I know there's an old Boer and Lycan paper about this very topic that I haven't read yet, so nothing I say here may be new.
I don't think that the linguistic phenomena of 'presupposition' really exists, but that in all cases what we're calling presupposition is really a mixture of semantic entailment, implicature, and possibly related discourse phenomena (such as the procedural knowledge that indefinites usually refer to things not introduced into the discourse yet).
First, the standard definition of presupposition doesn't work. Everybody holds that a sentence's presupposition is whatever information needs to been in the common ground in order for an assertion to be felicitous. So if I say, "Mary stopped smoking" it would only be a felicitous assertion if we both already know that Mary did smoke.
Except when it's not. In every single case of presupposition one can imagine cases where the sentence is felicitously asserted without the "presupposed' information being in the common ground. Typically, pragmatics folks say that in such cases the hearer "accommodates" the information by adding it into the common ground. But this is nothing more than an admission that the definition does not work. Since there is no generative theory of accommodation, all it does is function as a place to throw recalcitrant counterexamples. It's as though Newtonians had said that Mercury was accommodating the sun and that light accomodated different inertial frames and left it at that.
Second, the biggest reason one needs a different theory of presupposition is because of "presupposition holes," where people can reliably infer material that is provably not entailed. So, in most cases, if "Mary stopped smoking" is the antecedent of a material or subjunctive conditional, we can still infer that she used to smoke, even though nothing is entailed by the antecedent of a conditional.
But I think this is purely a case of conversational implicature (at least with stop/start and definites; I need to look at other constructions). Asserting a material conditional implicates that you don't know if the antecedent is true. Asserting a counterfactual conditional implicates that the antecedent is false.
To see how this removes the need for presupposition, consider definite descriptions. Assume that the Russellian analysis holds. Then, inote that negation has a strong tendency to narrowly scope (and Michael and Ann Hegarty have a very good explanation involving discourse effects of why this is the case). Then the implicature of a counterfactual conditional with the antecedent that "The present king of France is bald" is that there is a present king of France and he is not bald. So there is an implicature that there is a present king of France! Likewise if the information that we don't know narrowly scopes. Then the implicature for a material conditional with the antecedent that the present king of France is bald is that there is a present king of France and we don't know if he's bald. Moreover, the way these implicatures cancel I think explains the cases that linguists call "acccomodation." This clearly works with "stop/start" type presupposition because in a rich enough lexical decomposition you will get exactly analogous places for wide and narrow scope negation.
We'd need to look at the way "presupposition" works with relative clauses, and also see if the account can handle "filters" where a presupposition is blocked. These are canonically conjunctions where the second conjunct asserts (or directly entails?) the presupposition of the first conjunct. I know in some of these cases the second conjunct is providing information about scope, i.e. "The present king of France is not bald, and there is a present King of France." This works fine if uttered in a context (Sherlock Holmes type induction scenarios) where the person might have thought you meant the first conjunct to have wide scope negation. My prediction is that in other cases the second conjunct is in a quasi meta-linguistic way working to tell us something about what implicatures the hearer should be drawing, and that the entailment still goes through anyhow (see the nice discussion in the last chapter on presupposition in Kadmon's book about this). That may be completely wrong though.
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