This paper discusses similarities of both form and meaning between two symbolic structures: the Diagram of the Supreme Pole of Song Neo-Confucianism and the Kabbalistic Tree of medieval Jewish mysticism. These similarities are remarkable in the light of the many differences that exist between Chinese and Judaic thought, which also manifest in the two symbols. Intercultural influence might account for the similarities, but there is no historical evidence for such influence. An alternative explanation would attribute the similarities to the (...) ubiquity of religious-philosophical ideas about hierarchy, polarity, and macrocosm-microcosm parallelism, but this does not adequately account for the similar overall structure of the symbols. The question of how to understand these similarities remains open. (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)

Creationists who object to evolution in the science curriculum of public schools often cite Jonathan Well’s book Icons of Evolution in their support (Wells 2000). In the third chapter of his book Wells claims that neither paleontological nor molecular evidence supports the thesis that the history of life is an evolutionary process of descent from preexisting ancestors. We argue that Wells inappropriately relies upon ambiguities inherent in the term ‘Darwinian’ and the phrase ‘Darwin’s theory’. Furthermore, he does not accurately distinguish (...) between the overwhelming evidence that supports the thesis of common descent and controversies that pertain to causal mechanisms such as natural selection. We also argue that Wells’ attempts to undermine the evidence in support of common descent are flawed and his characterization of the relevant data is misleading. In particular, his assessment of the ‘Cambrian explosion’ does not do justice to the fossil record. Nor do his selective references to debate about molecular and paleontological phylogenies constitute a case against common descent. We conclude that the fossil and molecular evidence is more than sufficient to warrant science educators to present common descent as a well-established scientific fact. We also argue that diagrams depicting the ‘tree of life’ can be pedagogically useful as simplified representations of the history of life. (shrink)

In this paper, we present a survey of the development of the technique of argument diagramming covering not only the fields in which it originated - informal logic, argumentation theory, evidence law and legal reasoning – but also more recent work in applying and developing it in computer science and artificial intelligence. Beginning with a simple example of an everyday argument, we present an analysis of it visualised as an argument diagram constructed using a software tool. In the context of (...) a brief history of the development of diagramming, it is then shown how argument diagrams have been used to analyze and work with argumentation in law, philosophy and artificial intelligence. (shrink)

In a multi-study naturalistic quasi-experiment involving 269 students in a semester-long introductory philosophy course, we investigated the effect of teaching argument diagramming on students’ scores on argument analysis tasks. An argument diagram is a visual representation of the content and structure of an argument. In each study, all of the students completed pre- and posttests containing argument analysis tasks. During the semester, the treatment group was taught AD, while the control group was not. The results were that among the different (...) pretest achievement levels, the scores of low-achieving students who were taught AD increased significantly more than the scores of low-achieving students who were not taught AD, while the scores of the intermediate- and high-achieving students did not differ significantly between the treatment and control groups. The implication of these studies is that learning AD significantly improves low-achieving students’ ability to analyze arguments. (shrink)

Our aim here is to lay the groundwork for formal tree-ring analysis combining data from dendrochronology with formal techniques from semantics. We will present the basic syntax of, and basic compositional semantics of tree-ring structures.

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

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

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that (...) one of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

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

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)

Historians occasionally use timelines, but many seem to regard such signs merely as ways of visually summarizing results that are presumably better expressed in prose. Challenging this language-centered view, I suggest that timelines might assist the generation of novel historical insights. To show this, I begin by looking at studies confirming the cognitive benefits of diagrams like timelines. I then try to survey the remarkable diversity of timelines by analyzing actual examples. Finally, having conveyed this (mostly untapped) potential, I (...) argue that neglecting timelines might mean neglecting significant aspects of reality that are revealed only by those signs. My overall message is that once we accept that relations are as important for the mind as what they relate, we have to pay closer attention to any semiotic device that enables or facilitates the discernment of new relations. (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)

There are two important ways in which, when dealing with documents, we go beyond the boundaries of linear text. First, by incorporating diagrams into documents, and second, by creating complexes of intermeshed documents which may be extended in space and evolve and grow through time. The thesis of this paper is that such aggregations of documents are today indispensable to practically all complex human achievements from law and finance to orchestral performance and organized warfare. Documents provide for what we (...) can think of as a division of intellectual, instructional, and deontic labour, allowing plans, orders, and obligations to be enmeshed together in a way that often involves the use of diagrammatic elements, as for example in a musical score. (shrink)

A common complaint about Kantian ethics is that it cannot do justice to the social or intersubjective dimensions of human life – that, unlike Fichte, Hegel, or Marx, Kant remains caught within a fundamentally individualistic perspective on practical or moral questions. In this way, the objection goes, Kantian ethics leaves agents alienated from others around them and their larger community. While not entirely unnatural, I argue here that such concerns rest on a mischaracterization of where the most serious problems in (...) this region for Kant lie. Far from being too individualistic, the real worry about Kantian moral theory is that it may not be individualistic enough. In a contemporary context, where Kantian ethics is associated with the “self-constitution” of agents or the “separateness of persons”, the idea that it is insufficiently concerned with individual persons may come as a surprise. But it would not surprise Kant’s immediate successors like Fichte and Hegel. For this is where they often located the deepest problems facing Kant’s moral philosophy. As we will see, in making such points, they were in some ways closer to Kant’s own views than they recognized. Nonetheless, if this is right, Kantian ethics may in fact more anti-individualistic than many of the ethical systems that followed in his wake. And the most radical elements in Kant’s ethics might point, less to Fichte and Hegel, and more to figures like Schopenhauer who sought to develop a synthesis of Kant and ideas from South Asian philosophy. Perhaps, we might say, Kant has been sitting under the Bodhi tree all along? (shrink)

We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the four (...) classes of trees mentioned above. (shrink)

The argument diagramming method developed by Monroe C. Beardsley in his (1950) book Practical Logic, which has since become the gold standard for diagramming arguments in informal logic, makes it possible to map the relation between premises and conclusions of a chain of reasoning in relatively complex ways. The method has since been adapted and developed in a number of directions by many contemporary informal logicians and argumentation theorists. It has proved useful in practical applications and especially pedagogically in teaching (...) basic logic and critical reasoning skills at all levels of scientific education. I propose in this essay to build on Beardsley diagramming techniques to refine and supplement their structural tools for visualizing logical relationships in a number of categories not originally accommodated by Beardsley diagramming, including circular reasoning, reductio ad absurdum arguments, and efforts to dispute and contradict arguments, with applications and analysis. (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)

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.

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)

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)

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)

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)

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)

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 aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this (...) reason, experts must develop a specific form of enhanced manipulative imagination, in order to draw inferences from knot diagrams by performing epistemic actions. Moreover, it will be argued that knot diagrams not only can promote discovery, but also provide evidence. This case study is an experimentation ground to evaluate the role of space and action in making inferences by reasoning diagrammatically. (shrink)

While mechanistic explanation and, to a lesser extent, nomological explanation are well-explored topics in the philosophy of biology, topological explanation is not. Nor is the role of diagrams in topological explanations. These explanations do not appeal to the operation of mechanisms or laws, and extant accounts of the role of diagrams in biological science explain neither why scientists might prefer diagrammatic representations of topological information to sentential equivalents nor how such representations might facilitate important processes of explanatory reasoning (...) unavailable to scientists who restrict themselves to sentential representations. Accordingly, relying upon a case study about immune system vulnerability to attacks on CD4+ T-cells, I argue that diagrams group together information in a way that avoids repetition in representing topological structure, facilitate identification of specific topological properties of those structures, and make available to controlled processing explanatorily salient counterfactual information about topological structures, all in ways that sentential counterparts of diagrams do not. (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)

Logic diagrams have seen a resurgence in their application in a range of fields, including logic, biology, media science, computer science and philosophy. Consequently, understanding the history and philosophy of these diagrams has become crucial. As many current diagrammatic systems in logic are based on ideas that originated in the 18th and 19th centuries, it is important to consider what motivated the use of logic diagrams in the past and whether these reasons are still valid today. This (...) paper proposes that transcendental philosophy was a key inspiration for the development of logic diagrams and that such diagrams can be employed in transcendental arguments, even after the linguistic turn. (shrink)

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)

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)

Pereira’s “The Projective Theory of Consciousness” is an experimental statement, drawing on many diverse sources, exploring how consciousness might be produced by a projective mechanism that results both in private selves and an experienced world. Unfortunately, pulling together so many unrelated sources and methods means none gets full attention. Furthermore, it seems to me that the uncomfortable breadth of this paper unnecessarily complicates his project; in fact it may hide what it seeks to reveal. If this conglomeration of diverse sources (...) and methods were compared to trees, the reader may feel like the explorer who cannot see the forest for the trees. Then again, it may be the author who is so preoccupied with foreground figures that the everpresent background is ultimately obscured. (shrink)

Given a class of linear order types C, we identify and study several different classes of trees, naturally associated with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and completely determine the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We then obtain some general results about the axiomatization of the first-order theories of some of these classes of trees in terms of (...) the first-order theory of the generating class C, and indicate the problems obstructing such general results for the other classes. These problems arise from the possible existence of nondefinable paths in trees, that need not satisfy the first-order theory of C, so we have started analysing first order definable and undefinable paths in trees. (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)

The Tree of Life has traditionally been understood to represent the history of species lineages. However, recently researchers have suggested that it might be better interpreted as representing the history of cellular lineages, sometimes called the Tree of Cells. This paper examines and evaluates reasons offered against this cellular interpretation of the Tree of Life. It argues that some such reasons are bad reasons, based either on a false attribution of essentialism, on a misunderstanding of the problem (...) of lineage identity, or on a limited view of scientific representation. I suggest that debate about the Tree of Cells and other successors to the traditional Tree of Life should be formulated in terms of the purposes these representations may serve. In pursuing this strategy, we see that the Tree of Cells cannot serve one purpose suggested for it: as an explanation for the hierarchical nature of taxonomy. We then explore whether, instead, the tree may play an important role in the dynamic modeling of evolution. As highly-integrated complex systems, cells may influence which lineage components can successfully transfer into them and how they change once integrated. Only if they do in fact have a substantial role to play in this process might the Tree of Cells have some claim to be the Tree of Life. (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.

A detailed axiomatisation of diagrams (in affine geometry) is presented, which supports typing of geometric objects, calculation of geometric quantities and automated proof of theorems.

Many undergraduates, philosophy majors included, read philosophical texts similar to the way they read stories. One method for teaching students how to discern the argumentative structure of a philosophy text is through argument diagrams (text boxes used to represent claims with arrows and lines used to represent connections between these claims). This paper provides criteria for an ideal argument diagramming software and then reviews the strengths and weaknesses of such software currently available, e.g. Araucaria, Argutect, Athena Standard, Inspiration, and (...) Reason!Able. (shrink)

Do trees of life have roots? What do these roots look like? In this contribution, I argue that research on the origins of life might offer glimpses on the topology of these very roots. More specifically, I argue (1) that the roots of the tree of life go well below the level of the commonly mentioned ‘ancestral organisms’ down into the level of much simpler, minimally living entities that might be referred to as ‘protoliving systems’, and (2) that further (...) below, a system of roots gradually dissolves into non-living matter along several functional dimensions. In between non-living and living matter, one finds physico-chemical systems that I propose to characterize by a ‘lifeness signature’. In turn, this ‘lifeness signature’ might also account for a diverse range of biochemical entities that are found to be ‘less-than-living’ yet ‘more-than-non-living’. (shrink)

This paper will present two contributions to teaching introductory logic. The first contribution is an alternative tree proof method that differs from the traditional one-sided tree method. The second contribution combines this tree system with an index system to produce a user-friendly tree method for sentential modal logic.

The paper aims at critical reconsideration of a motif popular in Indian literary, ritual, and pictorial traditions – a tree goddess (yakṣī, vṛkṣakā) or a woman embracing a tree (śālabhañjīkā, dohada), which points to a close and intimate bond between women and trees. At the outset, I present the most important phases of the evolution of this popular motif from the ancient times to present days. Then two essential characteristics of nature recognized in Indian visual arts, literature, religions (...) and philosophy will be distinguished: (1) a dynamic, creative, self-sufficient and inexhaustible power, and (2) a passive, merely reproductive or vegetative, and dependent field of potentiality. The paper is to demonstrate the interdependence of the popular concepts of nature identified with femininity, and their iconic representations circulating for centuries in Indian culture, with a specific line of argument repeatedly used in social practices and public debates. While doing so, I consider the semiotic function of a cultural topos which proves to be an effective instrument for construing and supporting the gender roles and gender identities. As a modern example illustrating vitality and persuasive power of the motif of yakṣī and śālabhañjīkā, I refer to the Chipko Movement, a group of rural women based in the Garhwal Himalayas (state Uttarakhand), who fought against the mass cut of trees in the 1970s. They were involved in the wide-spread environmental campaign which significantly affected the ecological policy of the local and state authorities. Thus, a traditional motif of the visual arts has been revived and re-elaborated by the activists of this ecofeminist movement through converting the symbolic potential of yakṣī/śālabhañjīkā into social and political power. (shrink)

Logicians commonly speak in a relatively undifferentiated way about pre-euler diagrams. The thesis of this paper, however, is that there were three periods in the early modern era in which euler-type diagrams (line diagrams as well as circle diagrams) were expansively used. Expansive periods are characterized by continuity, and regressive periods by discontinuity: While on the one hand an ongoing awareness of the use of euler-type diagrams occurred within an expansive period, after a subsequent phase (...) of regression the entire knowledge about the systematic application and the history of euler-type diagrams was lost. I will argue that the first expansive period lasted from Vives (1531) to Alsted (1614). The second period began around 1660 with Weigel and ended in 1712 with lange. The third period of expansion started around 1760 with the works of Ploucquet, euler and lambert. Finally, it is shown that euler-type diagrams became popular in the debate about intuition which took place in the 1790s between leibnizians and Kantians. The article is thus limited to the historical periodization between 1530 and 1800. (shrink)

We investigate the formal cause of diagrams, initially realizing that diagrams have no obvious form. It is argued their form is to mimic expert perspectives. This perspective provides a organizational structure that represents the relations important in understanding the worldly situation. We then shift to a study of how we are to understand an expert perspective. Using the distinction between intuitive and formal logic, logica utens versus logica docens, we identify games of habituation: games of focus and distraction. (...) The skills required for games of focus and distraction are phenomenological in regulating the way we see, i.e., how we approach and analyze different situations. This unique phenomenological skill is characteristic of diagrammatic reasoning. (shrink)

In Sense and Content , Christopher Peacocke points out that two equally-sized trees at different distances from the perceiver are normally represented to be the same size, despite the fact that in a certain sense the nearer tree looks bigger ; he concludes on the basis of this observation that visual experiences possess irreducibly phenomenal properties. This argument has received the most attention of all of Peacocke’s arguments for separatism—the view that the intentional and phenomenal properties of experiences are (...) independent of one another. However, despite its notoriety, the argument is widely misunderstood and underappreciated. I argue that once the structure of the argument is clarified and the replies that have been offered are considered closely, one must conclude that the trees argument is successful. (shrink)

In this paper I begin by extending two results of Solovay; the first characterizes the possible Turing degrees of models of True Arithmetic (TA), the complete first-order theory of the standard model of PA, while the second characterizes the possible Turing degrees of arbitrary completions of P. I extend these two results to characterize the possible Turing degrees of m-diagrams of models of TA and of arbitrary complete extensions of PA. I next give a construction showing that the conditions (...) Solovay identified for his characterization of degrees of models of arbitrary completions of PA cannot be dropped (I showed that these conditions cannot be simplified in the paper. (shrink)

The purpose of this paper is to point the way to the money tree. Currently, almost all investment professionals think that outperformance requires an “edge,” that is, the ability to predict the future to some degree. In this paper, I suggest that money can be made in a 0, or even slightly negative, expected value environment by carefully choosing investment/bet sizes. Philosophical considerations are found mainly in Section 4.

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)