There is nothing philosophically new here, but I think this is kind of fun. If you click here, you will get a mathematical equation which you ought to have credence 1/2 in (at least immediately). Why? Because a visit to that link has a 50% chance of displaying a mathematical equation that is true and a 50% chance of displaying one that is false, and it's too complicated to see off-hand which it is (now if you have more time, you can do the calculation, and then revise your credence). So you click there, and you do the rational thing—you assign a credence of 1/2. And then, of course, there is a Dutch Book against you, since in fact the equation is either necessarily (and a priori) true or necessarily (and a priori) false.
If you'd like an equation you should be pretty sure about, click here—you'll have a 90% chance that the equation is true. And if you'd like something to speculate over, click here—you'll have a 10% chance of getting a true equation.
I'm pretty sure I've missed you point here, Alex. There's a Dutch Book against any bet that X is the case once you've worked out whether X or not X is the case, isn't there? I therefore don't see what necessity adds to this.
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