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Logics for Order-of-Magnitude Qualitative Reasoning: Formalizing Negligibility

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Ewa Orłowska on Relational Methods in Logic and Computer Science

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 17))

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Abstract

Qualitative reasoning deals with information expressed in terms of qualitative classes and relations among them, such as comparability, negligibility or closeness. In this work, we focus on the different logic-based approaches to the notions of negligibility developed by our group.

Partially supported by the Spanish Ministry of Economy and Competitiveness and the European Fund for Regional Development through project TIN15-70266-C2-P-1.

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Notes

  1. 1.

    Notice that this is a generalization of the intended structure used to define the qualitative classes.

  2. 2.

    \({\overrightarrow{d_\alpha }}^n\) is defined by \({\overrightarrow{d_\alpha }}^1={\overrightarrow{d_\alpha }}\) and \({\overrightarrow{d_\alpha }}^n={\overrightarrow{d_\alpha }}\circ {\overrightarrow{d_\alpha }}^{n-1}\), for \(n\in \mathbb N, n\ge 2\), being \(\circ \) the usual composition of relations.

  3. 3.

    This is the only axiom which is affected by our previous assumption that \(n=1\).

  4. 4.

    We could use any strict linearly ordered set with two internal operations \(+\) and \(\cdot \)

  5. 5.

    This is a well-known effect in marketing.

  6. 6.

    There are at least as many elements in \({\mathscr {C}}\) as qualitative classes.

  7. 7.

    Notice the use of \(\mathbb {OS}\) to denote the union of \(\mathbb {OV}\) and \(\mathbb {OC}\).

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Acknowledgements

Some excerpts of the following works have been reprinted with permission:

\(\bullet \) A. Burrieza et al. Order of Magnitude Qualitative Reasoning with Bidirectional Negligibility. Current Topics in Artificial Intelligence, Lecture Notes in Computer Science, Vol. 4177, pp. 370–378, Springer (2006). With permission of Springer.

\(\bullet \) A. Burrieza et al. A Logic for Order of Magnitude Reasoning with Negligibility, Non-closeness and Distance. Current Topics in Artificial Intelligence, Lecture Notes in Computer Science, Vol. 4788, pp. 210–219, Springer (2007). With permission of Springer.

\(\bullet \) A. Burrieza et al. A Propositional Dynamic Logic Approach for Order of Magnitude Reasoning. Advances in Artificial Intelligence—IBERAMIA 2008, Lecture Notes in Computer Science, Vol. 5290, pp. 11–20, Springer (2008). With permission of Springer.

\(\bullet \) A. Burrieza, M. Ojeda-Aciego. A Multimodal Logic Approach to Order of Magnitude Qualitative Reasoning with Comparability and Negligibility Relations. Fundamenta Informaticae 68(1-2), 21–46, IOS Press (2005). With permission of IOS Press.

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Authors and Affiliations

  1. Dept Filosofía, Universidad de Málaga, 29071, Málaga, Spain

    Alfredo Burrieza

  2. Dept Matemática Aplicada, Universidad de Málaga, 29071, Málaga, Spain

    Emilio Muñoz-Velasco & Manuel Ojeda-Aciego

Authors
  1. Alfredo Burrieza
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  2. Emilio Muñoz-Velasco
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  3. Manuel Ojeda-Aciego
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Correspondence to Manuel Ojeda-Aciego .

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Editors and Affiliations

  1. Institute of Philosophy, University of Warsaw, Warsaw, Poland

    Joanna Golińska-Pilarek

  2. Department of Logic and Methodology of Science, University of Lodz, Łódź, Poland

    Michał Zawidzki

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Burrieza, A., Muñoz-Velasco, E., Ojeda-Aciego, M. (2018). Logics for Order-of-Magnitude Qualitative Reasoning: Formalizing Negligibility. In: Golińska-Pilarek, J., Zawidzki, M. (eds) Ewa Orłowska on Relational Methods in Logic and Computer Science. Outstanding Contributions to Logic, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97879-6_8

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