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  1.  48
    A Diagrammatic Inference System with Euler Circles.Koji Mineshima, Mitsuhiro Okada & Ryo Takemura - 2012 - Journal of Logic, Language and Information 21 (3):365-391.
    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 (...)
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  2.  43
    Proof Theory for Reasoning with Euler Diagrams: A Logic Translation and Normalization.Ryo Takemura - 2013 - Studia Logica 101 (1):157-191.
    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 (...)
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  3.  10
    Counter-Example Construction with Euler Diagrams.Ryo Takemura - 2015 - Studia Logica 103 (4):669-696.
    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 (...)
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  4.  40
    A Generalized Syllogistic Inference System Based on Inclusion and Exclusion Relations.Koji Mineshima, Mitsuhiro Okada & Ryo Takemura - 2012 - Studia Logica 100 (4):753-785.
    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.
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  5.  11
    Completeness of Second-Order Intuitionistic Propositional Logic with Respect to Phase Semantics for Proof-Terms.Yuta Takahashi & Ryo Takemura - 2019 - Journal of Philosophical Logic 48 (3):553-570.
    Girard introduced phase semantics as a complete set-theoretic semantics of linear logic, and Okada modified phase-semantic completeness proofs to obtain normal-form theorems. On the basis of these works, Okada and Takemura reformulated Girard’s phase semantics so that it became phase semantics for proof-terms, i.e., lambda-terms. They formulated phase semantics for proof-terms of Laird’s dual affine/intuitionistic lambda-calculus and proved the normal-form theorem for Laird’s calculus via a completeness theorem. Their semantics was obtained by an application of computability predicates. In this paper, (...)
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  6.  6
    A Phase Semantics for Polarized Linear Logic and Second Order Conservativity.Masahiro Hamano & Ryo Takemura - 2010 - Journal of Symbolic Logic 75 (1):77-102.
    This paper presents a polarized phase semantics, with respect to which the linear fragment of second order polarized linear logic of Laurent [15] is complete. This is done by adding a topological structure to Girard's phase semantics [9]. The topological structure results naturally from the categorical construction developed by Hamano—Scott [12]. The polarity shifting operator ↓ (resp. ↑) is interpreted as an interior (resp. closure) operator in such a manner that positive (resp. negative) formulas correspond to open (resp. closed) facts. (...)
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