The dominant view in contemporary philosophy is that possible worlds are classically closed – closed under classical logic. (If A is true at a possible world, and B is a consequence of A according to classical logic, then B is true at the possible world too.) Another dominant view is that logical consequence is per the classical-logic account – that logical consequence is classical (-logic) consequence. In this paper, I challenge neither of the given dominant views, and indeed take them as true. There is a third – possibly even more dominant – view in contemporary philosophy: namely, that the idea of so-called paraconsistent consequence relations (paraconsistent entailment relations) is a highly radical one, demanding a vast shift in the dominant views of both logic and possible worlds. Such a view is the default one in contemporary philosophy, if not in print

(where standards of professional politeness still apply) then at least in conversation. Talk of paraconsistent consequence relations carries the burden of motivation: if we’re going to talk about them, we had better have very strong philosophical motivation to do so.

My principal aim in this paper is to refute the third of the canvassed mainstream views. The refutation is both simple and straightforward: I show that the standard conception of possible worlds (as closed under logic, where logic is classical logic) induces a paraconsistent consequence relation on our language. Moreover, the given relation is not just any old paraconsistent consequence (entailment) relation; it’s a very natural and very familiar paraconsistent consequence relation. The upshot is that either the dominant

view of paraconsistent consequence relations carries unnecessary baggage or

it is simply mistaken. Either way, the given view should be rejected.