ABSTRACT
A propositional language is characterized by the fact that aside from the category of sentences it only has categories of sentential operators (sometimes also called connectives). Each category of sentential operators is distinguished by an “arity” n, and there is always a syntactic rule mapping an n-ary operator plus n sentences on a sentence. From the calculus of intuitionistic propositional logic, which we already defined, we can get, by a slight tampering with the inferential rules for negation, the calculus of classical propositional logic. These calculi then can be modified and extended in various ways in the direction of relevant, modal, and other logics; and also to substructural logics. A related philosophical question is what might be the difference between modification and extension (when we modify an operator, do we not have a new one, no longer related to the original one?). Another question is how free are we in producing new operators in terms of inferential rules.
