and and A plausible explanation of some instances of belief bias (the effect of the believed truth-value of a conclusion on the judged validity of an argument) is that some subjects fail to screen out some of their beliefs (see Evans, Barston, and Pollard 1983). Consider counterfactual reasoning. When considering what it would be like if I were a professional truck-driver, I must screen out my belief that I am a philosopher living in Kokomo, Indiana. Now, when considering a counterfactual, subjects can screen out those beliefs that (with the antecedent of the counterfactual) imply contradictions. To use Nichols and Stich’s account, I suppose that I am a truck-driver, and then only allow this supposition to interact with those beliefs that do not imply contradictions. Consider counterpossible reasoning. Notice that many conceptual truths are loosened in certain genres of fiction. For example, Bugs Bunny might pick up a hole off the ground and throw it on a wall. It is not metaphysically possible to pick up a hole, but we are able to suppose that Bugs has picked up the hole and recognize that Bugs can now jump through the wall. Thus, we can imagine an impossible state of affairs and make judgments about what would obtain within that state of affairs. In representing the impossible state of affairs, we screen out those beliefs that would lead to outright contradiction. In Bugs’s hole case, we must screen out certain beliefs regarding the nature of holes (i.e. that they cannot be picked up). Counterpossible reasoning does not just occur within fanciful fiction. Similarly, philosophers engage in counterpossible reasoning. Suppose an endurantist who believes perdurantism to be necessarily false receives an undergraduate paper outlining how perdurantism fits with certain views of material constitution. How will the professor proceed? Hopefully this professor will do so by assuming that perdurantism is true and then seeing if the student’s solutions to problems surrounding material constitution are successful on that assumption. Thus, the endurantist philosopher engages in what that professor takes to be counterpossible reasoning.11 I suggest that the professor is able to do so by supposing that perdurantism is true and then screening out those beliefs that would render perdurantism necessarily false (e.g. that presentism is necessarily true). On this account of cognitive architecture, in order to consider an impossible state of affairs, subjects form a single ‘supposition’ and then screen out certain beliefs from that supposition. Some subset of the remaining beliefs, combined with the suppositions, are then cognitively quarantined. Subjects use these cognitively quarantined beliefs and suppositions in inferences resulting in further suppositions. My suggestion is that the representation of the state of affairs is constituted by the cognitively quarantined beliefs and suppositions. Counterfactual states of affairs can be maximal: for any proposition in a counterfactual state of affairs, that proposition is either true or false. Indeed, when we reason about a counterfactual state of affairs, it is best to do so in as maximal a way as we can. This is not so when we represent impossible states of affairs. The representation of the impossible state of affairs must be partial because some propositions (along with the denial of those propositions)