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2015 | OriginalPaper | Chapter

11. On Possibilistic Modal Logics Defined Over MTL-Chains

Authors : Félix Bou, Francesc Esteva, Lluís Godo

Published in: Petr Hájek on Mathematical Fuzzy Logic

Publisher: Springer International Publishing

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Abstract

In this chapter we revisit a 1994 chapter by Hájek et al. where a modal logic over a finitely-valued Łukasiewicz logic is defined to capture possibilistic reasoning. In this chapter we go further in two aspects: first, we generalize the approach in the sense of considering modal logics over an arbitrary finite MTL-chain, and second, we consider a different possibilistic semantics for the necessity and possibility modal operators. The main result is a completeness proof that exploits similar techniques to the ones involved in Hájek et al.’s previous work.

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previous chapter Semi-linear Varieties of Lattice-Ordered Algebras
next chapter The Quest for the Basic Fuzzy Logic
Footnotes
1
Actually, Hájek et al. (1994) was for F. Esteva and L. Godo the first joint paper with P. Hájek.
 
2
In the sense of Possibility Theory (Dubois and Prade 1988).
 
3
In fact, as it is explained in Sect. 11.2, two-valued possibility and necessity measures over classical propositions can be taken as an alternative semantics for the modal operators in the system KD45.
 
4
Other connectives are defined as usual in MTL, for instance \(\lnot \varphi \) is \(\varphi \rightarrow \overline{0}\), \(\varphi \vee \psi \) is \(((\varphi \rightarrow \psi ) \rightarrow \psi ) \wedge ((\psi \rightarrow \varphi ) \rightarrow \varphi )\), and \(\varphi \leftrightarrow \psi \) is \((\varphi \rightarrow \psi ) \wedge (\psi \rightarrow \varphi )\).
 
5
We use the notation \(e(w, \cdot )\) to denote the function \(p \in Var \longmapsto e(w,p) \in A\).
 
6
Here we use the fact the equation \(x \Rightarrow (y \Rightarrow z) = (x \odot y) \Rightarrow z\) holds in every MTL-chain.
 
7
Here we use the fact the equation \((x_1 \Rightarrow y) \wedge (x_2 \Rightarrow y) = (x_1 \vee x_2) \Rightarrow y\) holds in every MTL-chain.
 
8
Notice that these axioms could also be expressed as the following B-formulas:
$$ \begin{aligned} (r \odot s)(\overline{r} \& \overline{s}), \; (r \Rightarrow s)(\overline{r} \rightarrow \overline{s}), \; (\min (r,s))(\overline{r} \wedge \overline{s}), \; (\triangle r) \triangle \overline{r} \; \text {and} \; \bigvee \nolimits _{r \in A} (r) \varphi . \end{aligned}$$
However, the adopted formulation makes less use of the \(\triangle \) connective.
 
9
Recall that a complete theory is prime in the classical sense for B-formulas.
 
10
This definition is sound due to (c) of Lemma 11.3.
 
Literature
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Metadata
Title
On Possibilistic Modal Logics Defined Over MTL-Chains
Authors
Félix Bou
Francesc Esteva
Lluís Godo
Copyright Year
2015
Book
Petr Hájek on Mathematical Fuzzy Logic

Print ISBN: 978-3-319-06232-7

Electronic ISBN: 978-3-319-06233-4

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-06233-4_11

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