Suppressed premises come in several varieties. They often concern facts or conventions that might have been otherwise—that are contingent rather than necessary. Our example assumed that the dead are not eligible for the presidency, but we can imagine a society in which the deceased are elected to public office as an honor (something like posthumous induction into the Baseball Hall of Fame). Our national government is not like that, however, and this is something that most Americans know. This makes it odd to come right out and say that the deceased cannot hold public office. In most settings, this would involve a violation of the conversational rule of Quantity, because it says more than needs to be said.
Even though it would be odd to state it, this fact plays a central role in the argument. To assert the conclusion without believing the suppressed premise would involve a violation of the conversational rule of Quality, because the speaker would not have adequate reasons for the conclusion. Furthermore, if this suppressed premise were not believed to be true, then to give the explicit premise as a reason for the conclusion would violate the conversational rule of Relevance (just as it would be irrelevant to point out that Babe Ruth is dead when someone asks whether he is in the Baseball Hall of Fame). For these reasons, anyone who gives the original argument conversationally implies a commitment to the suppressed premise.