LOGIC, SCIENCE, AND ENGINEERING

My commentators have taken issue with me on matters falling into each of three areas. The areas thus provide an appropriate framework for my response. While the line between science and engineering may be a little fuzzy, I 1.ake the line between logic and science to be quite sharp. Nevertheless, I seem to draw it in a slightly different place from others. My concerns are logical concerns. But what is logic? I take logic to have been traditionally concerned with standards for valid inference. But it is more than this. If we were only concerned to ensure that the consequences of our inferences should not be false when our premises are true, we could attain that goal by a judicious universal suspension of belief in the results of argument. In traditional logic we want not only principles that will ensure validity but also principles that will be useful. Early in this century, one (controversial) goal was to provide logical principles from which all matheinatics could be derived. Whether that goal was achieved or not depends on what you mean by logical principles. If you include the axioms that characterize the membership relation as part of logic, then it can be argued that mathematics can be derived from logic. If you don’t, then you must clearly distinguish another set of truths, in addition to logical truths, namely, mathematical truths. We must, as Kant argued, accommodate “‘5 + 7 = 12,” which clearly cannot (contrary to Mill) be falsified by experience. There are other forms of usefulness, and indeed it can plausibly be argued that the reason that the development of a logic adequate for mathematical argument was important was that arguments in science and engineering make use of quantities that may be represented as mathematical objects. To argue that if A is five feet long, and B is seven feet long, and C is the collinear juxtaposition of A and B , then C is twelve feet long, is just the sort of argument that interests us. It is the sort that interests us precisely because it is the sort of argument that we can often characterize as sound. That is, not only does the conclusion follow from the premises, but the premises are true. Note the difference between ii rule of inference being sound (it preserves truth) and an argument being sound (its premises are true). Given inductive acceptance, we can’t be certain that an argument is sound, but we can surely have good reason to think it is. It is here that the alternative approach touted by Bacchus et al. comes to a screeching halt as a “logical” approach, The authors agree that given a prior distribution the probability of C given BK: A E is deductively determined, but argue that “the choice of a prior distribution is not deductive!” Quite so. In the tradition of logic, the inferences that interest us are those that are sound: They are valid, and they have true premises. For a particular inference to interest us, we must have some reason to believe that the premises are true. But not only do I find it hard to believe that any of the prior distributions suggested by Bacchus et al. are true, I find it hard to imagine what it would mean to call one of them true. I understand that such inferences can be valid; I denigrated them as “merely deductive” in part because I see no way in which they can be both interesting and sound. A number of these issues are also raised by Charles Morgan. It appears that for him, as I suspect it is for Bacchus and Halpern, probabilities have to do with belief, and are psychological. The “principles must be true of all probability distributions Prob. This observation corresponds well with our intuitive notion that the logical principles of inference