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The dynamic approach posits that a presupposition must be satisfied in its local context. But how is a local context derived from the global one? Extant dynamic analyses must specify in the lexical entry of any operator what its 'Context Change Potential' is, and for this very reason they fail to be sufficiently explanatory. To circumvent the problem, we revise two assumptions of the dynamic approach: we take the update process to be derivative from a classical, non-dynamic semantics -- which obviates the need for dynamic lexical entries; and we deny that a local context encodes what the speech act participants 'take for granted.' Instead, we take the local context of an expression E in a sentence S to be the smallest domain that one may restrict attention to when assessing E without jeopardizing the truth conditions of S. To match the results of dynamic semantics, local contexts must be computed incrementally, using only information about the expressions that precede E. This version of the theory can be shown to be nearly equivalent to the dynamic theory of Heim 1983 -- but unlike the latter, it is entirely predictive. We also suggest that local contexts can, at some cost, be computed symmetrically, taking into account information about all of S (except E); this leads to gradient predictions, whose assessment is left for future research.