  We study a natural variant of the implicational fragment of propositional
logic. Its formulas are pairs of conjunctions of positive literals, related
together by an implicational-like connective; the semantics of this sort of
implication is defined in terms of a threshold on a conditional probability of
the consequent, given the antecedent: we are dealing with what the data
analysis community calls confidence of partial implications or association
rules. Existing studies of redundancy among these partial implications have
characterized so far only entailment from one premise and entailment from two
premises, both in the stand-alone case and in the case of presence of
additional classical implications (this is what we call "relative entailment").
By exploiting a previously noted alternative view of the entailment in terms of
linear programming duality, we characterize exactly the cases of entailment
from arbitrary numbers of premises, again both in the stand-alone case and in
the case of presence of additional classical implications. As a result, we
obtain decision algorithms of better complexity; additionally, for each
potential case of entailment, we identify a critical confidence threshold and
show that it is, actually, intrinsic to each set of premises and antecedent of
the conclusion.
