Logic
(Adapted from Critchlow & Eck)
In a sense, we know a lot more than we realize, because everything that we know has consequences—logical consequences—that follow automatically. If you know that all humans are mortal, and you know that you are human, then in a sense you know that you are mortal, whether or not you have ever considered or wanted to consider that fact. This is an example of logical deduction: From the premises that "All humans are mortal: and "I am human," the conclusion that "I am mortal" can be deduced by logic.
Logical deduction is a kind of computation. By applying rules of logic to a given set of premises, conclusions that follow from those premises can be generated automatically, by a computational process which could be carried out by a computer. Once you know the premises, or are willing to accept them for the sake of argument, you are forced—by logic—to accept the conclusions. Still, to say that you "know" those conclusions would be misleading. The problem is that there are too many of them (infinitely many), and, in general, most of them are not particularly interesting. Until you have actually made the deduction, you don't really know the conclusion, and knowing which of the possible chains of deduction to follow is not easy. The art of logic is to find an interesting conclusion and a chain of logical deductions that leads from the premises to that conclusion. Checking that the deductions are valid is the mechanical, computational side of logic.
This chapter is mostly about the mechanics of logic. We will investigate logic as a branch of mathematics, with its own symbols, formulas, and rules of computation. Your object is to learn the rules of logic, to understand why they are valid, and to develop skill in applying them. As with any branch of mathematics, there is a certain beauty to the symbols and formulas themselves. But it is the applications that bring the subject to life for most people. We will, of course, cover some applications as we go along. In a sense, though, the real applications of logic include much of computer science and of mathematics itself.
Among the fundamental elements of thought, and therefore of logic, are propositions. A proposition is a statement that has a truth value: It is either true or false. "Grass is green" and "" are propositions. In the first part of this chapter, we will study propositional logic, which takes propositions as basic and considers how they can be combined and manipulated. This branch of logic has surprising application to the design of the electronic circuits that make up computers.
Logic gets more interesting when we consider the internal structure of propositions. In English, a proposition is expressed as a sentence, and, as you know from studying grammar, sentences have parts. A simple sentence like "Grass is green" has a subject and a predicate. The sentence says something about its subject. The subject of "Grass is green" is grass. The sentence says something about grass. The something that the sentence says about its subject is the predicate. In the example, the predicate is the phrase "is green." Once we start working with predicates, we can create propositions using quantifiers like "all," "some," and "no." For example, working with the predicate "is above average," we can move from simple propositions like "Johnny is above average" to "All children are above average" or to "No child is above average" or to the rather more realistic "Some children are above average." Logical deduction usually deals with quantified statements, as shown by the basic example of human mortality with which we began this chapter. Logical deduction will be a major topic of this chapter; under the name of proof, it will be the last major topic of this chapter, and a major tool for the rest of this book.