Although the beta chains of alpha beta+ T cells exhibit allelic exclusion it has recently been shown that approximately 20-30% of such cells express both alpha chain alleles on their surface. Potentially these cells have dual specificity, and it has been suggested that they may play a role in autoimmunity and alloreactivity. The analysis presented in this paper examines critically the possibility that the observed violation of allelic exclusion of alpha chain expression arises as a natural consequence of alpha chain rearrangement and intrathymic positive selection. It is concluded that, subject to certain identified conditions being satisfied, the observed frequency of T cells expressing two alpha chains is compatible with such a mechanism. However, it is an essential assumption of the theory that, of the two alpha beta chain complexes on these cells, only one mediates positive selection. It follows that to what extent both receptors are actually functional depends on whether an alpha beta chain complex that has failed to satisfy the criterion of positive selection can mediate antigen recognition in the periphery. In principle the answer to this question may be different if one is considering autoreactivity or alloreactivity. Finally, attention is drawn to the apparent discrepancy between the frequency of peripheral T cells expressing two alpha chains and the failure to find T cell clones of this phenotype. Possible explanations are discussed.