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.
Dedicated to Petr Hájek
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Notes
- 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.
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Acknowledgments
The authors are deeply indebted to an anonymous reviewer for the useful comments that have helped to significantly improve the layout of this chapter. The authors also acknowledge partial support of the Spanish projects EdeTRI (TIN2012-39348-C02-01) and Agreement Technologies (CONSOLIDER CSD2007-0022, INGENIO 2010), and the grant 2009SGR-1433 from the Catalan Government. Initially they were also supported by the projects TASSAT (TIN2010-20967-C04-01) and 2009SGR-1434.
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Bou, F., Esteva, F., Godo, L. (2015). On Possibilistic Modal Logics Defined Over MTL-Chains. In: Montagna, F. (eds) Petr Hájek on Mathematical Fuzzy Logic. Outstanding Contributions to Logic, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-06233-4_11
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