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Verifying Validity of Selected Forms of Syllogisms with Intermediate Quantifiers Using Peterson’s Rules

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Fuzzy Logic and Technology, and Aggregation Operators (EUSFLAT 2023, AGOP 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14069))

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Abstract

The most reliable method for how the validity of a logical syllogism can be verified is to formalize it and show that there is either a formal proof or it is true in any model. A specific method for proving validity is to use special rules that have been used by logicians. However, we cannot be sure that they indeed verify the validity of syllogisms. The goal of this paper is to show that the rules indeed work. In his book [15], Peterson studied syllogisms with intermediate quantifiers and suggested extension of the rules also to them. In this paper, we formalize them and prove that a logical syllogism of Figure I with intermediate quantifiers is valid iff it satisfies four extended Peterson’s rules.

OP PIK CZ.01.1.02/0.0/0.0/17147/0020575 AI-Met4Laser: Consortium for industrial research and development of new applications of laser technologies using artificial intelligence methods (10/2020–6/2023), of the Ministry of Industry and Trade of the Czech Republic.

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Notes

  1. 1.

    We write this interval as a union of two intervals to emphasize the role of the central point \(v_S\) which represents “typical medium”.

  2. 2.

    Note that (8) is a semantic interpretation of the formula \(\mathop { Ev}\nolimits ((\mu B)(B|z))\) occurring in (5) and (6).

  3. 3.

    Recall that the quantifiers \(\forall \) and \(\exists \) are in fuzzy logic interpreted by \(\bigwedge \) and \(\bigvee \), respectively.

  4. 4.

    Many authors speak about terms instead of formulas. We call S, P, M formulas as is common in logic.

References

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Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, NSC IT4Innovations, University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic

    Petra Murinová & Vilém Novák

Authors
  1. Petra Murinová
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  2. Vilém Novák
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Correspondence to Petra Murinová .

Editor information

Editors and Affiliations

  1. University of the Balearic Islands, Palma, Spain

    Sebastia Massanet

  2. University of Oviedo, Gijón, Spain

    Susana Montes

  3. University of the Balearic Islands, Palma, Spain

    Daniel Ruiz-Aguilera

  4. University of the Balearic Islands, Palma, Spain

    Manuel González-Hidalgo

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Murinová, P., Novák, V. (2023). Verifying Validity of Selected Forms of Syllogisms with Intermediate Quantifiers Using Peterson’s Rules. In: Massanet, S., Montes, S., Ruiz-Aguilera, D., González-Hidalgo, M. (eds) Fuzzy Logic and Technology, and Aggregation Operators. EUSFLAT AGOP 2023 2023. Lecture Notes in Computer Science, vol 14069. Springer, Cham. https://doi.org/10.1007/978-3-031-39965-7_30

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  • DOI: https://doi.org/10.1007/978-3-031-39965-7_30

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-39964-0

  • Online ISBN: 978-3-031-39965-7

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