Capturing the notion of pertinence or relevance in logic is usually attempted at the meta-level. It can be induced either by specific extra information, or by general philosophical principles. In this work we pay attention to both these origins. We present a semantic modal interpretation of the idea that there are two distinct relationships between a premise and a conclusion that are pertinent to each other: of semantic entailment in the forward direction from A to B, and of semantic constraint in the backward direction from B to A. Unpacking the notion of pertinence into these two semantic components yields a class of entailment relations with appealing properties.

We define an entailment relation via a modal logic, and investigate its behaviour as a viable candidate for capturing the notion of pertinence. This approach allows us to deal with a number of paradoxes of material and strict implication (e.g. positive paradox), as well as some counter-intuitive properties of classical (and modal) entailment (e.g. explosiveness and disjunctive syllogism), in a satisfactory way. Furthermore, the resulting logic is infra-modal, non-monotonic, and allows for non-trivial reasoning with inconsistencies.