Thus (171) will not have the presupposition (172) because the word claims will have an associated heritage expression that will block it:
(171) Nato claims that the nuclear deterrent is vital
(172) There exists a nuclear deterrent
As we noted above, it is not clear that plugs are a useful category, but if they are, here is a coherent way of modelling them. Similarly with the class of filters: each connective will have associated with it a heritage expression that will block the presuppositions of the lower constituent sentences just in case the filtering conditions in (137) and (138) are met. For example, the heritage expression that captures the filtering condition for conditionals can be thought of as something like (173):
(173) The conventional implicatures of if p then q (and also perhaps of p and q) are the conventional implicatures of p together with the expression ‘if p then the conventional implicatures of q’
To see how this works apply it to a case like (174) where the presupposition, (175), of the consequent is filtered:
(174) If John has children, all of john’s children must be away
(175) John has children
Here the presuppositions of the whole will be whatever the presuppositions of the antecedent are (e.g. John exists), plus the proposition that if John has children, then he has children. Since this proposition is tautologous, it is vacuous, and the speaker is specifically not committed to (175) even though the phrase all of John’s children presupposes (or conventionally implicates, in the terminology of this theory) (175).
For holes Karttunen & Peters can obviously just let the heritage expression allow the implicature expressions to ascend to become the conventional implicatures of the whole.
Thus, on this theory, presuppositions are not actually cancelled, they are blocked during the derivation of the sentence and simply do not arise from the whole. In many ways this is a highly sophisticated and carefully constructed model that can be fully formalized within what is perhaps the most rigorous of contemporary linguistic theories.