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An algebraic study of Peterson’s Intermediate Syllogisms

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Abstract

Peterson’s Intermediate Syllogisms, generalizing Aristotelian syllogisms by intermediate quantifiers ‘Many’, ‘Most’ and ‘Almost all’, are studied. It is demonstrated that, by associating certain values V, W and U on standard Łukasiewicz MV-algebra with the first and second premise and the conclusion, respectively, the validity of a corresponding intermediate syllogism is determined by a simple MV-algebra (in-)equation. Possible conservative extensions of Peterson’s system are discussed. Finally it is shown that Peterson’s bivalued intermediate syllogisms can be viewed as fuzzy theories in Pavelka’s fuzzy propositional logic, i.e. a fuzzy version of Peterson’s Intermediate Syllogisms is introduced.

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Notes

  1. An interesting historical detail is the fact that Jan Łukasiewicz whose multivalued logic’s algebraic counterpart MV-algebras are, was a leading figure in studies of syllogisms in the early twentieth century. Łukasiewicz published in 1951 his now famous monograph Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic. It is unknown if Łukasiewicz himself ever investigated intermediate syllogisms.

  2. This example is not based on the real linguistic analysis, only on our intuition.

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Acknowledgments

The author was supported by the Czech Technical University in Prague under project SGS12/187/OHK3/3T/13. The author is grateful for M. Navara, P. Murinová, T. Vetterlein, P. Cintula, C. Noguera and R. Horčík for their comments to improve the final version of this research.

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Authors and Affiliations

  1. Research Unit Computational Logic, Institute of Discrete Mathematics and Geometry, University of Technology, Vienna, Wiedner Hauptstrasse 8-10, 1040 , Vienna, Austria

    Esko Turunen

  2. Tampere University of Technology, Tampere, Finland

    Esko Turunen

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  1. Esko Turunen
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Correspondence to Esko Turunen.

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Communicated by V. Loia.

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Turunen, E. An algebraic study of Peterson’s Intermediate Syllogisms. Soft Comput 18, 2431–2444 (2014). https://doi.org/10.1007/s00500-013-1216-2

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