Karttunen & Peters call the meaning expressions that capture presuppositions implicature expressions or conventional implicatures, and the terminology overtly identifies presuppositions with those pragmatic inferences that Grice (1975) isolated as being conventional, non-cancellable and yet not part of the truth conditions. For on Karttunen & Peters’ theory, presuppositions (or, as they would have it, conventional implicatures) are in fact non-cancellable. But Karttunen is well aware of the defeasibility and projection properties of presuppositions — indeed he was the first to explore them in detail. How then can it be claimed that presuppositions are non-cancellable ?
The answer lies in the details of Karttunen & Peters’ system. The idea is that in addition to implicature expressions capturing the presuppositional content of each presupposition-triggering item, there will be associated with each constituent a heritage expression whose sole function will be to govern the projection of the pre-suppositions expressed in the implicature expressions. In this way, Karttunen’s (1973) classification of embedding constructions into plugs, filters and holes can be incorporated into the Montague grammar framework: for example, where an embedding complement is a plug it will have a heritage expression that will block the
presuppositions (expressed by the implicature expressions) from ascending to be presuppositions of the whole sentence.