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Boolian Algebra—Elementary Explanations

Fall 1886

Houghton Library

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 c is false, that is, one of those two, a and c, is false, so that a + ac is false whenever a is false and only then.

And on the same principle a + ab is equal to a, and thus the second member of the last conclusion reduces to a + bc, and the whole conclusion is (a + b)(a + c) equals a + bc, which is the very conclusion we had to prove. The two principles that (a + b)c = ac + bc, and a + bc equals (a + b)(a + c) are commonly referred to by saying that multiplication is distributive with reference to addition, and that addition is distributive with reference to multiplication. I shall now introduce two statements which have special symbols in this system; the first is $, and means any fact necessarily true. All facts that are necessarily true are equal, because we agreed that we should say that two statements are equal provided they are true and false together in all conceivable states of things. The other special symbol for a statement is 0. This signifies any statement that is false. All false statements are equal for the same reason that all true statements are equal. There are a number of rules facilitating the use of these symbols $ and 0. The first is a + 0 = a. This means that to say that either a is true, or else a false statement is true, is the same as to say at once that a is true. In like manner $a = a; for this means that to say a is true and also that any undesignated true statement is true, is no more than to say that a is true. Second, $ + # = $, for this means that to say that either a is true or something true is true, is no more than to say that something true is true, which is not saying anything at all. And in like manner 0a = 0; for this means that to say that a is true and that something false is true, is to say something false. Third, since every statement is either true or false, if we replace any letter, say a, by $ throughout any formula and find the formula is then necessarily true, and if, on afterwards replacing the same letter by 0, we find that the formula so resulting is true also, then the original formula must be true any way. This affords quite a valuable means of proving any doubtful formula. For instance, let us apply it to proving the formula demonstrated above, a + bc = (a + b)(a + c). First, replace a by $ and the formula becomes, $ + bc = ($ + b)($ + c). Now, $ added to anything gives $; so that $ + bc = $, $ + b = $, and $ + c = $. The whole formula thus reduces to $ = $$ which is true. Now replace a by 0 and the formula becomes, 0 + bc = (0 + b)(0 + c). Now 0 added to anything does not alter it, so that we may drop these added Os, and the formula reduces to bc = bc, which is true. Thus, it has been shown that the formula is true when a equals $ or a equals 0; and as a must equal one or the other, it is true any way. We must now introduce a new sign, a = $ is the same as a written alone; it means that the statement a is true. But we have as yet no simple expression for a = 0, meaning that the statement a is false. Let us denote this by making a line over the a; thus, a; this we call the negative or denial of a. There are several rules facilitating the use of denials. First, aā equals 0, or nothing can be true and false at the same time; this is called the principle of contradiction. Second, a + ā equals $, or everything is either true or false; this is called the principle of excluded middle.

We will now proceed to show this system of signs is to be used for the purpose of drawing conclusions from premises. The simplest possible kind of reasoning is the immediate application of a rule. Thus, a little girl says that whatever mamma forbids is wrong, but mamma forbids this, therefore, this is wrong. Let a mean that anything is forbidden by mamma, b that it is wrong. Then, to say that anything is forbidden by mamma is wrong is the same as to say that either it is not forbidden by mamma, or else it is wrong. This proposition is therefore written ā + b. The other proposition is a. These propositions are asserted to be both true and therefore, they must be multiplied together, and we have, a(ā + b). On performing this multiplication, that is, on applying the distributive principle, we get aā + ab, but aā is 0 by the principle of contradiction, and may therefore be dropped. We therefore have ab. ab is therefore asserted of both the propositions a and b. It therefore asserts b, and therefore the act in question is wrong. Now it would of course be perfectly ridiculous to use this cumbrous system of signs for the purpose of bringing out the conclusion of such a simple mode of argument as this, but it will be found that the system is well adapted to complicated cases but this very feature makes it cumbrous for simple ones.

It will be observed in the above example that after we get the conclusion ab, we drop the factor a, leaving only b. We obviously have the right to do this at any time. We are always entitled to drop a factor from any additive term, and we are also at liberty to add a term to any factor. In consequence of this, whenever we have given an expression in the form a(b + c) we are at liberty to drop the parenthesis and write ab + c. For the distributive principle gives us ab + ac, and on dropping the factor a from the last term, we get ab + c.

In my different publications I have used a sign like Y turned over on its side, , to signify the relation between the antecedent and consequent of an hypothetical proposition. It is a very convenient sign, but as I have no such sign on this typewriter, I shall use a colon for the same purpose, according to the practice of Mr. Hugh McColl. Then, we may write a:b = ā + b. But although the use of this sign does simplify some cases, and I have been one of its principal advocates, and it certainly is useful for a learner, yet it makes a good deal of difficulty in complicated problems. The best way for a beginner to do is to use this sign first, to write down the relations of antecedent and consequent, and afterwards to replace it by +, at the same time negativing the antecedent.

Let us now take a slightly more complicated kind of reasoning, the direct syllogism. If you tell one lie you will tell a hundred, and if you tell a hundred lies you will corrupt your integrity. Let a mean that you tell a lie, b that you tell a hundred, c that you corrupt your integrity. Then, the premises are a:b and b:c. Multiplying them together we have (a:b)(b:c). This is equivalent to (ā + b). Breaking down the first parenthesis, according to the rule just given, we have . Now breaking down the second parenthesis, according to the same rule, we have . But bb is 0 and may be dropped. Thus, we reach the conclusion ā + c or a:c, if you tell a lie you will corrupt your integrity.

I will now show how to treat a little more complicated arguments called indirect syllogisms. Take these premises: if Enoch and Elijah are mortal the Bible errs; but all men are mortal. Let a mean that any given person is Enoch or Elijah, b that he is a man, c that he is mortal, and d that the Bible errs. Then a:c means that if any person is Enoch or Elijah he is mortal, or what is the same thing, that Enoch and Elijah are mortal. Then, (a:c):d means that if Enoch and Elijah are mortal the Bible errs, which is the first premise. The second premise is b:c, or if any person is a man he is mortal. Multiplying the two premises together, we have [(a:c):d](b:c). Now a:c = ā + c and (ā + c):d is to be converted into the regular form by negativing the antecedent and putting a + instead of the colon. We have, therefore, to find the negative of ā + c. The rule for finding the negative of any expression is this: put a line over every letter that has no line over it, and take a line off every letter that has a line over it, and everywhere substitute multiplication for addition and addition for multiplication. Applying this rule, the negative of ā + c is . That is to say, to deny that anything is either not Enoch or Elijah, or else is mortal, is equivalent to asserting that something is Enoch or Elijah and at the same time is not mortal. To prove that this is so it will be sufficient to show that these two expressions satisfy the formulae of contradiction and excluded middle. By the principle of contradiction their product ought to vanish. Now their product is . By the distributive principle this is the same as , but aā = 0 and , so that the whole is . Now and 0a = 0, so that it comes to 0 + 0 which is 0.

Thus, the principle of contradiction is satisfied. According to the principle of excluded middle, the sum of any expression and its negative gives $. Adding the two expressions we have . By the distributive principle of addition with respect to multiplication this is the same as . Now ā + a = $ and , so that the whole becomes ($ + c)(ā + $).

But $ + c = $ and $ + ā = $, so that it reduces to $ + $ which is $; and thus, the principle of excluded middle is also satisfied. Our first premise, then, is , and the product of the two premises is . We may arrange this by the associative principle in the following order, . We now put in a new parenthesis, which we are entitled to do by the associative principle, so as to write . We now break down the inner parenthesis and thus have , and since this is , or (a:b):d, which means that if Enoch or Elijah is a man the Bible errs.

I will now give an example of another variety of indirect syllogism. Take the premises, Translated persons are not mortal and All men are mortal. Let a mean that any person is translated, b that he is a man, c that he is mortal. The first premise, no translated persons are mortal, might be written ac = 0. But if we take the negative of both members of this conclusion we have , or simply . The other premise is . The product of the two premises is this, . Treating this precisely as in the case of a direct syllogism we get the conclusion , or no translated persons are men.

We now go on to a slightly more complicated kind of reasoning, the dilemma. The dilemma is a reasoning in which you show that there are two (or more) possible alternatives, and then show that in either case a consequence follows. This kind of reasoning, although treated in books on Rhetoric, was first introduced into the treatises on logic about the year 1500 by Laurentius Valla. In point of fact the dilemma was very little used during the middle ages and it forms the most elementary example of the falsity of the traditional and Aristotelian notion that all reasoning is syllogistic. In the present system of signs, however, we fail to see anything peculiar about the dilemma, for the reason that we have in this system arbitrarily twisted every syllogism into a dilemmatic form, by writing all a is b in the form of either non-a or b. The truth is that this system of signs is altogether framed to meet the case of the dilemma. Syllogistic reasoning is so easy that it is got rid of by a little artifice. The old stock example of a dilemma is as follows: “It is not good to marry a wife, for if she be fair she will be common, if foul then loathesome” (Blundeville, Art of Logic, 1599). Let a mean that any object is a wife, b that she is fair, c that she is foul, d that she is eligible. Then, the first premise is ā + b + c, or any object is either a wife, or is fair, or is foul. The other two premises are ab:d and ac:d, that is, if any object is a fair wife it is not eligible, and if any object is a foul wife it is not eligible. These two propositions may be otherwise written, and . Now, multiplying the three premises we have . By the application of the distributive principle of addition with respect to multiplication, this is the same as and by the application of the distributive principle of multiplication with respect to addition, this is the same as . Again, applying the same principle, and striking out bb and , we have , and by the striking out of factors this reduces to ā + d + d, or ā + d, or if any object be a wife she is not eligible, or no wife is eligible.

Without the introduction of any further signs, the above kind of reasoning is all that this system can comprehend, but it is useful in two ways. First, the practice of the method gives us great facility in imagining facts in other logical relations, so that we reason much more easily than we did before, even without the use of the method. Secondly, the algebra is itself very useful whenever we meet with a very complicated state of things, provided it is not so excessively complicated as to be unmanageable even with this aid. I shall now add a number of examples sufficient to thoroughly exercise the student and give him a mastery of this really very simple system.