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## On fuzzy modal logics KD45(C)

### Rodríguez, Ricardo

\newtheorem{theorem}{Theorem:} \section{Introduction} The logic KD45($\mathcal{C}$) (where $\mathcal{C}$ stands for a recursively axiomatized fuzzy propositional logic extending the basic logic {\bf BL}) was studied by Hajek in Section 8.3 of his book \cite{Hajek98}. The language $\mathcal{L}_{\square \Diamond }(Var)$ is that of propositional calculus on $Var$ extended by modalities $\Box$ and $\Diamond$. Fixed a {\bf BL}-chain $A$, an $A$-Kripke model has the form $K=\langle W, \pi, e\rangle$ where $W$ is a non-empty set of worlds, $e: W \times Var \mapsto A$ an evaluation which associates a truth-value to each propositional variable in each possible world, and $\pi$ is a normalized non-empty fuzzy subset of $W$, i.e. $\pi : W \mapsto A$ and $\sup_{w \in W} \pi(w) =1$. The evaluation of formulae is extended in the usual way for connectives on $A$ and for $\Box$ and $\Diamond$ is as follows: $$e(v, \Diamond\varphi) = \sup_{w \in W} \{ \pi(w) \ast e(w,\varphi) \} \hspace{.5cm} e(v, \Box \varphi) = \inf_{w \in W} \{\pi(w) \to e(w,\varphi) \}$$ In \cite{Hajek10} the fuzzy modal logic S5($\mathcal{C}$) is introduced. In addition, it is proved that formulae of S5($\mathcal{C}$) are in the obvious one-one isomorphic correspondence with formulae of the monadic fuzzy predicate calculus $m\mathcal{C}\forall$ with unary predicates and just one object variable $x$, the atomic formula $P(x)$ corresponding to propositional variable $p$ and modalities corresponding to quantifiers. The correspondence maps tautologies of the modal logic to tautologies of the monadic predicate logic and the same for standard tautologies. By using this translation, Hajek is able to give an axiomatization for S5($\mathcal{C}$) and is also able to obtain several results on arithmetical complexity concerning these logics. The possibilistic semantics, mentioned at the begining, for KD45($\mathcal{C}$) suggests a translation from this logic into S5($\mathcal{C}$). Given a fixed propositional variable $c$, it defines inductively a map $\varphi \longmapsto \varphi ^{\ast }$ from $\mathcal{L}_{\square \Diamond }(Var)$ into $\mathcal{L}_{\square \Diamond}(Var\cup \{c\})$ as follows: \medskip $\varphi ^{\ast }=\varphi$ for $\varphi \in Var\cup \{\top ,\bot \}$ $(\varphi \circledast \psi )^{\ast }:=\varphi ^{\ast }\circledast \psi ^{\ast }$ for $\circledast \in \{\wedge ,\vee ,\rightarrow \}.$ $(\Box \varphi )^{\ast }:=\Box (c\rightarrow \varphi ^{\ast })$ $(\Diamond \varphi )^{\ast }:=\Diamond (c\wedge \varphi ^{\ast }).$ \medskip \noindent This is a faithful translation. That is, it preserves and reflects provability (or validity). \begin{theorem} \label{faithful}Let $c$ be a fixed propositional variable not occurring in $\varphi \in \mathcal{L}_{\square\Diamond }(Var)$ then \begin{equation*} \models _{KD45(\mathcal{C})}\varphi \text{ \ \ if and only if \ \ }\Diamond c\models _{S5(\mathcal{C})}\varphi ^{\ast }. \end{equation*} \end{theorem} Since the translation\ in Theorem \ref{faithful} is obviously polynomial, we are able to inherit some results on complexity from S5($\mathcal{C}$). In addition, according to this translation, we are also able to give an indirect axiomatization for KD45($\mathcal{C}$). We are going to focus our presentation in showing all these results. \begin{thebibliography}{99} \bibitem{Hajek98} {P. Hájek.} \textit{Metamathematics of Fuzzy Logic}. Trends in Logic, 4, Kluwer, 1998. \bibitem{Hajek10} P. Hájek, On fuzzy modal logics S5(C). Fuzzy Sets and Systems 161 (2010) 2389-2396. \end{thebibliography}

Autores: Rodríguez, Ricardo.