This study adopts a bidimensional solution to the belief projection puzzle of additive particles by proposing a Conventional Implicature operator (henceforth, CI), which bifurcates the input meaning into two independent dimensions: the at-issue dimension and the CI dimension. By attaching CI to additives under attitude verbs, the speaker-oriented construal is derived. The proposal of CI operator successfully extends to the derivation of the de re reading of appositives and expressives and does not overgenerate undesired de re readings for other presupposition triggers. The current study argues that the belief projection puzzle of additives does not harm the satisfaction theory of presuppositions.