Diagrammatic Representation and Inference

[9]summarised a program of research into diagrammatic reasoning based on the semantics of diagrams. It distinguished direct diagrams from indirect ones by whether there was a direct semantic interpretation of some spatial relations in the diagram or whether all the interpreted spatial properties were mediated by a concatenation relation, as is the case in written natural language. So an Euler diagram is direct because its spatial property of containment in a labelled closed curve is directly interpreted as membership in a set designated by the label. In a typical abstract network diagram, such as a semantic graph, the nodes-and-link configurations take the place of the concatenation operator (as well as other matters), and play similar roles in determining the semantics of the diagram indirectly as the concatenation operator does in written natural and logical languages. The hallmark of indirectness is that the concatenation relation (whether one- or two-dimensional) has itself no semantic interpretation—only a syntactic one.

The beauty of logical graphs consists in many facets, including notational simplicity, multi-modality and normativity. This paper aims at understanding the nature of Peirce’s graphical method and its implications to philosophy of logic.

Immanuel Kant had understood some deep facts about mathematical knowledge that have mostly been ignored by recent researchers on cognition, especially intrinsic connections between many everyday actions and mathematical competences, e.g. competences concerned with the fact that organisms inhabit environments with complex mathematical structures, some produced by activities of life forms, others not. I’ll present a variety of examples, to be discussed with the audience, with deep implications for future research in artificial and natural cognition, and raise questions about the diagram-like information structures many cognitive processes seem to be concerned with: “diagrams in the mind”. I suspect that Alan Turing’s 1952 paper on chemical morphogenesis, published two years before he died is connected with this. Perhaps if he had lived several more decades he would have worked on what I call the Meta-Morphogenesis project, which was inspired by Turing and has deep connections with mathematical structures important for animal cognition and future machine cognition. However it is an open question that current forms of (digital) computation will need to be enhanced using chemistry-based computation similar to sub-neural mechanisms in brains, or whether the required forms of reasoning can occur in virtual machines implemented in digital machinery. Von Neumann’s last little book, written as he was dying in 1958, raises similar questions.

How do we know what people perceive in a diagram? A diagram can be an excellent medium for communicating complex facts and relationships. Users may be able to learn a lot just from a quick glance at a well-designed diagram. Unfortunately, what users take from a diagram may not always be the same as what its designers intended to communicate. This tutorial explores the use of verbal protocol analysis in the area of diagram interpretation, and offer practical support for systematic analysis procedures. This includes a close look at the way people formulate their thoughts about a design, which can reveal underlying conceptualisations and perspectives that the speakers may not be aware of.

Charles Sanders Peirce (1839–1914) is one of the “grounding fathers” of mathematical logic, having developed all of the key formal results of modern logic. He did it firstly (from 1860 on) in the algebraic tradition of mathematical logic stemming from Boole, combining it with the logic of relations, explicitly developed by Augustus De Morgan. From this, Peirce obtained a system that included quantifiers—a term he seems to have invented—and relative predicates. Developing his own system of relative terms, Peirce started from Boole’s system, trying to apply it to De Morgan’s logic of relations. Indeed, Peirce’s aim is to include the logic of relations into the calculus of algebra using his own system of algebraic signs. On the one hand, Peirce’s algebraic notation will be presented, specially: (a) relative terms as iconic representations of logical relations; (b) Peirce’s quantifiers and the passage from a linear notation to a diagrammatic one. On the other hand, Peirce’s graphical notation will be presented, specially: (a) his Alpha and Beta systems, which are fully compatible with what is nowadays called first-order logic, (b) and his unfinished Gamma system, designed for second-order logic and modal logic.

We provide a self-contained introduction to quantum theory using a unique diagrammatic language. Far from simple visual aids, the diagrams we use are mathematical objects in their own right, which allow us to develop from first principles a completely rigorous treatment of ‘textbook’ quantum theory. Additionally, the diagrammatic treatment eliminates the need for the typical prerequisites of a standard course on the subject, making it suitable for a multi-disciplinary audience with no prior knowledge in physics or advanced mathematics.By subscribing to a diagrammatic treatment of quantum theory we place emphasis on quantum processes, rather than individual systems, and study how uniquely quantum features arise as processes compose and interact across time and space. We introduce the notion of a process theory, and from this develop the notions of pure and mixed quantum maps, measurements and classical data, quantum teleportation and cryptography, models of quantum computation, quantum algorithms, and quantum non-locality. The primary mode of calculation in this tutorial is diagram transformations, where simple local identities on diagrams are used to explain and derive the behaviour of many kinds of quantum processes.This tutorial roughly follows a new textbook published by Cambridge University Press in 2017 with the same title.

The use of diagrams in logic is old. Euler and Venn schemes are among the most popular. Carroll diagrams are less known but are occasionally mentioned in recent literature. The objective of this tutorial is to expose the working of Carroll’s diagrams and their significance from a triple perspective: historical, mathematical and philosophical. The diagrams are exposed, worked out and compared to Euler-Venn diagrams. These schemes are used to solve the problem of elimination which was widely addressed by early mathematical logicians: finding the conclusion that is to be drawn from any number of propositions given as premises containing any number of terms. For this purpose, they designed symbolic, visual and sometimes mechanical devices. The significance of Venn and Carroll diagrams is better understood within this historical context. The development of mathematical logic notably created the need for more complex diagrams to represent n terms, rather than merely 3 terms (the number demanded by syllogisms). Several methods to construct diagrams for n terms, with different strategies, are discussed. Finally, the philosophical significance of Carroll diagrams is discussed in relation to the use of rules to transfer information from a diagram to another. This practice is connected to recent philosophical debates on the role of diagrams in mathematical practices.

Euler diagrams are used for visualizing categorized data, with applications including crime control, bioinformatics, classification systems and education. Various properties of Euler diagrams have been empirically shown to aid, or hinder, their comprehension by users. Therefore, a key goal is to automatically generate Euler diagrams that possess beneficial layout features whilst avoiding those that are a hindrance. The automated layout techniques that currently exist sometimes produce diagrams with undesirable features. In this paper we present a novel approach, called iCurves, for generating Euler diagrams alongside a prototype implementation. We evaluate iCurves against existing techniques based on the aforementioned layout properties. This evaluation suggests that, particularly when the number of zones is high, iCurves can outperform other automated techniques in terms of effectiveness for users, as indicated by the layout properties of the produced Euler diagrams.

Diagrams and pictures are a powerful medium to communicate ideas, designs, and art. However, authors of pictures are forced to use rudimentary and ad hoc techniques in managing multiple variants of their creations, such as copying and renaming files or abusing layers in an advanced graphical editing tool. We propose a model of variational pictures as a basis for the design of editors and other tools for managing variation in pictures. This model enjoys a number of theoretical properties that support exploratory graphical design and can help systematize picture creators’ workflows.

Many visual languages based on node-link diagrams use edge labels. We describe different strategies of placing edge labels in the context of the layered approach to graph drawing and investigate ways of encoding edge direction in labels. We also report on the results of experiments conducted to investigate the effectiveness of the strategies.

Kolam-designs are diagrams used to decorate the floor, especially in front of a house in South India. Methods of generation of the kolam diagrams were developed based on two-dimensional picture generating models, broadly known as array grammars, introduced for the description and analysis of picture patterns. Rewriting array P system, a membrane computing model based on array rewriting has been developed to evolve picture arrays, based on context-free array rewriting rules. In contrast to this array P system, another P system model called contextual array P system (CAP) using contextual array rules for the evolution or generation of picture arrays has been proposed and its power in generating picture arrays investigated. Here we develop an application of CAP for the generation of the kolam diagrams. The advantage of using CAP is that kolam diagrams that cannot be handled by array grammars can be generated by the CAP model.

A knot diagram looks like a two-dimensional drawing of a knotted rubberband. Proving that a given knot diagram can be untangled (that is, is a trivial knot, called an unknot) is one of the most famous problems of knot theory. For a small knot diagram, one can try to find a sequence of untangling moves explicitly, but for a larger knot diagram producing such a proof is difficult, and the produced proofs are hard to inspect and understand. Advanced approaches use algebra, with an advantage that since the proofs are algebraic, a computer can be used to produce the proofs, and, therefore, a proof can be produced even for large knot diagrams. However, such produced proofs are not easy to read and, for larger diagrams, not likely to be human readable at all. We propose a new approach combining advantages of these: the proofs are algebraic and can be produced by a computer, whilst each part of the proof can be represented as a reasonably small knot-like diagram (a new representation as a labeled tangle diagram), which can be easily inspected by a human for the purposes of checking the proof and finding out interesting facts about the knot diagram.

In this paper, we develop and discuss a classification scheme that allows us to distinguish between the types of diagrams used in mathematical research based on the cognitive support offered by diagrams. By cognitive support, we refer to the gain that research mathematicians get from using diagrams. This support transcends the specific mathematical topic and diagram type involved and arises from the cognitive strategies mathematicians tend to use. The overall goal of this classification scheme is to facilitate a large-scale quantitative investigation of the norms and values governing the publication style of mathematical research, as well as trends in the kinds of cognitive support used in mathematics. This paper, however, focuses only on the development of the classification scheme.The classification scheme takes its point of departure from case studies known from the literature, but in this paper, we validate the scheme using examples from a preliminary investigation of developments in the use of diagrams. Building on these results, we discuss the potential and pitfalls in using one generic classification scheme, as done in this analysis. This approach is contrasted with attempts that respect and build on individual diagram types, and as part of this discussion, we report the problems we experienced when using that strategy. The paper ends with a description of possible next steps in using text corpora as an empirical approach to understanding the nature of mathematical diagrams and their relation to mathematical culture.

In a recent paper, De Toffoli and Giardino analyzed the practice of knot theory, by focusing in particular on the use of diagrams to represent and study knots [1]. To this aim, they distinguished between illustrations and diagrams. An illustration is static; by contrast, a diagram is dynamic, that is, it is closely related to some specific inferential procedures. In the case of knot diagrams, a diagram is also a well-defined mathematical object in itself, which can be used to classify knots. The objective of the present paper is to reply to the following questions: Can the classificatory function characterizing knot diagrams be generalized to other fields of mathematics? Our hypothesis is that dynamic diagrams that are mathematical objects in themselves are used to solve classification problems. To argue in favor of our hypothesis, we will present and compare two examples of classifications involving them: (i) the classification of compact connected surfaces (orientable or not, with or without boundary) in combinatorial topology; (ii) the classification of complex semisimple Lie algebras.

There is still debate as to whether Euclidean diagrams are symbols, indexes or icons, and of what sort. I hold them to be pictorial icons that reproduce at least some visual features of their objects. This hypothesis has been directly challenged by Sherry [36] and Panza [29] among others. My aim on this paper is defending this thesis against Macbeth’s [24–26] claim that if diagrams were pictures their content could not shift the way it does in Euclidean proof. To this goal I will present a broadly Gricean account of pictorial representation, where visual resemblance constraints but no fully determines reference, and then show how this account ratifies Macbeth’s insights about the importance of the author’s intentions in determining a diagram’s content, in a way that allows for the sort of content-shifting that she has identified as key to understanding the role of diagrams in Euclidean proof.

Mathematical problem solving typically involves manipulating visual symbols (e.g., equations), and prior research suggests that those symbols serve as diagrammatic representations (e.g., Landy and Goldstone 2010). The present work examines the ways that instructional design of student engagement with these diagrammatic representations may impact student learning. We report on two studies. The first describes systematic cross-cultural differences in the ways that teachers use mathematical representations as diagrammatic supports during middle school mathematics lessons, finding that teachers in two higher achieving regions, Hong Kong, and Japan, more frequently provided multiple layers of support for engaging with these diagrams (e.g. making them visible for a longer period, using linking gestures, and drawing on familiarity in those representations), than teachers in the U.S., a lower achieving region. In Study 2, we experimentally manipulated the amount of diagrammatic support for visually presented problems in a video-based fifth-grade lesson on proportional reasoning to determine whether these multiple layers of support impact learning. Results suggest that learning was optimized when supports were used in combination. Taken together, these studies suggest that providing visual, temporal, and familiarity cues as supports for learning from a diagrammatic representation is likely to improve mathematics learning, but that administering these supports non-systematically is likely to be overall less effective.

Automated geometry theorem provers start with logic-based formulations of Euclid’s axioms and postulates, and often assume the Cartesian coordinate representation of geometry. That is not how the ancient mathematicians started: for them the axioms and postulates were deep discoveries, not arbitrary postulates. What sorts of reasoning machinery could the ancient mathematicians, and other intelligent species (e.g. crows and squirrels), have used for spatial reasoning? “Diagrams in minds” perhaps? How did natural selection produce such machinery? Which components are shared with other intelligent species? Does the machinery exist at or before birth in humans, and if not how and when does it develop? How are such machines implemented in brains? Could they be implemented as virtual machines on digital computers, and if not what human engineered “Super Turing” mechanisms could replicate what brains do? How are they specified in a genome? Turing’s work on chemical morphogenesis, published shortly before he died suggested to me that he might have been considering such questions. Could deep new answers vindicate Kant’s claim in 1781 that at least some mathematical knowledge is non-empirical, synthetic and necessary? Discussions of mechanisms of consciousness should include ancient mathematical diagrammatic reasoning, and related aspects of everyday intelligence, usually ignored in AI, neuroscience and most discussions of consciousness.

Diagrams are frequently used in mathematics, not only in geometry but also in many other branches such as analysis or graph theory. However, the distinctive cognitive and methodological characteristics of mathematical practice with diagrams, as well as mathematical knowledge acquired using diagrams, raise some philosophical issues – in particular, issues that relate to the empiricism-realism debate in the philosophy of mathematics. On the one hand, it has namely been argued that some aspects of diagrammatic reasoning are at odds with the often assumed a priori nature of mathematical knowledge and with other aspects of the realist position in philosophy of mathematics. On the other hand, one can claim that diagrammatic reasoning is consistent with the realist epistemology of mathematics. Both approaches will be analyzed, referring to the use of diagrams in geometry as well as in other branches of mathematics.

A graphical abstract (GA) provides a concise visual summary of a scientific contribution. GAs are increasingly required by journals to help make scientific publications more accessible to readers. We characterize the design space of GAs through a qualitative analysis of 54 GAs from a range of disciplines, and descriptions of GA design principles from scientific publishers. We present a set of design dimensions, visual structures, and design templates that describe how GAs communicate via pictorial and symbolic elements. By reflecting on how GAs employ visual metaphors, representational genres, and text relative to prior characterizations of how diagrams communicate, our work sheds light on how and why GAs may be distinct. We outline steps for future work at the intersection of HCI, AI, and scientific communication aimed at the creation of GAs.

A systematic method is presented that describes comprehensively the very broad design space of visualizations and the interrelationships between their constituents. The framework offered here includes three representational modes and fifteen visual encodings principles as the proposed universal building blocks of all types of diagrams and information graphics. The framework provides: 1. A vocabulary and a method for thoroughly analyzing the full spectrum of visual representations of information. 2. A mechanism for exploring previously unexploited combinations of visual encoding principles for representing information. 3. A potential tool for creating alternative representations for any given visualization or data set.

Classifications are useful for describing existing phenomena and guiding further investigation. Several classifications of diagrams have been proposed, typically based on analytical rather than empirical methodologies. A notable exception is the work of Lohse and his colleagues, published in Communications of the ACM in December 1994. The classification of diagrams that Lohse proposed was derived from bottom-up grouping data collected from sixteen participants and based on 60 diagrams. Mean values on ten Likert-scales were used to predict diagram class. We follow a similar methodology to Lohse, using real-world infographics (i.e. embellished data charts) as our stimuli. We propose a structural classification of infographics, and determine whether infographics class can be predicted from values on Likert scales.

Vignelli’s 1972 diagrammatic subway map is hailed as a design classic, but was dropped by the Metropolitan Transportation Authority (MTA) after just seven years’ usage. Following an absence of a generation, a diagrammatic map of the New York City subway system has been reintroduced into the MTA’s information provision. A digital version came back in 2011 and continues in use with weekly updates on the MTA Weekender website; print editions were issued in 2012, 2014, 2015, and 2017 for special occasions and from 2017 onwards for travel advisory notices. To see this in context, we need to understand why New York City adopted a diagrammatic map (Salomon map 1958), route colour-coded it (D’Adamo map 1967), stylized it (Vignelli map 1972), replaced it with a geographic map (Tauranac map 1979), and re-imagined it for the digital era (Waterhouse-Cifuentes map 2011). Using primary sources, we characterise the birth, death, and rebirth of the diagrammatic map of the New York City subway.

Identifying auditory correlates of the graphic-linguistic distinction informs our design of an auditory display based on Charles Minard’s depiction of Napoleon’s Russia campaign – the gold standard for visual (graphic) information design and therefore a grand challenge for auditory display design. We identify viable alternatives to the text-only translations currently employed in making graphics accessible to blind and/or low-vision individuals by introducing sounds bearing strong ecological resemblances to Minard’s depictions. Our integration of theoretical work about classic distinctions with common properties across diagrammatic and auditory display communities reveals practical opportunities for designing inclusive and accessible graphics.

The aim of this paper is to lay out the foundations of a typology of diagrams in linguistics. We draw a distinction between linguistic parameters — concerning what information is being represented — and diagrammatic parameters — concerning how it is represented. The six binary linguistic parameters of the typology are: (i) mono- versus multilingual, (ii) static versus dynamic, (iii) mono- versus multimodular, (iv) object-level versus meta-level, (v) qualitative versus quantitative, and (vi) mono- versus interdisciplinary. The two diagrammatic parameters are (i) iconic/concrete versus symbolic/abstract representation and (ii) static versus dynamic representation. We briefly illustrate how different types of linguistic diagrams can be analysed in terms of the interaction between the linguistic and the diagrammatic parameters.

High-tech systems are ubiquitous and often safety and security critical: reasoning about their correctness is paramount. Thus, precise modelling and formal reasoning are necessary in order to convey knowledge unambiguously and accurately. Whilst mathematical modelling adds great rigour, it is opaque to many stakeholders which leads to errors in data handling, delays in product release, for example. This is a major motivation for the development of diagrammatic approaches to formalisation and reasoning about models of knowledge. In this paper, we present an interactive theorem prover, called iCon, for a highly expressive diagrammatic logic that is capable of modelling OWL 2 ontologies and, thus, has practical relevance. Significantly, this work is the first to design diagrammatic inference rules using insights into what humans find accessible. Specifically, we conducted an experiment about relative cognitive benefits of primitive (small step) and derived (big step) inferences, and use the results to guide the implementation of inference rules in iCon.

In developing mechanistic explanations for biological phenomena, researchers have their choice of several different types of diagrams. First, a mechanism diagram spatially represents a proposed mechanism, typically using simple shapes for its parts and arrows for their operations. Beyond this representational role, such diagrams can provide a platform for further reasoning. Published diagrams in circadian biology show how question marks support reasoning about the proposed molecular mechanisms by flagging where there are knowledge gaps or uncertainties. Second, an annotated mechanism diagram can support computational modeling of the dynamics of a proposed mechanism. Each variable and parameter needed for the model is added to the diagram adjacent to the appropriate part or operation. Anchoring the model in this way helps with its construction, revision, and interpretation. Third, a network diagram fosters a different approach to mechanistic reasoning. Layout algorithms are applied to data generated by high-throughput experiments to reveal modules that correspond to mechanisms. We present examples in which network diagrams enable viewers to advance hypotheses about previously unknown mechanisms or unknown parts and operations of known mechanisms as well as to develop new understanding about how a given mechanism is situated in a larger environment.

Navya-Nyāya, “The New Reasoning”, is a formal philosophical logic developed in India from the 11th to the 17th centuries CE, and which builds on the older traditions of Nyāya and Vaiśeṣika. Not surprisingly, Navya-Nyāya is fundamentally different from classical Western logic and from the meanings ascribed to traditional logical diagrams. For instance, although it is not entirely correct to describe Navya-Nyāya as extensional or intensional, it has an intensional flavour: abstractions are built up from concrete individuals of which we know only their possession, or not, of certain properties. In this paper we look at the implications of these semantics for the use of logical diagrams in Navya-Nyāya. We survey the use of diagrams in modern studies of Navya-Nyāya, notable examples having been produced by Wada, Das and Ganeri. We use notions of well-matchedness, iconicity and Cheng’s recent framework to analyse the effectiveness of the notations in the context of their intended purposes.

A single feature indicator system (SFIS) is a signaling system where a representation carries information through a one-to-one correspondence of the “values” taken by its elements to those taken by a set of represented objects. The purpose of this paper is to demonstrate that many common diagrammatic systems are either SFISs or have SFISs as their semantic basis. We take as examples several familiar diagrammatic systems with seemingly diverse semantic systems (tables, charts, connectivity diagrams) and show the fundamental similarities among them that put them all under the concept of SFIS. We then explore different ways in which an SFIS is extended to a new, perhaps more expressive representation system. The paper paves the way to an account of the functional commonality and diversity of diagrammatic systems in terms of the operations that generate them from some basic systems.

The ability of diagrams to convey information effectively in part comes from their ability to make facts explicit that would otherwise need to be inferred. This type of advantage has often been referred to as a free ride and was deemed to occur only when a diagram was obtained by translating a symbolic representation of information. Recent work generalised free rides to the idea of an observational advantage, where the existence of such a translation is not required. Roughly speaking, it has been shown that Euler diagrams without existential import are observationally complete as compared to symbolic set theory. In this paper, we explore to what extent Euler diagrams with existential import are observationally complete with respect to set-theoretic sentences. We show that existential import significantly limits the cases when observational completeness arises, due to the potential for overspecificity.

I distinguish two kinds of observational advantages: (i) a given representation is observationally advantageous over another if a logical consequence of the information represented in it is observable in the former but only inferable from the latter; (ii) a given representation is observationally advantageous over another if a logical equivalence is observable in the former but only inferable from the latter. The paper also discusses the following question: observing (vs inferring) a piece of information in a given representation is an advantage if the purpose of the system of representation is to directly observe what could otherwise be inferred. But if the purpose were to infer what could be otherwise be observed, then one should conversely speak of observational disadvantages.

This paper presents a comprehensive representation for video analysis combining qualitative reasoning with diagrammatic reasoning. The hybrid approach is motivated by the power of diagrams that allows explicit relational representation of entities involved. Perception of qualitative information over the underlying representation, employment of inter-diagrammatic reasoning approach and their combined relevance for temporal abstractions holds key to the analysis. Activity recognition over selected videos from J-HMDB dataset are performed and encouraging results are achieved.

We present a class of diagrams in which to reason about causation. These diagrams are based on a formal semantics called ‘system semantics’, in which states of systems are related according to temporal succession. Arguing from straightforward examples, we provide the truth conditions for causal claims that one may make about these diagrams.

Arcform is a notation for expressing diverse thoughts using nodes and arcs in a new graph-like network structure. The structure differs from directed graphs by including arcs that point from or to other arcs, and semi arcs where one end points from or to itself. This supports a new generative statement composition structure which allows expressive statements to be read as grammatically normal sentences while integrated into maps containing multiple statements. This paper describes this compositional structure with a special focus on a few patterns for assigning meaning to nodes and arcs that preserve the above characteristics while ensuring an even tighter integration of diverse statements into networks. A few additional features are considered before raising some far reaching questions about how it can support thought work.

Extension and intension are two ways of indicating the fundamental meaning of a concept. The extent of a concept, C, is the set of objects which correspond to C whereas the intent of C is the collection of attributes that characterise it. Thus, intension defines the set of objects corresponding to C without naming them individually. Mathematicians switch comfortably between these perspectives but the majority of logical diagrams deal exclusively in extension. Euler diagrams indicate sets using curves to depict their extent in a way that intuitively matches the relations between the sets. What happens when we use spatial diagrams to depict intension? What can we infer about the intension of a concept given its extension, and vice versa? We present the first steps towards addressing these questions by defining extensional and intensional Euler diagrams and translations between the two perspectives. We show that translation in either direction leads to a loss of information, yet preserves important semantic properties. To conclude, we explain how we expect further exploration of the relationship between the two perspectives could shed light on connections between diagrams, extension, intension, and well-matchedness.

Euler famously used diagrams to illustrate syllogisms in his Lettres à une princesse d’Allemagne [1]. His diagrams are usually seen as suffering from a fatal “ambiguity problem” [11]: as soon as they involve intersecting circles, which are required for the representation of existential statements, it becomes unclear what exactly may be read off from them, and as Hammer & Shin conclusively showed, any set of reading conventions can lead to erroneous conclusions. I claim that Euler diagrams can, however, be used rigorously, if they are read in conjunction with the premises they are supposed to illustrate. More precisely, I give rigorous “heterogeneous” inference rules (in the sense of Barwise and Etchemendy) – rules whose premises are a sentence and a diagram and whose conclusion is a sentence – which allow to use them safely. I conclude that one should abandon the preconception that diagrams can only be used rigorously if they can be given a context-independent semantics. Finally, I suggest that context-dependence is a widespread feature of diagrams: for instance, Mumma [12] noticed that what may be read off from a Euclidean diagram depends not only on the diagram’s appearance, but also on the way it was constructed.

Diagrams in mechanised reasoning systems are typically encoded into symbolic representations that can be easily processed with rule-based expert systems. This relies on human experts to define diagram-to-symbol mapping and the set of rules to reason with the symbols. We present a new method of using Deep artificial Neural Networks (DNN) to learn continuous, vector-form representations of diagrams without any human input, and entirely from datasets of diagrammatic reasoning problems. Based on this DNN, we developed a novel reasoning system, Euler-Net, to solve syllogisms with Euler diagrams. Euler-Net takes two diagrams representing the premises in a syllogism as input, and outputs either a categorical (subset, intersection or disjoint) or diagrammatic conclusion (generating an Euler diagram representing the conclusion) to the syllogism. Euler-Net can achieve 99.5% accuracy for generating syllogism conclusions, and learns meaningful representations. We propose that our framework can be applied to other types of diagrams, especially the ones we are less sure how to formalise symbolically.

Proof systems play a major role in the formal study of diagrammatic logical systems. Typically, the style of inference is not directly comparable to traditional sentential systems, to study the diagrammatic aspects of inference. In this work, we present a proof system for Euler diagrams with shading in the style of sequent calculus. We prove it to be sound and complete. Furthermore we outline how this system can be extended to incorporate heterogeneous logical descriptions. Finally, we explain how small changes allow for reasoning with intuitionistic logic.

Does the choice of colour-coding scheme affect the usability of metro maps, as measured by the accuracy and speed of navigation? Using colour to differentiate lines or services in maps of metro rail networks has been a common practice around the world for many decades. Broadly speaking, there are two basic schemes: ‘route colouring’, in which each end-to-end route has a distinct colour, and ‘trunk colouring’, in which each major trunk has a distinct colour, and the individual routes inherit the colour of the main trunk that they run along. A third, intermediate scheme is ‘shaded colouring’, in which each trunk has a distinct colour, and each route has a distinct shade of that colour. In this study, 285 volunteers in the US were randomised to these three colour-coding schemes and performed seventeen navigational tasks. Each task involved tracing a route in the New York City subway map. Overall, we found that route colouring was significantly more accurate than the trunk- and shaded-colouring schemes. A planned subset analysis, however, revealed major differences between specific navigational hazards: route colouring performed better only against certain navigational hazards; trunk colouring performed best against one hazard; and other hazards showed no effect of colour coding. Route colouring was significantly faster only in one subset.

We present a study that investigates how graph format and training can affect undergraduate psychology students’ ability to interpret three-variable bar and line graphs. A pre and post-test design was employed to assess 76 students’ conceptual understanding of three-variable graphs prior to and after a training intervention. The study revealed that significant differences in interpretation are produced by graph format prior to training; bar graph users outperform line graph users. Training also resulted in a statistically significant improvement in interpretation of both graph formats with effect sizes confirming the intervention resulted in substantial learning gains in graph interpretation. This resulted in bar graph users outperforming line graph users pre and post training making it the superior format even when training has occurred. The effect of graph format and training differed depending on task demands. Based on the results of this experiment, it is argued that undergraduate students’ interpretations of such three-variable data are more accurate when using the bar form. Findings also demonstrate how a brief tutorial can result in large gains in graph comprehension scores. We provide a test which can be used to assess students understanding of three-variable graphs and the tutorial developed for the study for educators to use.

How do you make sense of a graph that you have never seen before? In this work, we outline the types of prior knowledge relevant when making sense of an unconventional statistical graph. After observing students reading a deceptively simple graph for time intervals, we designed four instructional scaffolds for evaluation. In a laboratory study, we found that only one scaffold (an interactive image) supported accurate interpretation for most students. Subsequent analysis of differences between two sets of materials revealed that task structure–specifically the extent to which a problem poses a mental impasse–may function as a powerful aid for comprehension. We find that prior knowledge of conventional graph types is extraordinarily difficult to overcome.

Almost 100 years ago, Otto Neurath developed the Isotype (International System of Typographic Picture Education) method to communicate statistical information to the broad public in an intuitive, pictorial way. It translates numerical data into arrangements of repeated pictograms. This method is still well-used in information design and data journalism. Neurath’s original publications contained a lot of assumptions on how Isotype diagrams are processed by recipients: e.g. they can be understood easily, because pictograms are processed in the same way as everyday observations of the same concepts. But documented empirical proof was entirely missing. We present a model for the reception of Isotype-like diagrams from a cognitive perspective. This model includes Isotype’s positive effects of countability, iconicity and ancillary semantic information on graph comprehension. Positive effects on engagement and perceived attractiveness are included as additional factors commonly attributed to Isotype. We discuss existing empirical studies, point out research gaps and propose a roadmap for further research.

To date, research on the processing involved in comprehending and learning from animated diagrams has accorded a minor role only to perceptual operations in general and peripheral processing in particular. For those aspects where the role of perception is acknowledged, it is foveal rather than peripheral processing that is regarded as the main player. In this paper, we use the results from additional finer grained analysis of data collected in a recent empirical study to suggest that information from a viewer’s peripheral field can play a much more central role in animation processing than has previously been recognized. It appears that if the dynamic information comprising an animated diagram is presented in a suitable way, the resources available for visual perception can be partitioned so that responsibility is shared efficiently between foveal and peripheral processing. Implications with regard to elaboration of the Animation Processing Model and possible interventions for improving animation processing are discussed.

This paper explores how people understand and visualize externally a synchronous multi-threaded four-party conversation that was audio-recorded, and investigates whether conversation-analytical knowledge and/or digital skills with social media tools have an effect on the nature and complexity of the conversational structure depicted in the representation constructed. An experiment has been performed, in which 60 participants took part. Their task was to listen to a conversation, and to display it on a magnetic whiteboard in their own way. Predesigned conversation’s utterances and pictures of participants were provided, as well as markers of different colors. Both visualization process and product were coded. Coding of process included production time and relistening behavior. The product was analyzed with respect to the ordering of utterances. We used four characterizations of ordering: by chronology, by reply-to relationships, by topic, and by conversational participant. Production time and relistening behavior turn out to have varying effects on products. Results of the representations’ analysis suggest that conversation-analytical knowledge or experience with a variety of social media influence the type and the number of ordering principles used.

The study strategy of comparing-and-contrasting has been well validated for learning from text, but not from diagrams. As part of a semester-long study strategies intervention for undergraduate biology students, we created 4 short instructional videos demonstrating the strategy of comparing-and-contrasting within diagrams (CC DIA) and delivered these just before the first course exam. We hypothesized that this strategy would help students develop a deeper comprehension of the instructed biology content. Participants were 128 undergraduates in a 2nd semester introductory (molecular and cellular) biology course, who participated in exchange for extra course credit. Students who accessed our videos scored a significant 5.5% points higher on the first exam of the semester, compared to students in other conditions or non-viewers (d = .35). Our brief (approx. 10 min per week × 4 weeks) instruction in using diagrams to learn biology yielded significant gains in undergraduate achievement.

The way in which cueing a local element of a hierarchical diagram influences the distribution of visual attention was examined. Using a modified spatial cueing paradigm, the relation between the cue and the target was manipulated by two factors: the level at which the target was presented (higher, identical, lower), and the component to which the target belonged (same or different). The results showed an interaction between these two factors, and simple main effect analysis revealed that the detection time of the target was influenced by three factors: belongingness to the same component, geometrical collinearity of the nodes, and top bias, which regards the top of the diagram as being more informative. All of these factors are related to the conventional knowledge normally possessed about the particular category of the diagrams, and to how such knowledge affects the efficiency of the diagram comprehension process.

Among the growing body of research on the interpretation of diagrams there appears to have been relatively little attention paid to emotional or attitudinal responses, despite the fact that they may be significant for communicators aiming to stimulate interest, influence attitudes, or motivate action. This research explores the impact on affect of visual context in biological life cycle diagrams. In two qualitative studies, participants viewed decontextualized life cycle diagrams along with diagrams that included a contextual backdrop, and discussed their interpretations, associations, and attitudes toward the diagram content. Thematic analysis of the data revealed that context was associated with an elevated sense of empathy and concern for the animal, and a stronger perception of personal relevance—clear indications that diagram design can have important emotional and attitudinal impacts.

Latent Semantic Analysis is widely used for natural language processing, but is difficult to visualise and interpret. We present an interactive visualisation that enables the interpretation of latent semantic spaces. It combines a multi-dimensional scatterplot diagram with a novel clutter-reduction strategy based on hierarchical clustering. A study with 12 non-expert participants showed that our visualisation was significantly more usable than experimental alternatives, and helped users make better sense of the latent space.

A combination of node-link diagram and matrix seems to be beneficial since their respective strengths and weaknesses complement each other. However, users have to read both representations in different ways and switch between these representation styles. We conducted a user study to understand how users transform a node-link diagram to a matrix representation and vice versa. For this purpose we let participants draw node-link diagrams and matrices. The drawings were analyzed to identify strategies how user convert one visualization into the other one.

Using the proof of Peirce’s Law [{(x → y) → x} → x] as an example, I show how bilateral tableau systems (or “2-sided trees”) are not only more economical than rival systems of logical proof, they also better reflect the reasoning Peirce actually gives for securing the law’s acceptance as an axiom. Moreover, bilateral proof trees are readily adapted to Peirce’s own graphical notation, producing a proof system in that notation that is even more efficient and easier to learn than Peirce’s system of permissions. This is in part due to the fact that Peirce’s graphical notation is similarly bilateral. In effect bilateral proof trees in Peirce’s notation can be understood as representing the space of outcomes for a game very much like what Peirce envisions as his endopereutic, and they embody insights of certain expressions of the pragmatic maxim that Peirce offers around 1905. Taken together, this suggests to me that Peirce would have embraced such a system of logic, and so I find it especially unfortunate that he was evidently unaware of Lewis Carroll’s pioneering efforts to develop tree-like proof systems to solve logical puzzles with multiliteral sorites.

Peirce introduced the Alpha part of the logic of Existential Graphs (egs) as a diagrammatic syntax and graphical system corresponding to classical propositional logic. The logic of quasi-Boolean algebras (De Morgan algebras) is a weakening of classical propositional logic. We develop a graphical system of weak Alpha graphs for quasi-Boolean algebras, and show its soundness and completeness with respect to this algebra. Weak logical graphs arise with only minor modifications to the transformation rules of the original theory of egs. Implications of these modifications to the meaning of the sheet of assertion are then also examined.

Peirce’s graphical logic of Existential Graphs (egs) has no specific sign for assertion, although the notion is used virtually everywhere in Peirce’s logical theories. We outline the new system of Assertive Graphs (ags) that makes the embedded notion of assertions in egs explicit, and show how to inferentially transform ags to a classical graphical logic clag, without having to introduce polarities explicitly. We compare the philosophy of notation of ags to egs, where the latter has polarities both in its intuitionistic and classical cases. Our comparison is framed with respect to three different representations of implication, namely as cuts, boxes and scrolls. We also identify three fundamental differences in the meaning of the Sheet of Assertion and compare those with Peirce’s own proposed interpretation.

The aim of this paper is to show the symmetric organization of the transformation rules for erasing or inserting graphs in Existential Graphs, which Peirce constructed as the diagrammatical system of logic that uses graphical apparatus instead of formulae of standard formal logic. Many researchers have overlooked the symmetry of Peirce’s initial layout of the transformation rules of erasure and insertion. In this essay, I will symmetrically rearrange the rules of erasure and insertion on the basis of the accurate understandings of Peirce’s statements.

On the verge of the 20th century, Charles S. Peirce was convinced that his Existential Graphs were the best form of presenting every deductive argument. Between 1900 and 1909, Peirce chose the scroll as a basic sign in his Alpha system for Existential Graphs. According to a recent paper by Francesco Bellucci and Ahti-Veikko Pietarinen, the reason for this choice lies mainly in the non-analyzable nature of the scroll: Only one sign expresses the basic notion of illation. In this paper, some analogies between this early version of the Alpha system and Structural Reasoning (in the sense of Kosta Došen and Peter Schröder-Heister) are explored. From these analogies, it will be claimed that the system Alpha based on the scroll can be used as an accurate framework for (i) constructing basic structural deductions and (ii) accomplishing a diagrammatic interpretation of logical constants of First-Order Language. Moreover, EGs show cognitive advantages with respect to sequent systems. In this paper, the basic conception is outlined in an informal way, without making an exposition of the technical details.

In her writings, Shin points out that non-symbolic representation systems have so far been underestimated in favour of symbolic systems. Her alternative aims to overcome the shortcomings of diagrammatic systems while saving the benefits by formalising them in a way that takes advantage of the iconic properties of diagrammatic representations. Specifically, it proposes a natural system by providing a new formulation for reading algorithms and the inference rules of C. S. Peirce’s Existential Graphs (EG) that is easy to understand and use. In this study, I cover issues related to multiple readings. In their papers, Bellucci and Pietarinen critically examines Shin’s arguments from several points of view. According to Shin, multiple readings are an example that shows the typical characteristics of diagrammatic systems that are not possible in symbolic systems but possible in the alpha part of EG. According to Bellucci and Pietarinen, in contrast, the multiple readings argument is useless to distinguish diagrammatic systems from symbolic ones because it contains circular arguments. Through an examination of this issue, this study considers the diagram and language differences, and differences between the icons and symbols.

We attempt to develop a proof theory for heterogeneous logic combining first-order formulas and diagrams. In proof theory, normal proofs and normalization play a central role, which makes it possible to analyze and characterize the structure of proofs in a given system. In light of the difference between linguistic reasoning and diagrammatic reasoning, we apply the traditional proof theory developed in symbolic logic to heterogeneous logic, and we give a characterization of the structure of heterogeneous proofs based on our normalization theorem.

This paper provides a programmatic overview of a conception of iconic logic from a Wittgensteinian point of view (WIL for short). The crucial differences between WIL and a standard version of symbolic logic (SSL) are identified and discussed. WIL differs from other versions of logic in that in WIL, logical forms are identified by means of so-called ideal diagrams. A logical proof consists of an equivalence transformation of formulas into ideal diagrams, from which logical forms can be read off directly. Logical forms specify properties that identify sets of models (conditions of truth) and sets of counter-models (conditions of falsehood). In this way, WIL allows the sets of models and counter-models to be described by finite means. Against this background, the question of the decidability of first-order-logic (FOL) is revisited. In the last section, WIL is contrasted with Peirce’s iconic logic (PIL).

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

In this contribution we introduce a system that represents a modern version of syllogistic by exploiting an analogy with jigsaw puzzles.

Data privacy is a cross-cutting concern for many software projects. We reify a philosophically inspired model for data privacy into a concept diagram. From the concept diagram we extract the privacy constraints and demonstrate one mechanism for translating the constraints into executable software.

We introduce a general approach, based on diagrams, to the specification and construction of model checkers. This approach gives general model checkers that can be instantiated to a model checker for a specific modal logic with semantics described by graphical rules. This paper proposes a way of combining graphical and general approaches to model checking so that the instantiation to specific logics is user-friendly and natural.

An incessant debate that history of syllogistic reasoning has witnessed is on the status of fourth figure after its alleged invention. Commentators on Aristotle and several other logicians have advocated various approaches to include or abandon this last figure. However, in the middle of last century, the debate seemed to have reached quiescence with fifteen valid syllogisms present in four figures. Among this, some moods are distinct, i.e. they are valid in one figure whereas others are non-distinct as they are valid in multiple figures. In this paper, the notion of diagrammatic congruence for non-distinct syllogisms using Venn-Peirce diagrams is introduced. Consequently, we establish the equivalence of moods that are diagrammatically congruent. Furthermore, it is argued that the presence of a distinct mood is pivotal to recognize an arrangement as a separate figure, which is evident in Aristotle’s own treatment of figures. With this, the redundancy of fourth figure is demonstrated.

Ontology creation and editing involves multiple stakeholders, not all of whom may be mathematically trained. Whilst ontology editors, such as Protege, are extensible with visualisation tools to enhance understanding of the ontology, these tools are static representations only. We present initial work on creating editable visualisations for a fragment of OWL, which will in turn update the underlying ontology. The diagrams used are linear diagrams, which have previously been shown to aid comprehension of set-based data. In particular, we focus only on those OWL statements which do not include properties or datatypes.

Stock-flow (SF) systems are omnipresent in our lives while difficult to understand. An example is the amount of CO₂ in the atmosphere (stock) that changes in dependence of incoming CO₂ (inflow) and outgoing CO₂ (outflow). When participants are to deal with such tasks, they show poor performance. Despite several attempts to facilitate SF knowledge in participants, as far as we know, only one manipulation led to meaningfully increased SF performance: Changing the representation of the flows into pictograms. In the current study, we intend to modify these kind of diagrams so that they communicate SF information in a simple way. We tested whether the modified representation triggered basic SF understanding. Each participant worked on two tasks; one shown as line graph and one shown as diagram with pictograms. Getting the pictograms at first position led to strongly improved SF performance. A t-test revealed more correct solutions for pictograms than for line graph at first position. Again, the representation of the flows as pictograms led to better SF performance.

We present a method of digraphs for Syllogistic that uses only two rules for testing the validity of syllogisms without existential import and a third rule for cases in which the existential import of terms is used. This method derives from Martin Gardner’s network method for Classical Propositional Logic, preserving the iconicity features that Gardner attributes to the propositional case also in the case of Syllogistic. We will first present the graphical representations and the rules for manipulating these representations in the case of syllogisms in which the existential import of terms are not admitted. Then, we will extend the method with a graphical representation for the existential import and with a new rule for the manipulation of this representation. Finally, we will show some applications of the method. It was first presented in Portuguese in Cognitio 14(2013): 221–234.

Mathematical research is best characterized by problem- solving activities which make use of a variety of modes of representation. Against this background, my aim is to discuss the epistemic value of diagrammatic representation in problem-solving. To make my point, I consider a case study selected from Wallis’s work on the quadrature of conic sections. Wallis’s definition of conic sections is given in terms of algebraic equations setting them free from ‘the embrangling of the cone’. This suggests the aim to eliminate figures and other iconic elements with a view to attaining higher level of abstraction but, in Wallis’s work, geometric diagrams display relations that can be fruitfully used to calculate arithmetically the area of a figure. The use of displayed relations leads to the formulation of algebraic equations defining curves and it is also what makes room for arithmetical calculations. Accordingly, the notion of a general method of resolution is grounded on properties read off the diagram so that despite Wallis’s insistence on algebraic representation -I argue- diagrams remain essential working tools.

We propose an algorithmic procedure for the automatic analysis of syllogisms with negative terms based on the modified version of Shin’s Venn-I diagram. Our computational procedure can automatically generate all the possible conclusions derivable from the two premises of a given syllogism with negative terms. Our approach relies on the reformulation of the logic behind the relations between points, lines, and surfaces in the Venn diagram by employing conditional propagation rules.

It is little known that Schopenhauer (1788–1860) made thorough use of Euler diagrams in his works. One specific diagram depicts a high number of concepts in relation to Good and Evil. It is, hence, uncharacteristic as logicians of that time seldom used diagrams for more than three terms (the number demanded by syllogisms). The objective of this paper is to make sense of this diagram by explaining its function and inquiring whether it could be viewed as an early serious attempt to construct complex diagrams.

Multimedia learning research pointed out that adding a picture to a text is not systematically beneficial to learners. One of the most influential factors is the necessity for learners to identify mutually referring information in the written and pictorial representations. This study investigates how Cross-Representational Signaling (CRS) facilitates learning from multimedia document. In this study, CRS is implemented by mutually referring visual and verbal cues which highlight semantic links between text and picture. Two versions of the same multimedia document explaining the risks of being caught in a rapid, with or without CRS, are compared. The study that is still ongoing will provide data on online processing (eye-tracking data) and learning outcomes. The results will provide insights on the use of CRS to improve the design of instructional diagrams.

The impact of the dominant hand on the response time and precision in mental rotational tasks seems to be controversial. The goal of this study was to compare the differences in response times of mental rotation tasks when the task is performed with the dominant or non-dominant hand. In this study, 44 right-handers and 45 left-handers participated in mental rotation tests with 2-D and 3-D figures. Findings indicate that the right-handers had shorter response times than left-handers in tests with both types of figures.

Venn and Euler diagrams are valuable tools for representing the logical set relationships among events. Proportional Euler diagrams add the constraint that the areas of diagram regions denoting various compound and simple events must be proportional to the actual probabilities of these events. Such proportional Euler diagrams allow human users to visually estimate and reason about the probabilistic dependencies among the depicted events. The present paper focuses on the use of proportional Euler diagrams composed of rectangular regions and proposes an enhanced display format for such diagrams, dubbed “Euler boxes”, that facilitates quick visual determination of the independence or non-independence of two events and their complements. It is suggested to have useful applications in exploratory data analysis and in statistics education, where it may facilitate intuitive understanding of the notion of independence.

Symmetry is often considered a desirable feature of diagrams. However, quantifying the exact amount of symmetry present is often difficult. We propose a novel symmetry metric that can score the amount of rotational, translational, and reflective symmetry present in a graph or line diagram.

We present additions to the widely-used layout method for directed acyclic graphs of Sugiyama et al. that allow to better utilize a prescribed drawing area. The method itself partitions the graph’s nodes into layers. When drawing from top to bottom, the number of layers directly impacts the height of a resulting drawing and is bound from below by the graph’s longest path. As a consequence, the drawings of certain graphs are significantly taller than wide, making it hard to properly display them on a medium such as a computer screen without scaling the graph’s elements down to illegibility. We address this with the Wrapping Layered Graphs Problem (WLGP), which seeks for cut indices that split a given layering into chunks that are drawn side-by-side with a preferably small number of edges wrapping backwards. Our experience and a quantitative evaluation indicate that the proposed wrapping allows an improved presentation of narrow graphs, which occur frequently in practice and of which the internal compiler representation SCG is one example.

The semiotics of Charles Sanders Peirce was responsible for the blossoming of many semiotic approaches to music in the past few decades. Whilst it is clear that musicologists and philosophers of music have benefited from Peirce’s semiotics to better explain and discover aspects of music, the same cannot be said about Peirce’s philosophy of diagrams in particular, that seems to remain widely ignored by semioticians of music. Notwithstanding, music in general, and musical composition in particular, widely rely on schemas, rules, notational systems and other signs that might be understood as diagrams in some sense of the term. Following that clue, in this text we focus on the role played by the musical notation in the compositional process and argue that such role can be understood under the concept of diagram. We analyse an aspect of the compositional process of Beethoven’s Sehnsucht (WoO 146) and argue that the notational system functions here as a diagram that mediates a diagrammatic reasoning process.

The paper outlines the advantages and limits of the so-called ‘Calculus CL’ in the field of ontology engineering and automated theorem proving. CL is a diagram type that combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. Due to the simple taxonomical structures and intuitive rules of CL, it is easy to edit ontologies and to prove inferences.

In this contribution we present an extension of Englebretsen’s linear diagrams in order to deal with non-classical quantifiers.

In this contribution we review Murner’s syllogistic fragment as it appears in his Logica memorativa.

Human interaction is increasingly mediated through technological systems, resulting in the emergence of a new class of socio-technical systems, often called Social Machines. However, many systems are designed and managed in a centralised way, limiting the participants’ autonomy and ability to shape the systems they are part of.In this paper we are concerned with creating a graphical formalism that allows novice users to simply draw the patterns of interaction that they desire, and have computational infrastructure assemble around the diagram. Our work includes a series of participatory design workshops, that help to understand the levels and types of abstraction that the general public are comfortable with when designing socio-technical systems. These design studies lead to a novel formalism that allows us to compose rich interaction protocols into functioning, executable architecture. We demonstrate this by translating one of the designs produced by workshop participants into an a running agent institution using the Lightweight Social Calculus (LSC).

Design Thinking can be employed to define services, new product (features), innovative processes and disruptive business models collaboratively for digitization. Diagrammatic models play an essential role here as they capture relationships between different aspects of the problem. When computed by means of software they also explicitly show details which in design thinking tools users implicitly fill with their own world-understanding, thus fostering a clear and transparent representation of the problem space. In addition, diagrammatic models can be enriched by semantics and subsequently be queried, analysed and processed.The paper at hand shows the DigiTrans (http://www.interreg-danube.eu/approved-projects/digitrans) project approach for an automated transformation process of haptic storyboards into diagrammatic models by means of video-imaging and web-services.

In this contribution, we introduce an approach to visualize and analyze logical reasoning problems in a UML and OCL tool by using logical puzzles represented with UML diagrams. Logical reasoning is formalized as a UML class diagram model enhanced by OCL restrictions. Puzzle rules and questions are expressed as either partial object diagrams or OCL formulas within the model. Solutions can be found and explored by a tool as object diagrams.

In this study I aim to develop a cognitive evaluation of how semantically-driven and rule-based diagrammatic reasoning were psychologically plausible for particular cases of scientific practice: an actual trackless path reconstruction in bubble chamber experiments, as reported by Galison (1997). I will propose “cognitive imagery projection and manipulation” (CIPM) as the most plausible psychological (perceptual/attentional/cognitive) mechanism matching the specific explanatory requirements for the case study, outlining the most significant current theories about this mental phenomenon (Shimojima 2011).

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.