Abstract
This paper is a study of fuzzy type theory (FTT) with partial functions. Out of several possibilities we decided to introduce a special value “\(*\)” which represents “undefined”. In the interpretation of FTT, this value lays outside of the corresponding domain. In the syntax, it is naturally represented by the description operator acting on the empty (fuzzy) set which, of course, has no element and so, choosing an element from its kernel gives no result, i.e., it is undefined. We will demonstrate that our approach leads to reasonable characterization of the undefinedness. We will also show that any consistent theory of FTT has a model.
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Notes
- 1.
Note that this “\(*\)” is a different element from “\(*\)” introduced for truth values.
- 2.
Recall that the description operator represents, in fact, the defuzzification operation (cf. [14, Chapt. 3]).
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Acknowledgment
This paper was supported by the grant 16-19170S of GAČR, Czech Republic.
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Novák, V. (2018). Towards Fuzzy Type Theory with Partial Functions. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 643. Springer, Cham. https://doi.org/10.1007/978-3-319-66827-7_3
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DOI: https://doi.org/10.1007/978-3-319-66827-7_3
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