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Higher Degree Fuzzy Transform: Application to Stationary Processes and Noise Reduction Read chapter Read first chapter

2018 | OriginalPaper | Chapter

On Finite-Valued Bimodal Logics with an Application to Reasoning About Preferences

Authors : Amanda Vidal, Francesc Esteva, Lluis Godo

Published in: Advances in Fuzzy Logic and Technology 2017

Publisher: Springer International Publishing

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Abstract

In a previous paper by Bou et al., the minimal modal logic over a finite residuated lattice with a necessity operator \(\square \) was characterized under different semantics. In the general context of a residuated lattice, the residual negation \(\lnot \) is not necessarily involutive, and hence a corresponding possibility operator cannot be introduced by duality. In the first part of this paper we address the problem of extending such a minimal modal logic with a suitable possibility operator \(\Diamond \). In the second part of the paper, we introduce suitable axiomatic extensions of the resulting bimodal logic and define a logic to reason about fuzzy preferences, generalising to the many-valued case a basic preference modal logic considered by van Benthem et al.

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previous chapter Optimal Parameter Ranges in Fuzzy Inference Systems, Applied to Spatial Data
next chapter Improving Supervised Classification Algorithms by a Bipolar Knowledge Representation
Footnotes
1
For the sake of clarity, we use the same symbol (e.g. \(\odot , \rightarrow \)) both as syntactic connective in the language \(\mathbf {MFm}\) and as the corresponding algebraic operation.
 
2
This logical consequence is usually referred to as the local modal logic arising from a class of Kripke models, in contrast with the global one that considers truth in the whole model. It is out of the scope of this work to introduce and study the global modal logic over residuated lattices.
 
3
We do not detail this issue here due to lack of space and interest, but for the interested reader, it should be clear that the language from the right side of this equivalence counts with an extended -countable- set of variables that capture the modal formulas.
 
4
Note that for a finite set of formulas \(\varTheta \), \(\bigwedge \limits _{\theta \in \varTheta }\theta \) is a formula in the language too.
 
5
For the interested reader, see eg. [5] for an insight on the importance of this kind of algebras.
 
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van Benthem, J., Girard, P., Roy, O.: Everything else being equal: A modal logic for ceteris paribus preferences. J. Philos. Logic 38, 83–125 (2009)MathSciNetCrossRefMATH
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Bou, F., Cerami, M., Esteva, F.: Finite-valued Łukasiewicz modal logic is PSPACE-complete. In: Proceedings of IJCAI 2011, pp. 774–779 (2011)
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Bou, F., Esteva, F., Godo, L., Rodríguez, R.: On the minimum many-valued modal logic over a finite residuated lattice. J. Logic Comput. 21(5), 739–790 (2011)MathSciNetCrossRefMATH
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Marti, M., Metcalfe, G.: A hennessy-milner property for many-valued modal logics. In: Goré, R., Kooi, B.P., Kurucz, A. (eds.) Advances in Modal Logic, Vol. 10, pp. 407–420. College Publications (2014)
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Vidal, A.: On modal expansions of t-norm based logics with rational constants. Ph.D. Dissertation, University of Barcelona (2015)
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Metadata
Title
On Finite-Valued Bimodal Logics with an Application to Reasoning About Preferences
Authors
Amanda Vidal
Francesc Esteva
Lluis Godo
Copyright Year
2018
Book
Advances in Fuzzy Logic and Technology 2017

Print ISBN: 978-3-319-66826-0

Electronic ISBN: 978-3-319-66827-7

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-66827-7_47

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