Modal Consequence Relations

Logic is generally defined as the science of reasoning. Mathematical logic is mainly concerned with forms of reasoning that lead from true premises to true conclusions. Thus we say that the argument from σ0;σ1; · · · ;σn−1 to δ is logically correct if whenever σi is true for all i < n, then so is δ. In place of ‘argument’ one also speaks of a ‘rule’ or an ‘inference’ and says that the rule is valid. This approach culminated in the notion of a consequence relation, which is a relation between sets of formulae and a single formula. A consequence relation ` specifies which arguments are valid; the argument from a set Σ to a formula δ is valid in ` iff 〈Σ, δ〉 ∈ `, for which we write Σ ` δ. δ is a tautology of ` if ∅ ` δ, for which we also write ` δ. In the early years, research into modal logic was concerned with the question of finding the correct inference rules. This research line is still there but has been marginalized by the research into modal logics, where a logic is just a set of formulae; this set is the set of tautologies of a certain consequence relation, but many consequence relations share the same tautologies. The shift of focus in the research has to do in part with the precedent set by predicate logic: predicate logic is standardly axiomatized in a Hilbert-style fashion, which fixes the inference rules and leaves only the axioms as a parameter. Another source may have been the fact that there is a biunique correspondence between varieties of modal algebras and axiomatic extensions of K, which allowed for rather deep investigations into the space of logics, using the machinery of equational theories. This research lead to deep results on the structure of the lattice of modal logics and benefits also the research into consequence relations. Recently, however, algebraic logic has provided more and more tools that allow to extend the algebraic method to the study of consequence relations in general (see for example [59] and [14]). In particular the investigations into the Leibniz operator initiated by Blok and Pigozzi in [5] have brought new life into the discussion and allow to see a much broader picture than before. Now, even if one is comfortable with classical logic, it is not immediately clear what the correct inferences are in modal logic. The first problem is that it is not generally agreed what the meaning of the modal operator(s) is or should be. In fact, rather than a drawback, the availability of very many different interpretations is the strength of modal logic; it gives flexibility, however at the price that there is not one modal logic, there are uncountably many. For example, 2 as metaphysical necessity satisfies S5, 2 as provability in PA satisfies G, 2 as future necessity (arguably) satisfies S4.3, and so on. This is in part because the interpretation decides which algebras are suitable (intended) and which ones are not. However, there is another parameter of variation, and this is the notion of truth itself. In the most popular interpretation, truth is truth at a world; but we could also understand it