Real human agents, even when they are rational by everyday standards, sometimes assign different credences to objectively equivalent statements, such as ‘Orwell is a writer’ and ‘E.A. Blair is a writer’, or credences less than 1 to necessarily true statements, such as not-yet-proven theorems of arithmetic. Anna Mahtani calls this the phenomenon of ‘opacity’. Opaque credences seem probabilistically incoherent, which goes against a key modelling assumption of probability theory. I sketch a modelling strategy for capturing opaque credence assignments without abandoning probabilistic coherence. I draw on ideas from judgement-aggregation theory, where we face similar challenges of defining the ‘objects of judgement’.

In this paper, I develop an algorithmic impossible-worlds model of belief and knowledge that provides a middle ground between models that entail that everyone is logically omniscient and those that are compatible with even the most egregious kinds of logical incompetence. In outline, the model entails that an agent believes (knows) $\phi $ just in case she can easily (and correctly) compute that $\phi $ is true and thus has the capacity to make her actions depend on whether $\phi $. The model thereby captures the standard view that belief and knowledge ground are constitutively connected to dispositions to act. As I explain, the model improves upon standard algorithmic models developed by Parikh, Halpern, Moses, Vardi, and Duc, among other ways, by integrating them into an impossible-worlds framework. The model also avoids some important disadvantages of recent candidate middle-ground models based on dynamic epistemic logic or step logic, and it can subsume their most important advantages.

We present four classical theories of counterpossibles that combine modalities and counterfactuals. Two theories are anti-vacuist and forbid vacuously true counterfactuals, two are quasi-vacuist and allow counterfactuals to be vacuously true when their antecedent is not only impossible, but also inconceivable. The theories vary on how they restrict the interaction of modalities and counterfactuals. We provide a logical cartography with precise acceptable boundaries, illustrating to what extent nonvacuism about counterpossibles can be reconciled with classical logic.

Jonathan Schaffer argues against a necessary connection between properties and laws. He takes this to be a question of what possible worlds we ought to countenance in our best theories of modality, counterfactuals, etc. In doing so, he unfairly rigs the game in favor of contingentism. I argue that the necessitarian can resist Schaffer’s conclusion while accepting his key premise that our best theories of modality, counterfactuals, etc. require a very wide range of things called ‘possible worlds’. However, the necessitarian can and should insist that, in many cases, these worlds are not metaphysically possible. I will further argue that, having taken such a stance, the necessitarian has additional resources to respond to Schaffer’s other arguments against the view.