Modal $\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\tilde{\mathrm{l}}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{C}$ Logics and Predicate Superintuitionistic Logics: $\mathrm{C}_{0\Gamma \mathrm{r}\mathrm{e}..\mathrm{S}},\mathrm{p}$

In this note we deal with intuitionistic modal logics over $\mathcal{M}\mathcal{I}PC$ and predicate superintuitionistic logics. We study the correspondence between the lattice of all (normal) extensions of MTPC and the lattice of all predicate superintuitionistic logics. Let $\mathrm{L}_{Prop}$ denote a propositional language which contains two modal operators $\square$ and $\mathrm{O}$ , and $\mathrm{L}_{Pred}-\mathrm{a}$ first-order language. Formulas of $\mathrm{L}_{Prop}$ and $\mathrm{L}_{Pred}$ are built in a usual way. Let us denote the sets of all formulas of $\mathrm{L}_{Prop}$ and $\mathrm{L}_{Pred}$ by $\mathrm{F}\mathrm{o}\mathrm{R}\mathrm{M}(\mathrm{L}_{Prop})$ and $\mathrm{F}\mathrm{o}\mathrm{R}\mathrm{M}(\mathrm{L}_{p_{r\mathrm{e}d}})$ respectively. $\mathcal{M}\mathcal{I}PC$ (which was first introduced by A. Prior and R. Bull) is the least subset of $\mathrm{F}\mathrm{o}\mathrm{R}\mathrm{M}(\mathrm{L}_{P\mathrm{p}})r\circ$ which contains the propositional intuitionistic logic $\mathcal{I}N$, the formulas (1) $\square parrow p$ $parrow \mathrm{O}p$ (2) $\square (parrow q)arrow(\square parrow\square q)$ $\mathrm{O}(p\vee q)arrow(\mathrm{O}p\mathrm{O}q)$ (3) $\mathrm{O}parrow\square \mathrm{O}p$ $\mathrm{O}\square parrow\square p$ (4) $\square (parrow q)arrow(\mathrm{O}parrow \mathrm{O}q)$ and is closed under substitution, modus ponens and necessitation. A subset of $\mathrm{F}\mathrm{o}\mathrm{R}\mathrm{M}(\mathrm{L})Pr\circ \mathrm{p}$ which contains $\mathcal{A}\Lambda \mathcal{I}Pc$ and is closed with respect to those rules of inference is called an intuitionistic modal logic over $\mathcal{M}\mathcal{I}PC$ . Let us denote the lattice of all intuitionistic modal logics over $\mathcal{M}\mathcal{I}PC$ by $\Lambda(\mathcal{M}\mathcal{I}PC)$ . For any $\mathcal{L}\in\Lambda(\mathcal{M}\mathcal{I}PC)$ and $\Gamma\subseteq \mathrm{F}\mathrm{o}\mathrm{R}\mathrm{M}(\mathrm{L})Pr\circ p$ let $\mathcal{L}\oplus\Gamma$ denote the least logic in $\Lambda(\mathcal{M}\mathcal{I}PC)$ which contains both $\mathcal{L}$ and $\Gamma$ . We will denote by $Q-\mathcal{I}N$ the standard predicate intuitionistic logic. A subset of FoRM $(\mathrm{L})Pred$ which contains $Q-\mathcal{I}N$ and is closed with respect to the first-order rules of inference will be called a predicate superintuitionistic logic. Let us denote the lattice of all predicate superintuitionistic logics by $\Lambda(Q-\mathcal{I}N)$ . For any $S\in\Lambda(Q-\mathcal{I}N)$ and 数理解析研究所講究録 1010巻 1997年 1-6 1