A general treatment of equivalent modalities

The problem of the nonequivalent modalities available in certain systems is a classical problem of modal logic. In this paper we deal with this problem without referring to particular logics, but considering the whole class of normal propositional logics. Given a logic L let P( L ) (the m-partition generated by L ) denote the set of the classes of L -equivalent modalities. Obviously, different logics may generate the same m-partition; the first problem arising from this general point of view is therefore to determine the cardinality of the set of all m-partitions. Since, as is well known, there exist normal logics, and since one immediately realizes that there are infinitely many m-partitions, the problem consists in choosing (assuming the continuum hypothesis) between ℵ 0 and . In Theorem 1.2 we show that there are m-partitions, as many as the logics. The next problem which naturally arises consists in determining, given an m-partition P( L ), the number of logics generating P( L ) (in symbols, μ (P( L ))). In Theorem 2.1(ii) we show that ∣{P( L ): μ (P( L )) = }∣ = . Now, the set { L ′) = P( L )} has a natural minimal element; that is, the logic L * axiomatized by K ∪ { φ ( p ) ↔ ψ ( p ): φ, ψ are L -equivalent modalities}; P( L ) and L * can be, in some sense, identified, thus making the set of m-partitions a subset of the set of logics.