Koji Mineshima

A Diagrammatic Inference System with Euler Circles

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.

Diagrammatic Reasoning System with Euler Circles: Theory and Experiment Design

In this paper we are concerned with logical and cognitive aspects of reasoning with Euler circles. We give a proof-theoretical analysis of diagrammatic reasoning with Euler circles involving unification and deletion rules. Diagrammatic syllogisticreasoning is characterized as a particular class of the general diagrammatic proofs. Given this proof-theoretical analysis, we present some conjectures on cognitive aspects of reasoning with Euler diagrams. Then we propose a design of experiment for a cognitive psychological study.

Higher-order logical inference with compositional semantics

We present a higher-order inference system based on a formal compositional semantics and the wide-coverage CCG parser. We develop an improved method to bridge between the parser and semantic composition. The system is evaluated on the FraCaS test suite. In contrast to the widely held view that higher-order logic is unsuitable for efficient logical inferences, the results show that a system based on a reasonably-sized semantic lexicon and a manageable number of non-first-order axioms enables efficient logical inferences, including those concerned with generalized quantifiers and intensional operators, and outperforms the state-of-the-art firstorder inference system.

How Diagrams Can Support Syllogistic Reasoning: An Experimental Study

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.

Towards explaining the cognitive efficacy of Euler diagrams in syllogistic reasoning: A relational perspective

Although diagrams have been widely used as methods for introducing students to elementary logical reasoning, it is still open to debate in cognitive psychology whether logic diagrams can aid untrained people to successfully conduct deductive reasoning. In our previous work, some empirical evidence was provided for the effectiveness of Euler diagrams in the process of solving categorical syllogisms. In this paper, we discuss the question of why Euler diagrams have such inferential efficacy in the light of a logical and proof-theoretical analysis of categorical syllogisms and diagrammatic reasoning. As a step towards an explanatory theory of reasoning with Euler diagrams, we argue that the effectiveness of Euler diagrams in supporting syllogistic reasoning derives from the fact that they are effective ways of representing and reasoning about relational structures that are implicit in categorical sentences. A special attention is paid to how Euler diagrams can facilitate the task of checking the invalidity of an inference, a task that is known to be particularly difficult for untrained reasoners. The distinctive features of our conception of diagrammatic reasoning are made clear by comparing it with the model-theoretic conception of ordinary reasoning developed in the mental model theory.

A Generalized Syllogistic Inference System based on Inclusion and Exclusion Relations

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.

Context-Passing and Underspecification in Dependent Type Semantics

Dependent type semantics (DTS) is a framework of discourse semantics based on dependent type theory, following the line of Sundholm (Handbook of Philosophical Logic, 1986) and Ranta (Type-Theoretical Grammar, 1994). DTS attains compositionality as required to serve as a semantic component of modern formal grammars including variations of categorial grammars, which is achieved by adopting mechanisms for local contexts, context-passing, and underspecified terms. In DTS, the calculation of presupposition projection reduces to type checking, and the calculation of anaphora resolution and presupposition binding both reduce to proof search in dependent type theory, inheriting the paradigm of anaphora resolution as proof construction.

ccg2lambda: A Compositional Semantics System

We demonstrate a simple and easy-to-use system to produce logical semantic representations of sentences. Our software operates by composing semantic formulas bottom-up given a CCG parse tree. It uses flexible semantic templates to specify semantic patterns. Templates for English and Japanese accompany our software, and they are easy to understand, use and extend to cover other linguistic phenomena or languages. We also provide scripts to use our semantic representations in a textual entailment task, and a visualization tool to display semantically augmented CCG trees in HTML.

Human Reasoning with Proportional Quantifiers and Its Support by Diagrams

In this paper, we study the cognitive effectiveness of diagrammatic reasoning with proportional quantifiers such as most. We first examine how Euler-style diagrams can represent syllogistic reasoning with proportional quantifiers, building on previous work on diagrams for the so-called plurative syllogism (Rescher and Gallagher, 1965). We then conduct an experiment to compare performances on syllogistic reasoning tasks of two groups: those who use only linguistic material (two sentential premises and one conclusion) and those who are also given Euler diagrams corresponding to the two premises. Our experiment showed that (a) in both groups, the speed and accuracy of syllogistic reasoning tasks with proportional quantifiers like most were worse than those with standard first-order quantifiers such as all and no, and (b) in both standard and non-standard (proportional) syllogisms, speed and accuracy for the group provided with diagrams were significantly better than the group provided only with sentential premises. These results suggest that syllogistic reasoning with proportional quantifiers like most is cognitively complex, yet can be effectively supported by Euler diagrams that represent the proportionality relationships between sets in a suitable way.

A presuppositional analysis of definite descriptions in proof theory

In this paper we propose a proof-theoretic analysis of presuppositions in natural language, focusing on the interpretation of definite descriptions. Our proposal is based on the natural deduction system of Ɛ-calculus introduced in Carlstrom [2] and on constructive type theory [11,12]. Based on the idea in [2], we use the Ɛ-calculus as an intermediate language in the translation process from natural language into constructive type theory. Using this framework, we formulate the process of presupposition resolution as the process of searching for a derivation in a natural deduction system. In particular, we show how to treat presupposition projection and accommodation within our proof-theoretic framework.