Interrelation of Algebraic, Semantical and Logical Properties for Superintuitionistic and Modal Logics

We consider the families L of propositional superintuitionistic logics (s.i.l.) andNE(K) of normal modal logics (n.m.l.). It is well known that there is a duality between L and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also NE(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig’s interpolation property (CIP), the disjunction property (DP), the Beth property (BP), Hallden-completeness (HP) etc. have suitable properties of varieties as their images, and many natural algebraic properties are in accordance with natural properties of logics. For example, a s.i.l. L has CIP iff its associated variety V (L) has the amalgamation property (AP); L is Hallden-complete iff V (L) is generated by a subdirectly irreducible Heyting algebra. For any n.m.l. L, the amalgamation property of V (L) is equivalent to a weaker version of the interpolation property for L, and the superamalgamation property is equivalent to CIP; L is Hallden-complete iff V (L) satisfies a strong version of the joint embedding property. Well-known relational Kripke semantics for the intuitionistic and modal logics seems to be a more natural interpretation than the algebraic one. The categories of Kripke frames may, in a sense, be considered as subcategories of varieties of Heyting or modal algebras. We discuss the question to what extent one may reduce problems on properties of logics to consideration of their semantic models.