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Universal Statements

Arithmetic deals with particular relationships between particular numbers. For example, the statement \[ 2 + 3 = 5 \] tells us about a specific relationship between three specific numbers, $2$, $3$, and $5$. It says nothing at all about any other numbers.

Higher mathematics, beginning with algebra, deals with more general statements about whole classes of numbers. But as soon as we try to make general statements about numbers we find we need new symbols. The symbols used in arithmetic are specific numbers. The symbol $2$, for example, means only the number $2$. It does not mean "any number" or "any whole number" or "any even number." It means only the number $2$. Now, suppose we want to make a statement which applies to any even number. We cannot make the statement using the number symbols of arithmetic, because they are all symbols for specific numbers. We have to define new symbols to make a general statement.

The situation is very much the same as if we had only names for individuals but had no general word meaning "people." If we could use only proper names, communication would be very difficult. If the teacher of a class could not say, "Now, students...," but instead had to say, "Now, Mary, John Peter, William, Carol, Joan...," life would be very tedious.

Many times we want to say something about a person but do not know who the particular person is. For example, the father of a large family might have occasion to say, "Whoever has been using my chisel to dig in the garden is going to catch it." It would certainly take a lot of the force out of this statement if the father had to say, "If John, Patty, Stewart, Mary, Bruce, or Joan has been using my chisel to dig in the garden, then John, Patty, Stewart, Mary, Bruce, or Joan, as the case may be, is going to catch it!"

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