In the envisaged situation, the argument would be deductively sound, since it is deductively valid and the premises are true. But, although P2 is true, neither Bert nor Barney has any reason to think so. So, although the argument is deductively sound, and Bert and Barney can see that it is deductively valid, it gets them no closer to knowing the truth-value of the conclusion.
In such a situation, we say that the argument is rationally unper-suasive. More exactly we must say it is rationally unpersuasive for Barney, and also for Bert. Why we must put it this way - why we must relativise the notion of the rational persuasiveness of an argument, can be seen from a variation of the story.
Suppose that Barney, but not Bert, knows that the coin is weighted in such a way that it virtually always falls tails down. Then Barney, but not Bert, has a good reason to accept P2. Therefore Barney has a good reason to accept that the argument is sound, but not Bert. Therefore the argument is rationally persuasive for Barney, but not for Bert.
Before we give our official definition of rational persuasiveness, there is one last complication to be discussed. Looking over the example just given, you might think that an argument is rationally persuasive (for a person) if (i) it is inductively forceful or deductively valid and (ii) the person reasonably believes the premises. But this is not quite right.
For consider now the following example.
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