 Only a few of the many extensions of the Lewis systems that have been discussed in the literature are mentioned here. Alban (1943) introduces system S6 by adding to S2 the axiom \(\mvdash \Diamond \Diamond p\). Halldén (1950) calls S7 the system that adds the axiom \(\mvdash \Diamond \Diamond p\) to S3, and S8 the system that extends S3 with the addition of the axiom \(\mvdash \neg \Diamond \neg \Diamond \Diamond p\). While the addition of an axiom of universal possibility \(\mvdash \Diamond p\) would be inconsistent with all the Lewis systems, since they all contain theorems of the form \(\mvdash \Box p\), systems S6, S7 and S8 are consistent. Instead, the addition of either of these axioms to S4, and so also to S5, results in an inconsistent system, given that in S4 \(\mvdash \Diamond \Diamond p\Rightarrow \Diamond p\). Halldén also proved that a formula is a theorem of S3 if and only if it is a theorem of both S4 and S7 (1950: 231–232), thus S4 and S7 are two alternative extensions of S3.