Does the formation of delusions involve abnormal reasoning? According to the prominent ‘two-factor’ theory of delusions (e.g. Coltheart, 2007), the answer is yes. The second factor in this theory is supposed to affect a deluded individual's ability to evaluate candidates for belief. However, most published accounts of the two-factor theory have not said much about the nature of this second factor. In an effort to remedy this shortcoming, Coltheart, Menzies and Sutton (2010) recently put forward a Bayesian account of inference in delusions. I outline some criticisms of this important account, and sketch an alternative account of delusional inference that, I argue, avoids these criticisms. Specifically, I argue that the second factor in delusion formation involves a systematic deviation from Bayesian updating, a deviation that may be characterized as a bias towards ‘explanatory adequacy’. I present a numerical model of this idea and show that my alternative account is broadly consistent with prominent prediction error models of delusion formation (e.g. Corlett, Murray et al., 2007).