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Putting Logic in Its Place: Formal Constraints on Rational Belief
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This contribution is introductory only in the sense that it does not presume that the reader has read the relevant literature. It is not introductory in that the issues dealt with are highly specialized, and the general field of "logic" is not covered at all. Indeed, the author assumes we are interested in a particular form of modal logic in which truth is replaced by something like warranted belief.
This is a very interesting idea, leading to a host of interesting problems that do not arise in standard logic. For instance, Modus Ponens says that p and (p implies q) implies q. Not for Cristensen. If you believe p and you believe p implies q, but you don't believe q, then you have a problem. Perhaps you you stop believing p, because you are pretty hooked on q being true. Or, perhaps p does not imply q.
A key problem addressed by Christensen is the following. Suppose p stands for "the ball in in the red urn" and q stands for "the ball is in the green urn." If I know the ball is in one of the urns, then I believe p or q, but if I don't know which one it is in, then I don't believe p and I don't believe q. Now in standard logic p or q is true implies p is true or q is true. But in this modal logic, I can believe p or q, but I don't have to believe p or believe q.
Christensen uses this and other techniques to argue that we need a graded, probabilistic notion of belief, not a zero-one notion. This of course is what Bayesian have maintained all the time. Indeed, I even find the idea of believing something with probability one to be questionable, since there is no way to update if you are wrong (without having implausible hierarchies of belief).