Constructive geometrical reasoning and diagrams

Modern formal accounts of the constructive nature of elementary geometry do not aim to capture the intuitive or concrete character of geometrical construction. In line with the general abstract approach of modern axiomatics, nothing is presumed of the objects that a geometric construction produces. This study explores the possibility of a formal account of geometric construction where the basic geometric objects are understood from the outset to possess certain spatial properties. The discussion is centered around Eu, a recently developed formal system of proof (presented in Mumma (Synthese 175:255?C287, 2010)) within which Euclid??s diagrammatic proofs can be represented.     

Defeasible inheritance systems and reactive diagrams

Inheritance diagrams are directed acyclic graphs with two typesof connections between nodes: x y (read x is a y) and x y(read as x is not a y). Given a diagram D, one can ask the formalquestion of "is there a valid (winning) path between node xand node y?" Depending on the existence of a valid path we cananswer the question "x is a y" or "x is not a y". The answer to the above question is determined through a complexinductive algorithm on paths between arbitrary pairs of pointsin the graph. This paper aims to simplify and interpret such diagrams andtheir algorithms. We approach the area on two fronts. (1) Suggest reactive arrows to simplify the algorithms for the winningpaths. (2) We give a conceptual analysis of (defeasible or nonmonotonic)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 andtruth values, direct links as information, and valid paths asinformation channels and comparisons of truth values. This resultsin an upward chaining, split validity, off-path preclusion inheritanceformalism of a particularly simple type. We show that the small and big sets of preferential reasoninghave to be relativized if we want them to conform to inheritancetheory, resulting in a more cautious approach, perhaps closerto actual human reasoning. We will also interpret inheritance diagrams as theories of prototypicalreasoning, based on two distances: set difference, and informationdifference. We will see that some of the major distinctions between inheritanceformalisms are consequences of deeper and more general problemsof treating conflicting information. It is easily seen that inheritance diagrams can also be analysedin terms of reactive diagrams - as can all argumentation systems. AMS Classification: 68T27, 68T30 Received for publication 15 March 2007.     

Socratic Trees

The method of Socratic proofs (SP-method) simulates the solving of logical problem by pure questioning. An outcome of an application of the SP-method is a sequence of questions, called a Socratic transformation. Our aim is to give a method of translation of Socratic transformations into trees. We address this issue both conceptually and by providing certain algorithms. We show that the trees which correspond to successful Socratic transformations—that is, to Socratic proofs—may be regarded, after a slight modification, as Gentzen-style proofs. Thus proof-search for some Gentzen-style calculi can be performed by means of the SP-method. At the same time the method seems promising as a foundation for automated deduction.     

Aristotelian diagrams for semantic and syntactic consequence

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation.

     

Towards a model theory of diagrams

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.The authors are grateful to Jon Barwise and an anonymous referee for valuable comments and suggestions.     

Contingency diagrams as teaching tools

Contingency diagrams are particularly effective teaching tools, because they provide a means for students to view the complexities of contingency networks present in natural and laboratory settings while displaying the elementary processes that constitute those networks. This paper sketches recent developments in this visualization technology and illustrates approaches for using contingency diagrams in teaching.     

Solving path diagrams with EUREKA

To solve the path diagrams for a particular causal model, psychologists typically have made use of sophisticated programs such as LISREL. Such programs are difficult for the beginning student to use, and are occasionally awkward or inconvenient even for veteran causal modelers. It would be helpful to such users to have access to a method for conducting causal modeling on a microcomputer without requiring that they have any programming skills. Fortunately, such a procedure is now readily available in the form of EUREKA (1987). This paper describes a way of employing EUREKA so that it is of value not only to naive, but also to sophisticated, users.     

Scientific modelling with diagrams

Synthese - Diagrams can serve as representational models in scientific research, yet important questions remain about how they do so. I address some of these questions with a historical case study,...     

Animated diagrams in teaching statistics

In this study, we investigated whether computer-animated graphics are more effective than static graphics in teaching statistics. Four statistical concepts were presented and explained to students in class. The presentations included graphics either in static or in animated form. The concepts explained were the multiplication of two matrices, the covariance of two random variables, the method of least squares in linear regression, α error, β error, and strength of effect. A comprehension test was immediately administered following the presentation. Test results showed a significant advantage for the animated graphics on retention and understanding of the concepts presented.     

Trees with structure

We investigate M-trees, that is, trees with structure possible at each node or level. M is a mathematical structure such as a set or a Cartesian product. An extension of Pólya's theorem is proved which allows the number of M-trees for a given number of nodes to be counted. The special case of componential trees is investigated. Here M is a Cartesian product of 0's and 1's. A componential analysis is a componential tree of depth 1, that is, there is no hierarchy. We prove that for any componential tree there exists a componential analysis which makes the same predictions on a triad test of judged similarity. A brief empirical example is given, in which a componential tree is applied as a model of a sernantic domain.     

Exploring the fruitfulness of diagrams in mathematics

Synthese - The paper asks whether diagrams in mathematics are particularly fruitful compared to other types of representations. In order to respond to this question a number of examples of...     

Categorical reasoning from multiple diagrams.

Syllogistic reasoning from categorical premise pairs is generally taken to be a multistep process. Quantifiers (all, no, some, some ... not) must be interpreted, representations constructed, and conclusions identified from these. Explanations of performance have been proposed in which errors may occur at any of these stages. The current paper contrasts (a) representation explanations of performance, in which errors occur because not all possible representations are constructed, and/or mistakes are made when doing so (e.g., mental models theory), and (b) conclusion identification explanations, in which errors occur even when information has been correctly and exhaustively represented, due to systematic difficulties that people may have when identifying particular conclusions, or in identifying conclusions in particular circumstances. Three experiments are reported, in which people identified valid conclusions from diagrams analogous to Euler circles, so that the first two stages of reasoning from premise pairs were effectively removed. Despite this, several phenomena associated with reasoning from premise pairs persisted, and it is suggested that whereas representation explanations may account for some of these phenomena, conclusion identification explanations, which have never previously been considered, are required for others.     

Indexical Reference and bodily causal diagrams in intentional action

In this paper, completed only months before his death, the author studies a number of concepts of importance for the analysis of intentional action. Four themes in particular are discussed: the intentionality of action, the practical syllogism, what the author terms the practical causality of practical thinking, and the proximate cause of action. (K. S.)