Ryo Takemura

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.

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.

Proof Theory for Reasoning with Euler Diagrams: A Logic Translation and Normalization

Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables us to formalize and analyze free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs.

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.

Counter-Example Construction with Euler Diagrams

One of the traditional applications of Euler diagrams is as a representation or counterpart of the usual set-theoretical models of given sentences. However, Euler diagrams have recently been investigated as the counterparts of logical formulas, which constitute formal proofs. Euler diagrams are rigorously defined as syntactic objects, and their inference systems, which are equivalent to some symbolic logical systems, are formalized. Based on this observation, we investigate both counter-model construction and proof-construction in the framework of Euler diagrams. We introduce the notion of “counter-diagrammatic proof”, which shows the invalidity of a given inference, and which is defined as a syntactic manipulation of diagrams of the same sort as inference rules to construct proofs. Thus, in our Euler diagrammatic framework, the completeness theorem can be formalized in terms of the existence of a diagrammatic proof or a counter-diagrammatic proof.

An Indexed System for Multiplicative Additive Polarized Linear Logic

We present an indexed logical system MALLP( I) for Laurents multiplicative additive polarized linear logic ( MALLP ) [14]. The system is a polarized variant of Bucciarelli-Ehrhards indexed system for multiplicative additive linear logic [4]. Our system is derived from a web-based instance of Hamano-Scotts denotational semantics [12] for MALLP . The instance is given by an adjoint pair of right and left multi-pointed relations. In the polarized indexed system, subsets of indexes for Iwork as syntactical counterparts of families of points in webs. The rules of

A Logical Investigation of Heterogeneous Reasoning with Graphs in Elementary Economics

\sf MALLP({\it I})

Logical Investigation of Reasoning with Tables

describe (in a proof-theoretical manner) the denotational construction of the corresponding rules of MALLP . We show that

Proof-Theoretical investigation of venn diagrams: a logic translation and free rides

\sf MALLP({\it I})