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)

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)

Bayesian networks (BN) and argumentation diagrams (AD) are two predominant approaches to legal evidential reasoning, that are often treated as alternatives to one another. This paper argues that they are, instead, complimentary and proposes the beginnings of a method to employ them in such a manner. The Bayesian approach tends to be used as a means to analyse the findings of forensic scientists. As such, it constitutes a means to perform evidential reasoning. The design of Bayesian networks that accurately (...) and comprehensively represent the relationships between investigative hypotheses and evidence remains difficult and sometimes contentious, however. Argumentation diagrams are representations of reasoning, and are used as a means to scrutinise reasoning (among other applications). In evidential reasoning, they tend to be used to represent and scrutinise the way humans reason about evidence. This paper examines how argumentation diagrams can be used to scrutinise Bayesian evidential reasoning by developing a method to extract argument diagrams from BN. (shrink)

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)

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)

After determining one set of skills that we hoped our students were learning in the introductory philosophy class at Carnegie Mellon University, we designed an experiment, performed 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 Spring of 2004, and five in the Fall of 2004; and the students of Lecturer I 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. We were interested in whether this tool would help the students develop the skills we hoped they would master in this course. 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 able to construct argument diagrams gained significantly more than the other students. We conclude that learning how to construct argument diagrams improves a student's ability to analyze, comprehend, and evaluate arguments. (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)

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)

The importance of teaching critical thinking skills at the college level cannot be overemphasized. Teaching a subcategory of these skills—argument analysis—we believe is especially important for first-year students with their college careers, as well as their lives, ahead of them. The struggle, however, is how to effectively teach argument analysis skills that will serve students in a broad range of disciplines.

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)

The word “philosophy” comes from the Greek “philos” (meaning love) and “sophia” (meaning wisdom); thus philosophy literally is the “love of wisdom.” Whatever else philosophy may be, most people agree that it still retains this spirit of its etymological roots, and that when we are engaged in philosophy we are pursuing wisdom for the sake of itself. Wisdom, however, is not the same thing as knowledge or information. We aren’t merely trying to amass list of interesting ideas, or believe anything (...) that sounds good. Wisdom is, at least in part, the reflection on and critical evaluation of what we ourselves and others around us believe about some very heady topics. (shrink)

Research has shown that the construction of visual representations may have a positive effect on cognitive skills, including argumentation. In this paper we present a study on learning argumentation through computer-supported argument diagramming. We specifically focus on whether students, when provided with an argument-diagramming tool, create better diagrams, are more motivated, and learn more when working with other students or on their own. We use learning analytics to evaluate a variety of student activities: pre and post questionnaires (...) to explore motivational changes; the argument diagrams created by students to evaluate richness, complexity and completion; and pre and post knowledge tests to evaluate learning gains. (shrink)

There is substantial evidence from many domains that visual representations aid various forms of cognition. We aimed to determine whether learning to construct visual representations of argument structure enhanced the acquisition and development of argumentative writing skills within the context of first-year college writing 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 (...) first-year composition courses could provide substantial increases in student writing ability. (shrink)

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

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)

Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give (...) a natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs. (shrink)

The Toulmin diagram layout is very familiar and widely used, particularly in the teaching of critical thinking skills. The conventional box-and-arrow diagram is equally familiar and widespread. Translation between the two throws up a number of interesting challenges. Some of these challenges represent slightly different ways of looking at old and deep theoretical questions. Others are diagrammatic versions of questions that have already been addressed in artificial intelligence models of argument. But there are further questions that are posed as (...) a specific result of examining the diagram inter-translation problem. These three classes of problems are discussed. To the first class are addressed solutions based on engineering pragmatism; to the second class are addressed solutions drawn from the appropriate literature; and to the third class, fuller exploration is offered justifying the approaches taken in developing solutions that offer both pragmatic utility and theoretical interest. Finally, these solutions are explored briefly in the context of the Araucaria system, showing the ways in which analysts can tackle arguments either using one diagrammatic style or another, or even a combination of the two. (shrink)

Arguments with what are called "independent" or "convergent" premises are typically diagrammed by using an arrow between each premise and the conclusion. This makes diagramming objections to the reasoning difficult. It also obscures differences in argument structure. I suggest that a single arrow should be used for such arguments and that this is so even in the extreme form of independent premises when the argument is entirely unstructured. I then discuss the diagramming of objections.

The attempt to find usable diagrams fornterms of the sort devised by John Venn seems to have originated with Venn himself, who published diagrams for up to five classes (the fifth class, however, was shaped like a doughnut, and contained an area outside itself — like the hole in the doughnut). Venn then suggested that “if we wanted to use a diagram forsixterms (x, y, z, w, v, u) the best plan would probably be to taketwofive-term figures, one (...) for theupart and one for the non-upart of all the other combinations …” Such a method would, as Venn admits, be somewhat confusing to the eye, and thus fail to fulfil one of the major purposes of the diagram; nevertheless, it is a method which could be extended to covernclasses.Venn also suggests (but without illustration) another method for drawing diagrams fornterms: “… the rule of formation would be very simple. We should merely have to begin by drawing any closed figure, and then proceed to draw others subject to the one condition that each intersect once, and once only, all the existing subdivisions produced by those which had gone before.” Then, in a rather surprising footnote, he adds: “It will be found that when we adhere to continuous figures, instead of the discontinuous five-term figure given above, there is a tendency for the resultant outlines thus successively drawn to assume a comb-like shape after the first four or five. If we begin by circles or other rounded figures the teeth are curved, if by parallelograms then they are straight. (shrink)

In his well-known paper on Euclid’s geometry, Ken Manders sketches an argument against conceiving the diagrams of the Elements in ‘semantic’ terms, that is, against treating them as representations—resting his case on Euclid’s striking use of ‘impossible’ diagrams in some proofs by contradiction. This paper spells out, clarifies and assesses Manders’s argument, showing that it only succeeds against a particular semantic view of diagrams and can be evaded by adopting others, but arguing that Manders nevertheless (...) makes a compelling case that semantic analyses ought to be relegated to a secondary role for the study of mathematical practices. (shrink)

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

The Toulmin diagram layout is very familiar and widely used, particularly in the teaching of critical thinking skills. The conventional box-and-arrow diagram is equally familiar and widespread. Translation between the two throws up a number of interesting challenges. Some of these challenges (such as the relationship between Toulmin warrants and their counterparts in traditional diagrams) represent slightly different ways of looking at old and deep theoretical questions. Others (such as how to allow Toulmin diagrams to be recursive) are (...) diagrammatic versions of questions that have already been addressed in artificial intelligence models of argument. But there are further questions (such as the relationships between refutations, rebuttals and undercutters, and the roles of multiple warrants) that are posed as a specific result of examining the diagram inter-translation problem. These three classes of problems are discussed. To the first class are addressed solutions based on engineering pragmatism; to the second class, are addressed solutions drawn from the appropriate literature; and to the third class, fuller exploration is offered justifying the approaches taken in developing solutions that offer both pragmatic utility and theoretical interest. Finally, these solutions are explored briefly in the context of the Araucaria system, showing the ways in which analysts can tackle arguments either using one diagrammatic style or another, or even a combination of the two. (shrink)

This paper explains how to use a new software tool for argument diagramming available free on the Internet, showing especially how it can be used in the classroom to enhance critical thinking in philosophy. The user loads a text file containing an argument into a box on the computer interface, and then creates an argument diagram by dragging lines from one node to another. A key feature is the support for argumentation schemes, common patterns of defeasible reasoning (...) historically know as topics . Several examples are presented, as well as the results of an experiment in using the system with students in a university classroom. (shrink)

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless (...) diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: ( i ) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; ( ii ) EPG objects inherit some properties and relations from these diagrams. (shrink)

This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...) for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)

Scientists use a variety of modes of representation in their work, but philosophers have studied mainly sentences expressing propositions. I ask whether diagrams are mere conveniences in expressing propositions or whether they are a distinct, ineliminable mode of representation in scientific texts. The case of path analysis, a statistical method for quantitatively assessing the relative degree of causal determination of variation as expressed in a causal path diagram, is discussed. Path analysis presents a worst case for arguments against eliminability (...) since path diagrams are usually presumed to be mathematically or logically “equivalent” in an important sense to sets of linear path equations. I argue that path diagrams are strongly generative, i.e., that they add analytical power to path analysis beyond what is supplied by linear equations, and therefore that they are ineliminable in a strong scientific sense. (shrink)

Some very good arguments can be given in favor of the Augustinean wisdom, according to which it is impossible to provide a satisfactory definition of the concept of time. However, even in the absence of a proper definition, it is possible to deal with conceptual problems regarding time. It can be done in terms of analogies and metaphors. In particular, it is attractive to make use of Peirce's diagrams by means of which various kinds of conceptual experimentation can be (...) carried out. This paper investigates how Peircean diagrams can be used within the study of time. In particular, we discuss 1) the topological properties of time, 2) the implicative structure in tense logic, 3) the notions of open future and branching time models, and finally 4) tenselogical alternatives to branching time models. (shrink)

While it is customary for instructors when teaching a philosophical text to point to where a philosopher lays out their overall plan and then let students fill in the pieces, no such passage exists in Aristotle’s “Nicomachean Ethics.” While many philosophy courses focus on analyzing arguments, Aristotle’s work provides students a unique opportunity to learn how to assemble the parts into a coherent whole. This paper describes an assignment where students are asked to construct a diagram that visually represents the (...) structure of Aristotle’s work. In response to the tremendous success of this assignment, the paper examines the assignment’s precedents, suggests a theoretical basis for its success, and details its practical benefits. (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)

I will argue that the development of Feynman diagrams came from the physicist’s capacity of visualizing phenomena and that such visualization-skill contributed to the forming of a narrative explanation in the sense of Wise ( 2011 ) and Morgan ( 2001 ). The second part of the paper explores the extent to which Feynman diagrams can be considered as weak representations of quantum phenomena. I will review some of the most common arguments in support of the instrumentalist view (...) and I will suggest that a form of weak representation that does not imply ontological commitment can be applied to the diagrams. Such a form of weak representation will be characterized as non-denotative, intentional, and as conveying a physical interpretation through narrative explanations. (shrink)

In this paper we would like to attempt to shed some light on the way in which diagrams enter into the practice of ancient Greek geometrical analysis. To this end, we will first distinguish two main forms of this practice, i.e., trans-configurational and intra-configurational. We will then argue that, while in the former diagrams enter in the proof essentially in the same way they enter in canonical synthetic demonstrations, in the latter, they take part in the analytic (...) class='Hi'>argument in a specific way, which has no correlation in other aspects of classical geometry. In intra-configurational analysis, diagrams represent in fact the result of a purely material gesture, which is not codified by any construction canon, but permitted only by the practice of the method of analysis and synthesis. (shrink)

Based on a typology of five basic forms of abduction, I propose a new definition of abductive insight that empha sizes in particular the inferential structure of a belief system that is able to explain a phenomenon after a new, abductive ly created component has been added to this system or the entire system has been abductively restructured. My thesis is, first, that the argumentative structure of the pursued problem solution guides abductive creativity and, second, that diagrammatic reasoning—if conceptualized according (...) to the requirements defined by Charles Peirce—can support this guidance. This support is mainly possible based on the normative power of the system of representation that has to be used to construct diagrams and to perform experiments with them. (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)

Logicians have often suggested that the use of Euler-type diagrams has influenced the idea of the quantification of the predicate. This is mainly due to the fact that Euler-type diagrams display more information than is required in traditional syllogistics. The paper supports this argument and extends it by a further step: Euler-type diagrams not only illustrate the quantification of the predicate, but also solve problems of traditional proof theory, which prevented an overall quantification of the predicate. (...) Thus, Euler-type diagrams can be called the natural basis of syllogistic reasoning and can even go beyond. In the paper, these arguments are presented in connection with the book Nucleus Logicae Weisaniae by Johann Christian Lange from 1712. (shrink)

I give a brief introduction to how Feynman diagrams are used. I review arguments to the effect that they are only used as calculation tools and should not be interpreted as representations of physical processes. Against these arguments, I propose to regard Feynman diagrams as visual models that explain, in some respects, how elementary particles interact.

Diagrams in medieval manuscripts of Greek mathematical and astronomical works can seem peculiar for a modern reader, given their persistent and widespread tendency to represent more geometric regularity than the argument requires and their usual visual inaccuracy in depicting the mathematical objects discussed in the text. Although most scholars believe that these tendencies go back to the original Greek authors, in a recent paper I argued that these odd features should not be attributed to Greek authors, but to (...) transmission. My main argument there is based on experiments made with university students who played the role of copyists. In this paper, I show that the vestiges of the changes caused by transmission are present in the very diagrams of the extant manuscripts. I analyze four diagrams taken from Aristarchus's On Sizes. (shrink)

Understanding the language of rejections and objections is an important part of the analysis and practice of argument. In order to strengthen this understanding, we might turn to diagramming, as it has been shown to have the virtue of improving critical thinking skills. This paper discusses what reliable meaning can be taken from words and phrases related to rejections and objections, and then how to diagram them.

This review article of Frederik Stjernfelt’s Sheets, Diagrams, and Realism (2022) argues that Peirce’s theory of iconicity with its subdivision into the image-diagram-metaphor triad must not be reduced to diagrammatic iconicity. The foundation of the triadic subdivision of the icon is not in Peirce’s diagrammatic logic but in Peirce’s cenopythagorean categories. A focus is on misinterpretations of Peirce’s concept of thirdness in the firstness of the icon. The paper argues that not only metaphors, but also comparisons, analogies, analogic arguments, (...) and examples belong to the third class of icons. (shrink)

This article aims to examine the relationship between image and narrative by means of Peirce’s first trichotomy of qualisign-sinsign-legisign or, for the purposes of the current argument, image-diagram-metaphor. It is argued that narrative, as an extended metaphor, can be examined in three modes: in the image; schematically, in the imagination; and allegorically or in a thought experiment, through hypothetic interpretation. The article outlines two kinds of diagrammatic reasoning emphasized by Peirce: corollarial deduction in which the image is ‘literally seen’ (...) and the reasoning steps are manifest in its conclusion; and theorematic deduction where the conclusion in a diagram is subject to a hypothesis which transforms the image into something new. Demonstrating the breadth of diagrammatic reasoning with reference to the 2018 film, The Shape of Water, the article seeks to explore how allegory and diagram are mutually cooperative, based on three ontological modes: the expressive, the cognitive, and the symbolic. Its primary focus, then, is not so much on the story events of the narrative, as the way that they are visualized and characterized as the fairy story unfolds. It is suggested that the interpreting activity involved in allegory and diagram ties interpretation to metacognition, ultimately (re)recognizing the image in The Shape of Water in an attempt to ascertain the meaning of love. (shrink)

While Deleuze and Guattari's Anti-Oedipus is quite rightly considered a fully fledged response to May ’68 and as one with the radical politics of the 1970s, their 1980 follow-up, A Thousand Plateaus, has tended to provoke a more perplexed reaction. In this article, I will argue that we can nonetheless extract a definite line of argumentation serving a precise political end if we relate the text back to Foucault's mid-1970s output on power/knowledge. In particular, I will emphasise Deleuze and Guattari's (...) appropriation of the Foucaultian notion of dispositif via their concept of the assemblage, the former being understood as a concrete articulation of lines of power, knowledge and subjectivation, as well as the Foucaultian ‘diagram’, the latter being a more abstract or indeterminate stage of the dispositif whose relative indeterminacy, for Deleuze and Guattari, offers a means of escape. I will show that, making room for the assemblage's opening back onto the relative indeterminacy of its generative stages, the assemblage incorporates into itself a more immanent alternative to the dispositif that is focused on collective desire rather than power, within which resistance becomes a primary and generative dimension rather than a counter-attack. In the first section, I will outline Foucault's approach to power, knowledge and subjectivation, emphasising Deleuze's reading of Foucault though without trying to overdetermine my reading in this way. Next, I will turn to the Foucaultian diagram. In the third section, I will focus on A Thousand Plateaus, demonstrating how the notion of assemblage developed in this text responds to and builds upon Foucault's approach to power/knowledge and subjectivation in order to reconceive resistance. (shrink)

Starting from Poincaré’s assignment of an algebraic object to a topological manifold, namely the fundamental group, this article introduces the concept of categories and their language of arrows that has, since their mid-20th-century inception, altered how large areas of mathematics, from algebra to abstract logic and computer programming, are conceptualized. The assignment of the fundamental group is an example of a functor, an arrow construction central to the notion of a category. The exposition of category theory’s arrows, which operate at (...) three distinct but deeply interconnected levels, is framed by a comparison with the language and outlook of set theory founded on the concept of membership; sets and their theorization having provided, famously through the Bourbaki initiative, the basic ontological and epistemological vocabulary for defining and handling all mathematical entities. The comparison with sets emphasizes how categories offer a form of diagrammatic argument and thought against set theory’s fidelity to syntax-based proofs; how categories invert set theory’s priority of objects and their attributes over relations by making the relations of an object to others of its kin primary; and how categories replace identity, that is, equality, between objects, by the weaker notion of isomorphism, restricting equality to identity between arrows. The article concludes with a return to topology and some remarks about the question of its possible use in articulating/characterizing cultural dynamics. (shrink)

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

Inheritance diagrams are directed acyclic graphs with two types of connections between nodes: x → y and x ↛ y . Given a diagram D, one can ask the formal question of “is there a valid path between node x and node y?” Depending on the existence of a valid path we can answer the question “x is a y” or “x is not a y”. The answer to the above question is determined through a complex inductive algorithm on (...) paths between arbitrary pairs of points in the graph. This paper aims to simplify and interpret such diagrams and their algorithms. We approach the area on two fronts. Suggest reactive arrows to simplify the algorithms for the winning paths. We give a conceptual analysis of inheritance diagrams, and compare our analysis to the “small” and “big sets” of preferential and related reasoning. In our analysis, we consider nodes as information sources and truth values, direct links as information, and valid paths as information channels and comparisons of truth values. This results in an upward chaining, split validity, off-path preclusion inheritance formalism of a particularly simple type. We show that the small and big sets of preferential reasoning have to be relativized if we want them to conform to inheritance theory, resulting in a more cautious approach, perhaps closer to actual human reasoning. We will also interpret inheritance diagrams as theories of prototypical reasoning, based on two distances: set difference, and information difference. We will see that some of the major distinctions between inheritance formalisms are consequences of deeper and more general problems of treating conflicting information. It is easily seen that inheritance diagrams can also be analysed in terms of reactive diagrams - as can all argumentation systems. AMS Classification: 68T27, 68T30. (shrink)

In a series of joint papers, Teppo Felin and Nicolai J. Foss recently launched a microfoundations project in the field of strategic management. Felin and Foss observe that extant explanations in strategic management are predominantly collectivist or macro. Routines and organizational capabilities, which are supposed to be properties of firms, loom large in the field of strategic management. Routines figure as explanantia in explanations of firm behavior and firm performance, for example. Felin and Foss plead for a replacement of such (...) macro-explanations by micro-explanations. Such a replacement is needed, Felin and Foss argue, because macro-explanations are necessarily incomplete: they miss out on crucial links in the causal chain that connect macro phenomena with each other. I argue that this argument is flawed. It is based on a doubtful if not outright incorrect understanding and use of Coleman’s diagram. In a sense to be explained below, only if Coleman’s diagram is squared it can accurately account for the relations between individual action and interaction, routines and firm behavior and firm performance. Once Coleman’s diagram is squared, one can see why and how macro-explanations need not miss out on any link in the causal chains that connect macro phenomena. Micro-analyses are still needed, not to highlight and specify causal links that macro-explanations miss out on, but to check whether the many properties that are ascribed to routines in macro-explanations of firm behavior are warranted. (shrink)

That diagrams are analog, i.e., homomorphic, representations of some kind, and sentential representations are not, is a generally held intuition. In this paper, we develop a formal framework in which the claim can be stated and examined, and certain puzzles resolved. We start by asking how physical things can represent information in some target domain. We lay a basis for investigating possible homomorphisms by modeling both the physical medium and the target domain as sets of variables, each with a (...) constraint structure. When a homomorphism exists, the causality of the physical medium can provide a “free ride,” automatically determining the information that is needed to solve a problem. The modeling technique enables us to show how in sentential representations the structure of the 2-D space is used non-homomorphically to define tokens, and to provide a minimal homomorphism with respect to the ordering of the arguments for n-ary functions, predicates, and operators, but otherwise there is no homomorphism with the target domain. Our treatment also characterizes diagrams as a type of a more general class of physical representations in which physical causality plays a role in providing the information needed to solve a problem. (shrink)

A logical system is studied whose well-formed representations consist of diagrams rather than formulas. The system, due to Shin [2, 3], is shown to be complete by an argument concerning maximally consistent sets of diagrams. The argument is complicated by the lack of a straight forward counterpart of atomic formulas for diagrams, and by the lack of a counterpart of negation for most diagrams.

It is commonplace to view the rigor of the mathematics in Euclid's Elements in the way an experienced teacher views the work of an earnest beginner: respectable relative to an early stage of development, but ultimately flawed. Given the close connection in content between Euclid's Elements and high-school geometry classes, this is understandable. Euclid, it seems, never realized what everyone who moves beyond elementary geometry into more advanced mathematics is now customarily taught: a fully rigorous proof cannot rely on geometric (...) intuition. In his arguments he seems to call illicitly upon our understanding of how objects like triangles and circles behave rather than grounding everything rigorously in axioms.Though widespread, the attitude is in a historical sense puzzling. For over two millenia, mathematicians of all levels studied the arguments in Elements and found nothing substantial missing. The book, on the contrary, represented the limit of mathematical explicitness. It served as the paradigm for careful and exact reasoning. How it could enjoy this reputation for so long is mysterious if careful and exact reasoning demands that all inferences be grounded in a modern axiomatic theory in the way Hilbert did in his famous Foundations of Geometry. By these standards, Euclid's work is deeply flawed. The holes in his arguments are not minor and excusable, but massive and cryptic.With his book Euclid and His Twentieth Century Rivals, Nathaniel Miller makes substantial progress in clearing this mystery up. The book is an explication of FG , a formal system of proof developed by Miller which reconstructs Euclid's deductions as essentially diagrammatic. The holes in Euclid's arguments are taken to appear precisely at those steps which are unintelligible without an accompanying geometric diagram. Interpreting the reasoning in the Elements in terms of a modern axiomatization , …. (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)

Previous research on the use of diagrams for argumentation instruction has highlighted, but not conclusively demonstrated, their potential benefits. We examine the relative benefits of using diagrams and diagramming tools to teach causal reasoning about public policy. Sixty-three Carnegie Mellon University students were asked to analyze short policy texts using either: 1) text only, 2) text and a pre-made, correct diagram representing the causal claims in the text, or 3) text and a diagramming tool with which to construct (...) their own causal diagram. After a pretest and training, we tested student performance on a new policy text and found that students given a correct diagram significantly outperformed the other groups. Finally, we compared learning by testing students on a third policy problem in which we removed all diagram or tool aids and found that students who constructed their own diagrams learned the most. We describe these results and interpret them in a way that foreshadows work we now plan for a cognitive-tutor on causal diagram construction. (shrink)