Logical-Truth Calculation by Algebra
One of the more efficient ways of reasoning is often called the algebra of logic. This algebra is a part of a new science called mathematical logic. Mathematical logic is a science that has the following characteristics:
It studies chiefly nonnumerical reasoning.
It seeks accurate meanings and necessary consequences.
Its chief instruments are efficient symbols.
Mathematical logic studies especially the logical relations expressed in such words as “or,” “and,” “not,” “else,” “if,” “then,” “only,” “the,” “of,” “is,” “every,” “all,” “none,” “some,” “same,” “different,” etc. The algebra of logic studies especially only the first seven of these words.
The great thinkers of ancient Greece first studied the problems of logical reasoning as these problems turned up in philosophy, psychology, and debate. Aristotle originated what was called formal logic. This was devoted mainly to variations of the logical pattern shown above called the syllogism. In the last 150 years, the fine symbolic techniques developed by mathematicians were applied to the problems of the calculation of logical truth, and the result was mathematical logic, much broader and much more powerful than formal logic. A milestone in the development of mathematical logic was The Laws of Thought, written by George Boole, a great English mathematician, and published in 1854. Boole introduced the branch of mathematical logic called the algebra of logic, also called Boolean algebra. In late years, all the branches of mathematical logic have been improved and made easier to use.
We can give a simple numerical example of Boolean algebra and how it can calculate logical truth. Suppose that we take the truth value of a statement as 1 if it is true and 0 if it is false. Now we have numbers 1 and 0 instead of letters T and F. Since they are numbers, we can add them, subtract them, and multiply them. We can also make up simple numerical formulas that will let us calculate logical truth. If P and Q are statements, and if p and q are their truth values, respectively, we have Table 7.