Part 18

The simplest computer circuitry performs additions in a serial manner, that is, one operation at a time. This is obviously a slow way to do business, and by adding components so that there are enough to handle the digits in each row simultaneously the arithmetic operation is greatly speeded. This is called parallel addition. Both operations are done by parts understandably called adders, which are further broken down into half-adders.

There are refinements to basic binary computation, of course. By using a decimal point, or perhaps a binary point, fractions can be expressed in binary code. If the position to the left of the point is taken as 2 to the zero power, then the position just to the right of the point is logically 2 to the minus one, which if you remember your mathematics you’ll recognize as one-half. Two to the minus two is then one-fourth, and so on. While we are on the subject of the decimal point, sophisticated computers do what is called “floating-point arithmetic,” in which the point can be moved back and forth at will for much more rapid arithmetical operations.

No matter how many adders we put together and how big the computer eventually gets, it is still operating in what seems an awkward fashion. It is counting its fingers, of which it has two. The trick is in the speed of this counting, so fast that one million additions a second is now a commonplace. Try that for size in your own decimally trained head and you will appreciate the computer a little more.

The Logical Algebra

We come now to another most important reason for the effectiveness of the digital computer; the reason that makes it the “logical” choice for not only mathematics but thinking as well. For the digital computer and logic go hand in hand.

Logic, says Webster, is “the science that deals with canons and criteria of validity in thought and demonstration.” He admits to the ironic perversion of this basic definition; for example, “artillery has been called the ‘logic of kings,’” a kind of logic to make “argument useless.” Omar Khayyám had a similar thought in mind when he wrote in The Rubáiyát,

The grape that can with logic absolute, The Two-and-Seventy Sects confute.

Other poets and writers have had much to say on the subject of logic through the years, words of tribute and words of warning. Some, like Lord Dunsany, counsel moderation even in our logic. “Logic, like whiskey,” he says, “loses its beneficial effect when taken in too large quantities.” And Oliver Wendell Holmes asks,

Have you heard of the wonderful one-hoss shay That was built in such a logical way It ran a hundred years to the day?

The words logic and logical are much used and abused in our language, and there are all sorts of logic, including that of women, which seems to be a special case. For our purposes here it is best to stick to the primary definition in the dictionary, that of validity in thought and demonstration.

Symbolic logic, a term that still has an esoteric and almost mystical connotation, is perhaps mysterious because of the strange symbology used. We are used to reasoning in words and phrases, and the notion that truth can be spelled out in algebraic or other notation is hard to accept unless we are mathematicians to begin with.

We must go far back in history for the beginnings of logic. Aristotelian logic is well known and of importance even though the old syllogisms have been found not as powerful as their inventors thought. Modern logicians have reduced the 256 possible permutations to a valid 15 and these are not as useful as the newer kind of logic that has since come into being.

Leibniz is conceded to be the father of modern symbolic logic, though he probably neither recognized what he had done nor used it effectively. He did come up with the idea of two-valued logic, and the cosmological notion of 1 and 0, or substance and nothingness. In his Characteristica Universalis he was groping for a universal language for science; a second work, Calculus Ratiocinator, was an attempt to implement this language. Incidentally, Leibnitz was not yet twenty years old when he formulated his logic system.

Unfortunately it was two centuries later before the importance of his findings was recognized and an explanation of their potential begun. In England, Sir William Hamilton began to refine the old syllogisms, and is known for his “quantification of the predicate.” Augustus De Morgan, also an Englishman, moved from the quantification of the predicate to the formation of thirty-two rules or propositions that result. The stage was set now for the man who has come to be known as the father of symbolic logic. His name was George Boole, inventor of Boolean algebra.

In 1854, Boole published “An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities.” In an earlier pamphlet, Boole had said, “The few who think that there is that in analysis which renders it deserving of attention for its own sake, may find it worth while to study it under a form in which every equation can be solved and every solution interpreted.” He was a mild, quiet man, though nonconformist religiously and socially, and his “Investigation” might as well have been dropped down a well for all the immediate splash it made in the scientific world. It was considered only academically interesting, and copies of it gathered dust for more than fifty years.

Only in 1910 was the true importance given to Boole’s logical calculus, or “algebra” as it came to be known. Then Alfred North Whitehead and Bertrand Russell made the belated acknowledgment in their Principia Mathematica, and Russell has said, “Pure mathematics was discovered by Boole, in a work he called ‘The Laws of Thought.’” While his praise is undoubtedly exaggerated, it is interesting to note the way in which mathematics and thought are considered inseparable. In 1928, the first text on the new algebra was published. The work of Hilbert and Ackermann, Mathematical Logic, was printed first in German and then in English.

What was the nature of this new tool for better thinking that Boole had created? Its purpose was to make possible not merely precise, but exact analytical thought. Historically we think in words, and these words have become fraught with semantic ditches, walls, and traps. Boole was thinking of thought and not mathematics or science principally when he developed his logic algebra, and it is indicative that symbolic logic today is often taught by the philosophy department in the university.

Russell had hinted at the direction in which symbolic logic would go, and it was not long before the scientist as well as the mathematician and logician did begin to make use of the new tool. One pioneer was Shannon, mentioned in the chapter on history. In 1938, Claude Shannon was a student at M.I.T. He would later make scientific history with his treatise on and establishment of a new field called information theory; his early work was titled “A Symbolic Analysis of Relay and Switching Circuits.” In it he showed that electrical and electronic circuitry could best be described by means of Boolean logic. Shannon’s work led to great strides in improving telephone switching circuits and it also was of much importance to the designer of digital computers. To see why this is so, we must now look into Boolean algebra itself. As we might guess, it is based on a two-valued logic, a true-false system that exactly parallels the on-off computer switches we are familiar with.

The Biblical promise “Ye shall know the truth, and the truth shall make you free” applies to our present situation. The best way to get our feet wet in the Boolean stream is to learn its so-called “truth tables.”

Conjunctive Boolean Operation

A and B equal C A B C (A · B = C) ——— 0 0 0 1 0 0 0 1 0 1 1 1

Disjunctive Boolean Operation

A or B equals C A B C (Ā ∨ B = C) ——— 0 0 0 1 0 1 0 1 1 1 1 1

In the truth tables, 1 symbolizes true, 0 is false. In the conjunctive AND operation, we see that only if both A and B are true is C true. In the disjunctive OR operation, if either A or B is true, then C is also true. From this seemingly naïve and obvious base, the entire Boolean system is built, and digital computers can perform not only complex mathematical operations, but logical ones as well, including the making of decisions on a purely logical basis.

Before going on to the few additional conditions and combinations that complete the algebra, let’s study some analogies that will make clear the AND/OR principles of operation. We can think of AND as two bridges in sequence over two rivers. We can reach our destination only if both bridges are working. However, suppose there are two parallel bridges and only one river. We can then cross if either or both of the bridges is working. A closer example is that of electrical switches. Current will flow through our AND circuit if—and only if—both switches are closed. When the switches are in parallel—an OR circuit—current will flow if either, or both, are closed.

The truth tables resemble the bridge or switch arrangements. We can proceed across the line of 1’s and 0’s in the first table only if both switches are closed. The symbol 1 means that the switch is closed, so we can cross only the bottom line. In the second table, we are told we can proceed across the line if either switch is closed. Thus we can cross lines 2, 3, and 4. We can use many symbols in our two-valued system.

Symbol

Bridge No Bridge