Abstract
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.
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References
Allwein, G., and J. Barwise, (eds.), Logical reasoning with diagrams, Oxford Studies in Logic snd Computation Series, Oxford University Press, Oxford, 1996.
Barwise, J., and J. Etchemendy, Tarski’s world, CSLI, Stanford, 1993.
Barwise, J., and J. Etchemendy, Hyperproof: For Macintosh, vol. 42 of The Center for the Study of Language and Information Publications Lecture notes, Cambridge University Press, Cambridge, 1995.
Barwise, J., and J. Etchemendy, Logic, Proof, and Reasoning, manuscript.
Chen, J., A. G. Cohn, D. Liu, S. Wang, J. Ouyang, and Q. Yu, A survey of qualitative spatial representations, The Knowledge Engineering Review 1–31, 2013. doi:10.1017/S0269888913000350.
Cohn A. G., Hazarika S. M.: Qualitative spatial representation and reasoning: An overview. Fundamenta Informaticae 46(1–2), 1–29 (2001)
Cohn, A. G., and J. Renz, Qualitative spatial representation and reasoning, in F. van Hermelen, V. Lifschitz, and B. Porter, (eds.), Handbook of knowledge representation, Elsevier, New York, 2008, pp. 551–596.
Hammer E., Shin S.: Euler’s visual logic. History and Philosophy of Logic 19, 1–29 (1998)
Howse J., Stapleton G., Taylor J.: Spider diagrams. LMS Journal of Computation and Mathematics 8, 145–194 (2005)
Mineshima, K., M. Okada, and R. Takemura, Conservativity for a hierarchy of Euler and Venn reasoning systems, in Proceedings of visual languages and logic 2009, vol. 510 of CEUR Series, 2009, pp. 37–61.
Mineshima K., Okada M., Takemura R.: A diagrammatic inference system with Euler circles. Journal of Logic, Language and Information 21(3), 365–391 (2012)
Peirce, C. S., in C. Hartshorne and P. Weiss, (eds.), Collected papers of Charles Sanders Peirce, vol. 4, Harvard University Press, Cambridge, MA, 1933.
Shin S.-J.: The logical status of diagrams. Cambridge University Press, Cambridge (1994)
Randell, D. A., Z. Cui, and A. G. Cohn, A spatial logic based on regions and connection, in Proceedings of 3rd international conference on knowledge representation and reasoning, Morgan Kaufmann, San Mateo, 1992, pp. 165–176.
Takemura R.: Completeness of an Euler diagrammatic system with constant and existential points, Journal of Humanities and Sciences Nihon University. College of commerce Nihon University 19(1–2), 23–40 (2013)
Takemura R.: Proof theory for reasoning with Euler diagrams: a logic translation and normalization. Studia Logica 101(1), 157–191 (2013)
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Takemura, R. Counter-Example Construction with Euler Diagrams. Stud Logica 103, 669–696 (2015). https://doi.org/10.1007/s11225-014-9584-x
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DOI: https://doi.org/10.1007/s11225-014-9584-x