How can Euler diagrams support non-consequence inferences? Although an inference to non-consequence, in which people are asked to judge whether no valid conclusion can be drawn from the given premises, is one of the two sides of logical inference, it has received remarkably little attention in research on human diagrammatic reasoning; how diagrams are really manipulated for such inferences remains unclear. We hypothesized that people naturally make these inferences by enumerating possible diagrams, based on the logical notion of self-consistency, in (...) which every Euler diagram is true in a set-theoretical interpretation. The work is divided into three parts, each exploring a particular condition or scenario. In condition 1, we asked participants to directly manipulate diagrams with size-fixed circles as they solved syllogistic tasks, with the result that more reasoners used the enumeration strategy. In condition 2, another type of size-fixed diagram was used. The diagram layout change interfered with accurate task performances and with the use of the enumeration strategy; however, the enumeration strategy was still dominant for those who could correctly perform the tasks. In condition 3, we used size-scalable diagrams, which reduced the interfering effect of diagram layout and enhanced participants’ selection of the enumeration strategy. These results provide evidence that non-consequence inferences can be achieved by diagram enumeration, exploiting the self-consistency of Euler diagrams. An alternate strategy based on counter-example construction with Euler diagrams, as well as effects of diagram layout in inferential processes, are also discussed. (shrink)

Explanations in the life sciences frequently involve presenting a model of the mechanism taken to be responsible for a given phenomenon. Such explanations depart in numerous ways from nomological explanations commonly presented in philosophy of science. This paper focuses on three sorts of differences. First, scientists who develop mechanistic explanations are not limited to linguistic representations and logical inference; they frequently employ diagrams to characterize mechanisms and simulations to reason about them. Thus, the epistemic resources for presenting mechanistic explanations are (...) considerably richer than those suggested by a nomological framework. Second, the fact that mechanisms involve organized systems of component parts and operations provides direction to both the discovery and testing of mechanistic explanations. Finally, models of mechanisms are developed for specific exemplars and are not represented in terms of universally quantified statements. Generalization involves investigating both the similarity of new exemplars to those already studied and the variations between them. (shrink)

This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic and introduce (...) an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences. (shrink)

Semantic studies on diagrammatic notations (Barwise & Etchemendy, ; Shimojima, ; Stenning & Lemon, ) have revealed that the “non-deductive,” “emergent,” or “perceptual” effects of diagrams (Chandrasekaran, Kurup, Banerjee, Josephson, & Winkler, ; Kulpa, ; Larkin & Simon, ; Lindsay, ) are all rooted in the exploitation of spatial constraints on graphical structures. Thus, theoretically, this process is a key factor in inference with diagrams, explaining the frequently observed reduction of inferential load. The purpose of this study was to examine (...) the empirical basis for this theoretical suggestion, focusing on the reality of the constraint-exploitation strategy in actual practices of diagrammatic reasoning. Eye movements were recorded while participants used simple position diagrams to solve three- or four-term transitive inference problems. Our experiments revealed that the participants could exploit spatial constraints on graphical structures even when (a) they were not in the position of actually manipulating diagrams, (b) the semantic rule for the provided diagrams did not match their preferences, and (c) the constraint-exploitation strategy invited a partly adverse effect. These findings indicate that the hypothesized process is in fact robust, with the potential to broadly account for the inferential advantage of diagrams. (shrink)

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and (...) Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. (shrink)

In Volume 33:1 of the Notre Dame Journal of Formal Logic, a system for diagramming syllogistic inferences using straight line segments is presented by Englebretsen. In light of recent research on the representational power of diagrammatic representation systems by the authors, we point out some problems with the proposal, and indeed, with any proposal for representing logically possible situations diagrammatically. We shall first outline the proposed linear diagrammatic system of Englebretsen, and then show by means of counterexamples that it is (...) inadequate as a representation scheme for general logical inferences (the task for which the system is intended). We also show that modifications to the system fail to remedy the problems. The considerations we present are not limited to the particular proposal of Englebretsen; we thus draw a more general moral about the use of spatial relations in representation systems. (shrink)