Writings of Charles S. Peirce: A Chronological Edition, Volume 6. Charles S. Peirce

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      Boolian Algebra—Elementary Explanations

      There is a very convenient system of signs by which very intricate problems of reasoning can be solved. I shall now introduce you to one part of this system only, and after you are well exercised in that, we will study some additional signs which give the method increased range and power. We use letters in this system to signify statements or facts, real or fictitious. We change their signification to suit the different problems. Two statements a and b are said to be equivalent when equal, provided that in every conceivable state of things in which either is true, the other is true, so that they are true and false together, and we then use a sign of equality between them, and write a = b. We use the words addition, sum, etc., and the symbol + in such a sense that, if a is one fact, say that the moon is made of green cheese, and b is another fact, say that some nursery tales are false, that is a + b, or a added to b, or the sum of a and b, signifies that one or the other (perhaps both) of the facts added are true, so that a + b is a statement; true if one or both of the statements a and b are true and false if both are false. Giving to a and b the above significations, it would mean that the moon is made of green cheese, or some nursery tales are false, or both. In translating it into ordinary language, you generally omit the words “or both” as unnecessary.

We use the words multiplication, product, factor, etc., and the signs of multiplication, or we write the two factors one after the other with no sign between them to mean that both of the two statements multiplied are true, so that ab is a statement which is true only if both the statements a and b are true, and false if either a or b is false. With the above significations it would mean that the moon is made of green cheese, and that some nursery tales are false. When we wish to signify the multiplication of a whole sum by any factor, we write that sum in parenthesis. Thus, (a + b)c would mean the product of a + b into c while a + bc would mean the sum of a and of the product of b and c; giving the above significations to a and b, and letting c mean some proverbs were false, (a + b)c, there we signify the combined statements of, some proverbs are false, and that either the moon is made of green cheese, or some nursery tales are false, while a + bc would mean that either the moon is made of green cheese, or else some proverbs and some nursery tales are false. There are certain rules which facilitate the application of these symbols to reasoning. Thus, a + a will mean neither more nor less than a written alone, so that we may write a + a = a, for a + a, according to what has been said, is that statement which is true if a is true, and is false only if a is false.

      The statement aa is also the same as a standing alone, for it merely asserts the fact a twice over so that we may write aa = a. We also say that a + b is the same as b + a and that ab is the same as ba. This is usually expressed by saying that addition and multiplication are commutative operations. Also that (a + b) + c is the same as a + (b + c), and (ab)c is the same as a(bc). This is usually expressed by saying that addition and multiplication are associative operations. We also have (a + b)c = (ac + bc), for if we say that c is true and also that either a or b is true, we state neither more nor less than if we say that either both a and c are true, or both b and c are true. In like manner we have a + bc = (a + b)(a + c), for if we say that either a is true, or else both b and c are true, we state neither more nor less than if we say that either a or b is true, and also that either a or c is true. As this is perhaps not quite evident, I will give a proof of it. We have seen already that (a + b)c = ac + bc. Now this has nothing to do with the particular letters used, but will be as true for any other three letters. We will therefore write (a + b)x = ax + bx. Now x may be any statement whatever. Let it then be the statement a + c and substitute this in the place of x in the conclusion; then we get (a + b)(a + c) = a(a + c) + b(a + c). Now, on the same principle the first term of the second member of this conclusion a(a + c) is equal to aa + ac, and aa we have just seen to be equal to a, so that the first term is a + ac; the second term b(a + c) is equal to ba + bc, so that the whole expression (a + b)(a + c) equals a + ac + ab + bc. Now it is plain that a + ac equals a, for a + ac is only false if both a and either a or c are false. Now if a is false, plainly, either a or СКАЧАТЬ