Recent proposals for the analysis of the equivalents of Hebrew 'aval,and 'ela’ (in particular, English 'but’ and French 'mais’) are reviewed and it is suggested that they cannot provide a satisfactory account of the full range of uses of 'aval’ and 'ela’ as exemplified in this paper. It is further argued that in order to account for the linguistic facts of 'aval’ and 'ela’ we must allow for a broader definition of 'utterance meaning’ which would include both semantic and pragmatic notions, as these are traditionally classified. Aval,is shown to operate on two different layers of meaning, while 'ela’ always operates on the same layer. However, they perform similar cancellation jobs in all cases. General conditions governing the cancellation function of 'aval’ and 'ela’ are proposed in terms of such a unified treatment of the various uses of these particles. The meaning of an utterance is viewed as made up of hierarchically ordered 'layers of meaning’ (some of which contain partially ordered sublayers), from an inner 'core’ of propositional content to an outer 'shell’ of conversational implicature, via such layers as modality, illocutionary force, felicity conditions, etc.