Adding the Everywhere Operator to Propositional Logic

Abstract

Sound and complete modal propositional logic C is presented, in which "P" has the interpretation "P is true in all states". The interpretation is already known as the Carnapian extension of S5. A new axiomatization for C provides two insights. First, introducing an inference rule "textual substitution" allows seamless integration of the propositional and modal parts of the logic, giving a more practical system for writing formal proofs. Second, the two following approaches to axiomatizing a logic are shown to be not equivalent: (i) give axiom schemes that denote an infinite number of axioms and (ii) write a finite number of axioms in terms of propositional variables and introduce a substitution inference rule.