 In general, an abstract consequence relation on a set \(A\) (not necessarily the set of formulas of some formal language) is a relation \(\vdash\) between subsets of \(A\) and elements of \(A\) that satisfies conditions (1)–(3) above. If it also satisfies (4) it is said to be finitary. If \(\vdash\) is an abstract consequence relation and \(X \vdash a\), then we can say that \(X\) is a set of premises or hypothesis with conclusion \(a\) according to \(\vdash\) and that \(a\) follows from \(X\), or is entailed by \(X\) (according to \(\vdash)\). The abstract consequence relations correspond to Koslow’s implication structures; see Koslow 1992 for the closely related but different approach to logics (in a broad sense) as consequence relations introduced by that author.