The first two chapters of Probability Theory: The Logic of Science, by E. T. Jaynes, covers reasoning with incomplete information very nicely. I notice readers

The first two chapters of Probability Theory: The Logic of Science, by E. T. Jaynes, covers reasoning with incomplete information very nicely. I notice readers rushing ahead to Bayes Rule and paying insufficient attention to two preliminaries. First you have to fuzz the basic conjunctions not the propositions. If your have incomplete information about whether Sally has brown hair or red, whether Sally is short or tall, whether Sally is thin or fat, well, everybody starts off trying to use three numbers, one representing the brown/red uncertainty, one representing short/tall uncertainty, one representing thin/fat uncertainty. That never works out. You need seven numbers (eight that sum to one). You have a number for Sally is a tall, thin, red-head, another for Sally is a tall, thin, brunette, another for Sally is a tall, fat, red-head, etc. Second, you have to use all the information you have all the time. The great thing about logic is that it is monotonic. Suppose that you have a pile of accurate facts about the world (so we know that they are consistent). You can work through them, one by one. If, half way through, you come up with a useful deduction, you know that you get to keep it. It will still be true at the end. Reasoning with incomplete information is non-monotonic. Half way through you are confident that it has been raining earlier in the day because the lawn is wet. Later you trip over the lawn sprinkler and see that it has been used. Later facts can explain away early facts, and the strengths of conclusions can go down as well as up. Notice the horrible impracticality of doing exact reasoning with incomplete information. There are too many basic conjunctions and you have to pay attention to all the information all of the time :-( Real world reasoning is going to have to be approximate, so you know from the start that there is going to be trouble if you push practical reasoning too hard. The article talks about mathematical induction. That provides a good example of pushing approximate reasoning too hard. We've agreed that if a bald man (meaning a few hairs) had one more hair, he would still be bald. But there is a lurking ambiguity. Are we asserting that as a mathematical truth, hard edged, absolutely true, use it as many times as you like, nothing will go wrong? We had better be, if we are going to use it as a premise for mathematical induction. Or are we assenting to the claim for social and practical reasons? We don't want be awkward by refusing assent to a proposition that is kind-of true and insisting on actually counting the hairs is unreasonable. Real world reasoning nearly always involves a degree of fudge and muddle and we don't want to insist on a degree of exactitude that we will not adhere to in our own reasoning. Especially as we are damn sure that a quest for exactitude will slow us down and make use miss important deadlines.