Abstract: We compare four recent theories which
seek to predict how any operator projects presuppositions once its
syntax and bivalent semantics have been specified. These theories
differ along four dimensions: 1. their reliance on local contexts; 2.
their use of trivalence; 3. their semantic or pragmatic nature;
4. the strength of the left-right bias they posit for presupposition
projection. They also differ empirically: although they typically agree
in the propositional case, they make conflicting predictions with
respect to quantified examples. The debate is constrained by some new
experimental results.