The set theoretic latent trait model proposed in this paper assumes that: 1) subjects and items are represented by sets of discrete abilities, 2) the sets of abilities are partially ordered with respect to their inclusion relations, and 3) the probability of correct responses is defined on the partially ordered structure as a set function of subjects and items. This model is a modified latent class model incorporating order relations among latent classes. At the same time it is a generalization of ordinary item response theory in that subjects are represented in a partially ordered structure rather than a totally ordered structure (dimension). A computer program which estimates parameters of the model and determines set structures was developed, and three simulation studies suggested the viability of the model.