Symbolic logic is the branch of mathematics that makes use of symbols to express logical ideas. This method makes it possible to manipulate ideas mathematically in much the same way that numbers are manipulated.
Most people are already familiar with the use of letters and other symbols to represent both numbers and concepts. For example, many solutions to algebraic problems begin with the statement, "Let x represent.…" That is, the letter x can be used to represent the number of boxes of nails, the number of sheep in a flock, or the number of hours traveled by a car. Similarly, the letter p is often used in geometry to represent a point. P can then be used to describe line segments, intersections, and other geometric concepts.
In symbolic logic, a letter such as p can be used to represent a complete statement. It may, for example, represent the statement: "A triangle has three sides."
Mathematical operations in symbolic logic
Consider the two possible statements:
"I will be home tonight" and "I will be home tomorrow."
Let p represent the first statement and q represent the second statement. Then it is possible to investigate various combinations of these two statements by mathematical means. The simplest mathematical possibilities are to ask what happens when both statements are true (an AND operation) or when only one statement is true (an OR operation).
One method for performing this kind of analysis is with a truth table. A truth table is an organized way of considering all possible relationships between two logical statements, in this case, between p and q. An example of the truth table for the two statements given above is shown below. Notice in the table that the symbol ∧ is used to represent an AND operation and the symbol ∨ to represent an OR operation:
p q p∧q p∨q
T T T T
T F F T
F T F T
F F F F
Notice what the table tells you. First, if "I will be home tonight" (p) and "I will be home tomorrow" (q) are both true, then the statement "I will be home tonight and I will be home tomorrow"—(p) and (q)— also must be true. In contrast, look at line 3 of the chart. According to this line, the statement "I will be home tonight" (p) is false, but the statement "I will be home tomorrow" (q) is true. What does this tell you about p∧q and p∨q?
First, p∧q means that "I will be home tonight" (p), and "I will be home tomorrow" (q). But line 3 says that the first of these statements (p) is false. Therefore, the statement "I will be home tonight and I will be home tomorrow" must be false. On the other hand, the condition p∨q means that "I will be home tonight or I will be home tomorrow." But this statement can be true since the second statement—"I will be home tomorrow"—is true.
The mathematics of symbolic logic is far more complex than can be shown in this book. Its most important applications have been in the field of computer design. When an engineer lays out the electrical circuits that make up a computer, or when a programmer writes a program for using the computer, many kinds of AND and OR decisions (along with other kinds of decisions) have to be made. Symbolic logic provides a precise method for making those decisions.