Conditional probability of conclusion relative to premises
Figure 3.1 Arguments on a scale of conditional probability white shoes. But the probability of that conclusion, relative to the premise, is only slightly better than 1/2; so the argument is just barely inductively forceful. It is inductively forceful, but only to a low degree.
4 Whether or not an argument is deductively valid does not depend on whether anyone thinks it is. Likewise, probability, in our sense, is the degree to which it is rational or reasonable to expect something to be true (given a certain set of premises), irrespective of how likely we actually think it to be. When we claim that an argument is inductively forceful, we are not always correct. We may think that a set of premises (our evidence) makes it reasonable to accept a conclusion, when in fact it is not. Rational expectation concerns what is in fact reasonable, not what any particular person thinks is reasonable.
We can reinforce this point by considering a case in which everyone will agree that rational expectation depends on proportion. Suppose that Fiona is asked to pick a card at random. Her chance of getting, say, a heart, is exactly one in four. Her chance of not getting a heart, then, is three in four, or 3/4. So probably she will not get a heart. If she is perfectly rational and well-informed, she will be 3/4 certain that she won't get a heart. Suppose, however, that Fiona is not perfectly rational. She thinks that since she has recently fallen in love, she probably will choose a heart. She might express her thinking so by saying, 'I'll probably choose a heart'. Nevertheless, the probability that she will not choose a heart is
Logic: inductive force
greater - indeed three times as great - as the probability that she will. Fiona's actual degree of expectation is higher than the degree of rational expectation. More generally, we can observe that different people, going on the same set of premises, can have different degrees of expectation that something is or will be true; but there is only one correct answer as to how reasonable it is to infer the conclusion from the premises.
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