On some substitution instances of ${\bf R1}$ and ${\bf L1}$.
Wolfgang Lenzen · Notre Dame Journal of Formal Logic · 1978
A study of the epistemic correlates of the modal systems between S4 and S5, [4], has drawn my interest to certain modifications of the "factoring" axioms (c/.[ll]) 1which characterize S4.4 and S4.04, respectively.The following substitution instances turned out to be particularly interesting:In this note I want to investigate the results of adding these formulae as new axioms to the base of S4 (with a primitive rule of Necessitation).It will be shown that (i) S4 + R1.1 is deductively equivalent to S4.2; (ii) S4 + R1.2 is deductively equivalent to S4.3.2;(iii) S4 + R1.3 is a new system properly between S4.4 and S4.1.2,or else R1.3 is a new proper axiom of S4.1.2.(iv) S4 + L1.2 is a new system properly between S4.04 and S4 and properly between S4.3.2 and S4; (v) S4 + L1.3 is a new system properly between S4.04 and S4.02, or else L1.3 is a new proper axiom of S4.04.(a) It is well known (cf.[l], p. 252) that in the field of S4 the proper axiom of S4.2, 1. 1 assume the reader is familiar with the literature cited in this note, especially with [5] and [6].