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,
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)
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