Believing in conspiracy theories has often been found to be associated with having incoherent beliefs, although different measures of association and of incoherence have led to somewhat varied findings. Here we formalise pairs of incompatible conspiracy beliefs as logical contraries. Contraries cannot both be true but they can both be false, the latter being the case e.g. when the official version of events holds. For example, it cannot both be true that (a) Diana faked her own death and that (b) she was killed by the British secret service. But (a) and (b) can both be false if Diana's death instead resulted from a car accident. We operationalise coherence as probabilistic consistency in a Bayesian framework, and measure extent of incoherence as the Cartesian distance to the nearest response that would have been coherent. These measures are arguably more precise and less susceptible to confounds than previous proposals. Based on these measures we reanalysed the data from 8 studies involving 8590 participants, using zero-inflated generalized linear mixed models with individuals and studies as aggregation levels, and including the maximum random effects structure justified by the design. Our findings replicate the positive association between beliefs in conspiracy theories and incoherent beliefs - both in terms of the relative frequency of incoherence and in terms of the distance from the nearest coherent response. We discuss potential implications for ways of conceptualising and addressing conspiracy beliefs.