Inferring and Explaining

28 InferrIng and exPlaInIng Logical Connection We’ve said a bit about the top and the bottom in our schematic representation of an argument. What about that conspicuous, big, fat line? In good arguments, the conclusion follows from the premises; the evidence supports the theory. What exactly is this relationship of support or following from? Tat turns out to be a very con- troversial issue in both philosophy and math- ematical logic. In some cases, the relationship is semantic . If we just understood enough about the mean- ings of all the words in the premises, we would see that the conclusion has to be true. Ofen the examples are pretty trivial. e 1 . The number is even. e 2 . The number is greater than seventeen. t 0 . The number is not prime. Other times, however, there’s quite a bit of information hiding in the premises, and the conclusions are a little surprising and quite signifcant. e 1 . The fgure is a plane triangle. t 0 . The interior angles of the fgure equal exactly 180°. Arguments of the previous type have a tech- nical name. Tey are called deductive arguments . In a successful deductive argument, the relation- ship between the premises and conclusion (it’s artifcial here to call them evidence and theory) is a very special one. Logicians call it validity . Valid arguments are ones where if the premises are true, the conclusion has to be true . Many col- leges and universities have whole courses on deductive (or symbolic) logic. Very sophisticated techniques are developed for determining valid- ity. We will not spend time reviewing this mate- rial because as interesting (and just plain fun) as it is, one almost never fnds deductive arguments being put forward outside of academic philoso- phy and mathematics. A second way of connecting premises to con- clusions relies on the technical felds of mathe- matics and statistics. We cannot as conveniently ignore these arguments, since they play huge roles in contemporary science. Our approach to them, however, will be a little indirect. Rather than going through the basics of probability theory and then developing statistical tests for making sense of numerical data, we will treat these arguments as special cases of inductive arguments . Tis latter jargon simply means that the argument claims that the conclusion follows from the premises but not deductively—that is, it is possible for the premises to be true, yet the conclusion turns out to be false. Now, of course, it should be relatively rare that in good induc- tive arguments, the premises would be true and the conclusion false; otherwise these arguments will not be very useful. It is a matter of great controversy in logic, philosophy, and even the sciences as to how we describe this relationship between evidence and theories. Te rest of this book is devoted to showing you one way of char- acterizing this relationship. Inference to the Best Explanation Consider the three short examples mentioned previously. We had purported evidence that