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11.1: Informal Introduction

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  • The point of metatheory is to establish facts about logic, as distinguished from using logic. Sentence and predicate logic themselves become the object of investigation. Of course, in studying logic, we must use logic itself. We do this by expressing and using the needed logical principles in our meta-language. It turns out, however, that to prove all the things we want to show about logic, we need more than just the principles of logic. At least we need more if by 'logic' we mean the principles of sentence and predicate logic which we have studied. We will need an additional principle of reasoning in mathematics called Mathematical Induction.

    You can get the basic idea of mathematical induction by an analogy. Suppose we have an infinite number of dominos, a first, a second, a third, and so on, all set up in a line. Furthermore, suppose that each domino has been set up close enough to the next so that if the prior domino falls over, it will knock over its successor. In other words, we know that, for all n, if the nth domino falls then the n + 1 domino will fall also. Now you know what will happen if you push over the first domino: They will all fall. To put the idea more generally, suppose that we have an unlimited or infinite number of cases, a first case, a second, a third, and so on. Suppose that we can show that the first case has a certain property. Furthermore, suppose that we can show, for all n, that if the nth case has the property, then the n + 1 case has the property also. Mathematical induction then licenses us to conclude that all cases have the property. If you now have the intuitive idea of induction, you are well enough prepared to read the informal sections in chapters 12 and 13. But to master the details of the proofs in what follows you will need to understand induction in more detail.

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