We show which implicational relations hold between the three principles ex falso sequitur quodlibet, tertium non datur, and double negation elimination, on the basis of minimal logic.

We show that provability in the implicational fragment of relevance logic is complete for doubly exponential time, using reductions to and from coverability in branching vector addition systems.

We present an alternative, purely semantical and relatively simple, proof of the Statman’s result that both intuitionistic propositional logic and its implicational fragment are PSPACE -complete.

Implicational bases are objects of interest in formal concept analysis and its applications. Unfortunately, even the smallest base, the Duquenne-Guigues base, has an exponential size in the worst case. In this paper, we use results on the average number of minimal transversals in random hypergraphs to show that the base of proper premises is, on average, of quasi-polynomial size.