In this paper, we show (among other things) that the conditional in Frege's Begriffsschrift is ambiguous.

It is well-known that adding to a lattice the usual relative meet complement is not conservative, in the sense that distributivity is implied. In this paper we consider a weak relative meet complement that does not have the mentioned effect. We mostly study the mentioned operation from an algebraic point of view. However, we also provide a Hilbert-style axiomatization for its corresponding assertional logic.