Names and natural-kind terms are not indexicals (Nimtz and Beckermann forthcoming; Soames 2005): Two-dimensionalism entails that terms such as 'water' are really disguised indexicals that can pick out different referents in different contexts.
Response: Epistemic two-dimensionalism does not entail that names and natural kind terms are disguised indexicals, and it is consistent with the claim that any utterance of the English term 'water' refers to H2O. If primary intensions were Kaplanian characters or contextual intensions, then the claim that 'water' refers to H2O in any context would be inconsistent with the two-dimensionalist claim that the primary intension of 'water' picks out XYZ in the Twin Earth world.
Two-dimensionalism entails that names and natural kind terms are disguised rigidified descriptions (of the form 'the actual phi', for some phi).
Response: Two-dimensionalism does not entail that names and natural kind terms are rigidified descriptions.
Furthermore, as noted above, it is consistent with two-dimensionalism to hold that names and natural kind terms, unlike rigidified descriptions, have the same referent in any context of utterance.
It is also consistent with two-dimensionalism to hold that the primary intension of a name or natural kind term may vary between speakers.
Two-dimensionalism cannot handle belief ascriptions (Soames 2005): It is natural for two-dimensionalists to hold that 'x believes that S' is true when the subject has a belief whose primary intension is the primary intension of S.
Primary intensions are not linguistic meaning: Different speakers can use the same name ('Fred') or natural kind term ('water') with quite different cognitive roles, and with distinct patterns of epistemic evaluation.
But a philosopher, who purposes only to represent the common sense of mankind in more beautiful and more engaging colours, if by accident he falls into error, goes no farther; but renewing his appeal to common sense, and the natural sentiments of the mind, returns into the right path, and secures himself from any dangerous illusions.
The most perfect character is supposed to lie between those extremes; retaining an equal ability and taste for books, company, and business; preserving in conversation that discernment and delicacy which arise from polite letters; and in business, that probity and accuracy which are the natural result of a just philosophy.