From Symbolic Logic by Clarence Irving Lewis and Cooper Harold Langford, 1932/1959. Primitive connectives, ~, , and . Disjunction is defined as usual, and strict implication is defined by (P Q) abbreviates ~(P~ Q). (The notation here is not the same as the original.) Also define strict equivalence by P = Q abbreviates (P Q) (Q P).
This much gives system S1. Adding the following gives S2. (P Q) P
S4 is the set 1-7, together with P P It includes S2.
Material implication can be introduced by setting P Q to mean ~ (P~ Q). It will have its classical properties, but is different than strict implication.