Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in (...) the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem. (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)
We introduce a simple inference system based on two primitive relations between terms, namely, inclusion and exclusion relations. We present a normalization theorem, and then provide a characterization of the structure of normal proofs. Based on this, inferences in a syllogistic fragment of natural language are reconstructed within our system. We also show that our system can be embedded into a fragment of propositional minimal logic.
There is a widely held view that visual representations (images) do not depict negation, for example, as expressed by the sentence, “the train is not coming.” The present study focuses on the real-world visual representations of photographs and comic (manga) illustrations and empirically challenges the question of whether humans and machines, that is, modern deep neural networks, can recognize visual representations as expressing negation. By collecting data on the captions humans gave to images and analyzing the occurrences of negation phrases, (...) we show some evidence that humans recognize certain images as expressing negation. Furthermore, based on this finding, we examined whether or not humans and machines can classify novel images as expressing negation. The humans were able to correctly classify images to some extent, as expected from the analysis of the image captions. On the other hand, the machine learning model of image processing was only able to perform this classification at about the chance level, not at the same level of performance as the human. Based on these results, we discuss what makes humans capable of recognizing negation in visual representations, highlighting the role of the background commonsense knowledge that humans can exploit. Comparing human and machine learning performances suggests new ways to understand human cognitive abilities and to build artificial intelligence systems with more human-like abilities to understand logical concepts. (shrink)
This paper provides a detailed comparison between discourse representation theory and dependent type semantics, two frameworks for discourse semantics. Although it is often stated that DRT and those frameworks based on dependent types are mutually exchangeable, we argue that they differ with respect to variable handling, more specifically, how substitution and other operations on variables are defined. This manifests itself in two recalcitrant problems posed for DRT; namely, the overwrite problem and the duplication problem. We will see that these problems (...) still pose a challenge for various extended compositional systems based on DRT, while they do not arise in a framework of DTS where substitution and other operations are defined in the standard type-theoretic manner without stipulating any additional constraints. We also compare the notions of contexts underlying these two kinds of frameworks, namely, contexts represented as assignment functions and contexts represented as proof terms, and see what different predictions they make for some linguistic examples. (shrink)