Presuppositions triggered in the left argument of binary connectives like 'if' and 'or' typically become presuppositions of the complex sentence. By contrast, presuppositions triggered in the right argument varyingly project conditionally or unconditionally depending on the context — a phenomenon often referred to as the ‘proviso problem’. Theories diverge on how to account for this variation. Some propose that speakers accommodate stronger propositions than the conditional presuppositions directly derived from the semantics. Others argue that presuppositions are semantically unconditional but can be cancelled when
incompatible with the context. Both views thus account for the data, but rely on fundamentally different mechanisms. This paper investigates the underlying mechanism by leveraging the processing signatures of presupposition cancellation. When a sentence presupposes p, contexts incompatible with p
should trigger cancellation of that presupposition, though sentences associated with a weaker presupposition ‘If A, then p’, need not. Across three experiments, we detected a ‘cancellation cost’ during processing when presuppositions were triggered in the left argument of conditionals and disjunctions, though not in the right argument. These results suggest that presuppositions in the right argument project in a weaker form compared to those in the left.