On the extensions of $S5$.
Peter Gärdenfors · Notre Dame Journal of Formal Logic · 1973
1 The purpose of this paper is to investigate the extensions of the Lewis system S5. To some extent this is meant to be a complement to what is shown in Scroggs [7]. We will show that any formula containing only one variable, if added to S5, will give an inconsistency or make the system collapse into classical propositional calculus (PC). We then examine the proper extensions of S5 obtained by adding formulas containing more than one variable. We describe Kripke-type semantics for these systems and prove their completeness. 2 A normal extension of S5 is an extension which is closed under the rules of substitution and (material) detachment. A proper extension of S5 is a normal extension where some formula not valid in S5 is derivable, but the formula p — • Lp is not derivable. Theorem 1: If any wff, containing only one propositional variable, is added as a neiυ axiom to S5, then the system thus obtained is not a proper extension of S5.