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Publisher Summary This chapter presents axiomatics for B (the Brouwerian system), S4, S5, J(B), and J(S5), specially appropriate to reflect their mutual inclusion and non-inclusion relations, and focuses on the axiomatics in order to obtain a natural one for the discussive logic associated with the predicate logic S5 (with the Barcan formulas). The propositional logic J(S5) is not closed under all rules of inference that are universally valid in the classical propositional logic, not even under all those that are valid in the positive propositional logic. The logics are to be thought as formulated in a language whose primitive symbols are denumerably many propositional letters, the connectives ┐ (negation) and V (disjunction), the necessity operator L, and parentheses. The letters A, C, and D , with or without numerical subscripts is employed as syntactical variables for formulas. The chapter also discusses axioms schemas and rules, the axiom system J*, theorems, and lemmas. |
