 The equivalence between Peters’ connectives and Karttunen’s model paved the way for a more complete reformulation of the Karttunen model in Karttunen and Peters (1977; 1979), where certain types of presupposition (which Karttunen and Peters regard as conventional implicatures rather than presuppositions) are treated in a compositional grammar fragment. This fragment uses two dimensions of meaning, one for presupposition and one for assertion, and is effectively an implementation of the Peters connectives in a four-valued logic; see Krahmer (1994); Krahmer (1998), Beaver (2001), and Beaver and Krahmer (2001) for discussion, and the latter for a fragment that mirrors that of Karttunen and Peters, but allows for richer interactions between presuppositions and  quantifiers.[17]