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Substitution Rules with Respect to a Context

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AETA 2018 - Recent Advances in Electrical Engineering and Related Sciences: Theory and Application (AETA 2018)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 554))

Abstract

In this paper, we deal with Leibniz’s rule of substitution of identicals, and describe how the rule can be applied in the TIL-Script language. The main goal is to introduce the algorithm of valid application of the substitution rules in all the three kinds of context that we distinguish in the TIL-Script language. The language is a computational variant of TIL, which is a hyperintensional, partial typed -calculus. Hyperintensional, because the meaning of TIL -terms are procedures producing functions rather than the denoted functions themselves. Partial, because TIL is a logic of partial functions, and typed, because all the entities of TIL ontology receive a type. Based on the results of context recognition the algorithm makes it possible to validly apply the substitution rules and derive relevant new pieces of analytic information.

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Notes

  1. 1.

    See [2, 15].

  2. 2.

    Recent years have seen a significant rise of interest in hyperintensional concepts, see, for instance [13]. For a summary of -calculi, see for instance [1] or [14]. These calculi are characterized as hyperintensional, because individuation of functions is not reduced to set-theoretical mappings. Rather, they operate with Church’s functions-in-intensions individuated more finely than mappings. However, these functions-in-intensions (or rules for producing mappings) cannot be displayed as objects on which other procedures operate. Thus, TIL indeed introduces another higher level of abstraction which is the hyperintensional level of displayed procedures.

  3. 3.

    For details and definition of procedural isomorphism, see [6].

  4. 4.

    For details, see [2, Sect. 2.6].

  5. 5.

    Faroldi in [11] makes a similar point. See also [5].

  6. 6.

    For more details on this tool and the algorithm of context recognition see [7].

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Acknowledgments

This research was supported by the Grant Agency of the Czech Republic, project No. GA18-23891S “Hyperintensional Reasoning over Natural Language Texts”, by the internal grant agency of VSB-Technical University of Ostrava, project No. SP2018/172, “Application of Formal Methods in Knowledge Modelling and Software Engineering”, and also by the Moravian-Silesian region within the program “Support of science and research in Moravian-Silesian region 2017” (RRC/10/2017).

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Authors and Affiliations

  1. Department of Computer Science FEI, VŠB-Technical University of Ostrava, 17. listopadu 15, 708 33, Ostrava, Czech Republic

    Michal Fait & Marie Duží

Authors
  1. Michal Fait
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  2. Marie Duží
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Correspondence to Michal Fait .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Faculty of Electrical Engineering, VŠB-TUO, Ostrava-Poruba, Czech Republic

    Ivan Zelinka

  2. Department of Electronics, Faculty of Electrical Engineering, VŠB-TUO, Ostrava, Czech Republic

    Pavel Brandstetter

  3. International Cooperation, Research and Training Institute, Ton Duc Thang University, Ho Chi Minh, Vietnam

    Tran Trong Dao

  4. Department of Automatic Control, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Vo Hoang Duy

  5. Department of Mechanical Design Engineering, Pukyong National University, Busan, Korea (Republic of)

    Sang Bong Kim

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Fait, M., Duží, M. (2020). Substitution Rules with Respect to a Context. In: Zelinka, I., Brandstetter, P., Trong Dao, T., Hoang Duy, V., Kim, S. (eds) AETA 2018 - Recent Advances in Electrical Engineering and Related Sciences: Theory and Application. AETA 2018. Lecture Notes in Electrical Engineering, vol 554. Springer, Cham. https://doi.org/10.1007/978-3-030-14907-9_6

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  • DOI: https://doi.org/10.1007/978-3-030-14907-9_6

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