-fold classification, as described in the Syllabus. The second is 3N3, a computer program that builds equivalent diagrams for any n-trichotomic classification of signs. We are specially interested in how a graphic design methodology, associated with computer graphic resources and techniques, can contribute to the construction of interactive models that serve as tools for the investigation of C. S. Peirce’s theory of signs. (shrink)

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of (...) natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology. (shrink)

You ask if metaphysics is possible after modernity, or after Barth and Wittgenstein and Derrida and the critique of foundationalism? May I invite you, by way of response, to listen in on a conversation? It is a dialogue between what I will call a postcritical philosopher ("P") and a postcritical scriptural theologian —— I'll label the latter a "textualist" ("T"). What I mean by "postcritical" would be displayed as the pattern of inquiry traced by this dialogue. I take the term (...) "postcritical" from George Lindbeck, whose theological work is described by his commentators as more properly "textualist" rather than "philosophic." I believe my usage is, however, true to Lindbeck's discourse. This means that one of the arguments of this essay is that there is a mode of philosophic inquiry proper to the postcritical orientation in theology that we may associate with such Christian theologians as Hans Frei, etc., as well as Lindbeck. A second argument, offered indirectly, is that there is a family of Jewish thinkers whose work may also be labeled "postcritical" on the model of these Christian thinkers, and that this Jewish postcritical theology would also be displayed in dialogues between philosophic and textualist tendencies. A third argument is that, if we listen in on the philosophic side of any of these dialogues, we will hear a postcritical philosophy. By philosophy, I mean primarily logic, defined as an inquiry that examines problematic rules of reasoning and recommends testable ways of correcting them. Secondarily, philosophy includes metaphysics, defined as an inquiry that, for the sake of assuring people about the reliability of a given logic, constructs pictures of the world whose existence would warrant this logic. We need not assume either that logic and metaphysics can be conducted only in the manner of modern and scholastic foundationalism, or that non-foundationalist theology must be "textualist" in a manner that excludes any philosophic generalization. I will argue that there are, indeed, non-foundationalist ways of conceiving of both logical and metaphysical inquiry and that these ways contribute to the dialogic activity of postcritical inquiry. (shrink)

We advance a theory of the representational role of Euclidean diagrams according to which they are samples of co-exact features. We contrast our theory with two other conceptions, the instantial conception and Macbeth’s iconic view, with respect to how well they accommodate three fundamental constraints on theories of the Euclidean diagrammatic practice— that Euclidean diagrams are used in proofs whose results are wholly general, that Euclidean diagrams indicate the co-exact features that the geometer is allowed to infer from them and (...) that Euclidean diagrams play the same role in both direct proofs and indirect proofs by reductio—and argue that our view is the one best suited to account for them. We conclude by illustrating the virtues of our conception of Euclidean diagrams as samples by means of an analysis of Saccheri’s quadrilateral. (shrink)

This paper presents an analysis of Seneca's 58th letter to Lucilius and Porphyry's Isagoge, which were the origin of the tree diagrams that became popular in philosophy and logic from the early Middle Ages onwards. These diagrams visualise the extent to which a concept can be understood as a category, genus, species or individual and what the method of dihairesis (division) means. The paper explores the dissimilarities between Seneca's and Porphyry's tree structures, scrutinising them through the perspective of modern graph (...) theory. Although traditional trees can be interpreted in a modern way, their terms and definitions defy modern standardisation. These distinctions are not only useful for differentiating between Seneca's and Porphyry's trees, but also for comprehending the evolution of this philosophical tradition. Thus the paper highlights the continued significance of these tree diagrams in analysing concepts and categories in philosophy and structuring objects and information in modern fields such as artificial intelligence, computer science, bioinformatics, graph theory, and many more. (shrink)

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

If there is to be any progress in the debate about what sort of positive moral status Kant can give the emotions, we need a taxonomy of the terms Kant uses for these concepts. It used to be thought that Kant had little room for emotions in his ethics. In the past three decades, Marcia Baron, Paul Guyer, Barbara Herman, Nancy Sherman, Allen Wood and others have argued otherwise. Contrary to what a cursory reading of the Groundwork may indicate, Kant (...) thinks the emotions play an important role in the moral life. I want to extend the work of Baron, Guyer, Herman, Sherman and Wood in three ways. First, I will set out in a diagram Kant's taxonomy of feelings and emotions. Agreement on such a taxonomy should make it easier to evaluate debates about Kant and the emotions. Second, I will focus on a certain subclass of emotions – reason-caused affects – that have previously received little attention, even from these Kant scholars. Third, these scholars base much of their defence of Kant on his later works – especially the Metaphysics of Morals and the Anthropology – but Kant's fairly rich taxonomy of the emotions, including reason-caused affects, is clearly in place at least as early as the Critique of Judgment . I believe that the Critique of Judgment is an importantly ignored resource for understanding the moral role of the emotions for Kant. The third Critique makes positive, philosophically interesting claims about the emotions and morality. Kant emphasizes certain roles for emotions in this work that he develops to the same extent nowhere else. Nevertheless, the Critique of Judgment goes all but unmentioned by many who write on these issues. In what follows, I will defend as many of my claims as possible using the third Critique. (shrink)

I examine the passages where Aristotle maintains that intellectual activity employs φαντάσματα (images) and argue that he requires awareness of the relevant images. This, together with Aristotle’s claims about the universality of understanding, gives us reason to reject the interpretation of Michael Wedin and Victor Caston, on which φαντάσματα serve as the material basis for thinking. I develop a new interpretation by unpacking the comparison Aristotle makes to the role of diagrams in doing geometry. In theoretical understanding of mathematical and (...) natural beings, we usually need to employ appropriate φαντάσματα in order to grasp explanatory connections. Aristotle does not, however, commit himself to thinking that images are required for exercising all theoretical understanding. Understanding immaterial things, in particular, may not involve employing phantasmata. Thus the connection that Aristotle makes between images and understanding does not rule out the possibility that human intellectual activity could occur apart from the body. (shrink)

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

This article presents the first results of a research on visual models for the classifications of semiotic processes. The main issue discussed is how a graphic design methodology, associated with computer graphics resources, may contribute to the construction of interactive models, that can be used as tools for the investigation of C. S. Peirce theory of signs. Two models are presented: the first is an interactive 3-D model of Peirce's 3-trichotomic classification; the second is a computer program that builds diagrams (...) for any n-trichotomic classification of signs, and facilitates the analysis and comparison of different hypothesis regarding their structures. (shrink)

A concise open-access textbook intended for an undergraduate audience, which brings together essential selections from Gandhi on nonviolence with supplementary materials, including: a preface; boxes providing examples, historical notes, extended explanations, and related philosophical work; overviews of post-Gandhian developments in nonviolence; diagrams, tables, and photos; discussion questions; reading and viewing suggestions; and a glossary.

I propose in this text a rhythmic theory of signs drawn from the thought of Gilles Deleuze and Félix Guattari. I name this theory a semiorhythmology. I suggest that the theory of rhythm developed in A Thousand Plateaus (1980) can be understood, in part, as the culmination of the diverse set of inquiries into signs that both Deleuze and Guattari undertook, individually and together, beginning in the 1960s. I first outline Deleuze’s theory of signs as a theory of encounter as (...) developed in Proust and Signs (1964) and Difference and Repetition (1968), following which I sketch Guattari’s engagements with signs and semiotics throughout the 1960s and 1970s, particularly through his notion of “a-signifying semiotics” and the concept of the “diagram” he adapts from the semiotics of Charles Sanders Peirce. I close by showing how these heterogeneous theories of the sign are drawn together in A Thousand Plateaus through the Spinozist reading of the ethology of Jakob von Uexküll and the theorisation of rhythm in the form of the refrain. (shrink)

In his 1953 lectures at the College de France, Merleau-Ponty dedicated much effort to further developing his idea of embodied subject and interpreted fresh sources that he did not use in Phenomenology of Perception. Notably, he studied more in depth the neurological notion of "body schema". According to Merleau-Ponty, the body schema is a practical diagram of our relationships to the world, an action-based norm with reference to which things make sense. Merleau-Ponty more precisely tried to describe the fundamentally (...) dynamic unity of the body, i.e. the fact there are various possibilities how the practical "diagram" of body schema could be de-differentiated (in pathology) or further refined (via cognitive and cultural superstructures, symbolic systems). This chapter summarises Merleau-Ponty's interpretation of the notion, while contrasting it to the more traditional understanding of the body in phenomenology and recent philosophical texts dealing with body schema. (shrink)

Without doubt, there is a great diversity of scientific images both with regard to their appearances and their functions. Diagrams, photographs, drawings, etc. serve as evidence in publications, as eye-catchers in presentations, as surrogates for the research object in scientific reasoning. This fact has been highlighted by Stephen M. Downes who takes this diversity as a reason to argue against a unifying representation-based account of how visualisations play their epistemic role in science. In the following paper, I will suggest an (...) alternative explanation of the diversity of scientific images. This account refers to processes which are caused by the social setting of science. What exactly is meant by this, I will spell out with the aid of Ludwik Fleck’s theory of the social mechanisms of scientific communication. (shrink)

My first section considers Walter J. Ong’s influential analyses of the logical method of Peter Ramus, on whose system Milton based his Art of Logic. The upshot of Ong’s work is that philosophical logic has become a kind monarch over all other discourses, the allegedly timeless and universal method of mapping and diagramming all concepts. To show how Milton nevertheless resists this tyrannical result in his non-Logic writings, my second section offers new readings of Milton’s poems Il Penseroso and Sonnet (...) 16: “On His Blindness”, along with his prose epilogue to his elegies (and thereby the entire collection entitled Poems). These readings attempt to show (1) the original admixing of philosophy and poetry (under the heading of “thoughtfulness”), (2) the shadow-hidden superiority of poetry in connection to the effeminising disability of blindness, and (3) the potential irony of an apology that arguably suggests poetry’s superiority to philosophy. Finally, I rest my case for Milton’s rebellion by offering an interpretation of Paradise Lost which affirms the character of Satan qua dark, queer, poetic figure of classical republicanism. (shrink)

This article reconstructs Gilles Deleuze’s engagement with phenomenology, and with the phenomenological problematic of sensation, in his Francis Bacon: The Logic of Sensation. Considering Deleuze’s adoption, from the phenomenology of art, of notions of sensation and rhythm, it examines how Deleuze complexifies these phenomenological notions by aligning them with his profoundly non-phenomenological notion of the body without organs, as well as with the concepts of modulation and the diagram. In mapping Deleuze’s complexification of rhythm and his development of a (...) logic of rhythmic modulation, this article shows how Deleuze immanently refines an approach to working beyond phenomenology, through phenomenology. (shrink)

Scholars have thoroughly discussed the visual aspects of Foucault’s archaeological and genealogical methods, as well as his own emphasis on how sight functions and what contexts and conditions shape how we see and what we can see. Yet while some of the images and visual devices he uses are frequently discussed, like Las Meninas and the panopticon, his diagrams in The Order of Things have received little attention. Why does Foucault diagram historical ways of thinking? What are we supposed (...) to see and understand through these diagrams? To examine the role of the diagram in Foucault’s archaeological method, this paper provides a close reading of how the classical quadrilateral visualizes the structure, function, content, principles, and underlying assumptions of language and thought. In analyzing the diagram as a way for visualizing history, this paper demonstrates how Foucault enacts a new visual language that emphasizes the contingency of thought. (shrink)

The present paper deals thus with some fundamental agreements and disagreements between Peirce and James, on crucial issues such as perception and consciousness. When Peirce first read the Principles, he was sketching his theory of the categories, testing its applications in many fields of knowledge, and many investigations were launched, concerning indexicals, diagrams, growth and development. James's utterances led Peirce to make his own views clearer on a wide range of topics that go to the heart of the foundations of (...) psychology and that involve the relationship between perception and logic, between consciousness and the categories, between abstraction and the 'stream of thought'. The idea is to show that Peirce detected important discoveries and insights in the Principles, but felt that James could not make proper use of them because of logical confusions, and also because of his "clandestine" metaphysics. The point in this essay is thus not to look for remains of psychologism in Peirce's writings,13 but to look at Peirce's comments about James's psychology in an attempt to identify where and why Peirce amended James's views. Since the project to provide some insight on Peirce's extensive reading ofJames's Principles of Psycho/.ogy would deserve a full volume, I shall focus here on three occasions where Peirce explicidy commented on Jarnes's Principles. In the first section, I shall consider bis assessment of James's chapter on space, which was published as a series of articles in 1887, in Mind. I shall then turn to the 1891 review of the Principles in The Nation for important complements on perception as inference. In the third section, I shall deal with Peirce's manuscript "Questions on James's Principles"(Rl099). These "Questions" reveal a deep interest in psychological problems and suggest different ways along which Peirce's new advances in the field of the categories, of continuity, and abstraction could provide a proper basis for the philosophy of mind. (shrink)

There are no established conventions for, and few examples of, indexing visual material on the basis of its form. Most image databases use keywords to describe the form or function, and access data by text-based retrieval of these keywords. An image-based approach would order the data by appearance, e.g. Shepherd (1971) and Dreyfuss (1972). A taxonomy must be created in order to apply this technique to a new data set. Previous applications have been aided by certain limiting factors on the (...) possible range of images indexed thus providing the key to a taxonomy, e.g. international pictorial signs in Dreyfuss. A fundamental principle which may be inferred from these studies is that the taxonomy needs to be related to the needs of the user, and not solely to abstract features of design. This is reflected in guidelines issued by the UK organisation The Society of Indexers. The studies also suggest the need for several approaches to classification within the same data set, e.g Dreyfuss, in which signs are located under more than one heading. (shrink)

This book is written for those who wish to learn some basic principles of formal logic but more importantly learn some easy methods to unpick arguments and assess their value for truth and validity. -/- The first section explains the ideas behind traditional logic which was formed well over two thousand years ago by the ancient Greeks. Terms such as ‘categorical syllogism’, ‘premise’, ‘deduction’ and ‘validity’ may appear at first sight to be inscrutable but will easily be understood with examples (...) bringing the subjects to life. Traditionally, Venn diagrams have been employed to test arguments. These are very useful but their application is limited and they are not open to quantification. The mid-section of this book introduces a methodology that makes the analysis of arguments accessible with the use of a new form of diagram, modified from those of the mathematician Leonhard Euler. These new diagrammatic methods will be employed to demonstrate an addition to the basic form of syllogism. This includes a refined definition of the terms ‘most’ and ‘some’ within propositions. This may seem a little obscure at the moment but one will readily apprehend these new methods and principles of a more modern logic. (shrink)

A particularly promising trail on the search for forgotten logic diagrams leads to Upper Lusatia in the 17th century, more precisely to Christian Weise and his students. Samuel Grosser, who later became rector in Görlitz, and Johann Christian Lange, who later became professor of logic at the University of Gießen, are the most prominent to have published remarkable logic diagrams. Even more remarkable, however, is the fact that Lange's interest in these diagrams ultimately gave rise to the idea of building (...) a logic machine. (shrink)

This chapter is a rethinking of my earlier “The Ages of Beauty” which investigated Charles Hartshorne’s Diagram of Aesthetic Values. The argument is placed in a long history of beauty being considered as the middle between extremes. It slowly develops into a structure not merely of aesthetic experience but of existence itself, making it a competitor of Heidegger’s fourfold.

In this paper, we will make explicit the relationship that exists between geometric objects and geometric figures in planar Euclidean geometry. That will enable us to determine basic features regarding the role of geometric figures and diagrams when used in the context of pure and applied planar Euclidean geometry, arising due to this relationship. By taking into account pure geometry, as developed in Euclid’s Elements, and practical geometry, we will establish a relation between geometric objects and figures. Geometric objects are (...) defined in terms of idealizations of the corresponding figures of practical geometry. We name the relationship between them as a relation of idealization. This relation, existing between objects and figures, is what enables figures to have a role in pure and applied geometry. That is, we can use a figure or diagram as a representation of geometric objects or composite geometric objects because the relation of idealization corresponds to a resemblance-like relationship between objects and figures. Moving beyond pure geometry, we will defend that there are two other ‘layers’ of representation at play in applied geometry. To show that, we will consider Euclid’s Optics. (shrink)