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(9) Edward didn't regret that Margaret had failed because he didn't like her.
The "intuitive" difference can be explained only from a textual point of view, assuming the concept of presupposition as part of a background frame. In a dialogue, sentences such as (8) can occur only as an objection to assumptions made by some other speaker, in some previous sequence of the dialogue. 4 In other words, to negate the presupposition is to negate the background frame that another speaker has tried to impose on the discourse. These negations are corrections of the other speaker's words, hence used as a quotation of a previous sentence, since it is impossible for a single subject to utter a sentence which at the same time imposes and denies a textual background frame. Moreover, during a real communicative exchange, such counterexamples as (8) or (10) or (11) seem rather astonishing:
(10) Since the Big Bad Wolf does not exist, it is impossible that he has stolen your skateboard.
(11) I am not aware (or I don't know) that Mary is allowed to use my office.
It is quite improbable (according to everybody's intuition) to hear such sentences uttered in the course of an everyday conversation and expressed in natural language. They are frequently quoted in academic literature, because they belong to an artificial language that we label as "examplese." Usually, something more or less similar to the intended meaning of (8), (10), or (11) would be reformulated as a reaction of A to a previous assertive statement of S, in the following terms:
(8a) Are you crazy? Why did you say that Edward "regretted" that Margaret had failed? Don't you know that she passed?
(10a) What do you mean? You still believe in the Big Bad Wolf? First of all, are you sure that somebody really stole your skateboard? If so, let us try to find out who really could have done it. . . .
(11a) I was not aware that Mary was allowed to use my office. But if you say so. . . .
In (8a) A challenges the right S had to use a certain p-term, since A refuses the background information imposed by S. In (10a) A accepts

 
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