Logic Taught by Love/Chapter 9
BOOLE AND THE LAWS OF THOUGHT
"Hear, O Israel, the Lord our God is Unity."
Sixty years ago, logicians had concluded that it would be convenient to express ordinary statements about facts in some sort of Arithmetical or Algebraic notation, so as to be able to work out the logical consequences of premises with the same ease as we work sums. Many attempts were made to create such a notation; but none of them proved satisfactory. There was in Lincoln a school-master in humble circumstances, who had, when a mere lad, solved for himself a problem of a very different order. It had occurred to him that the Scripture writers must have had some reason for laying so much stress on the command never to think of The Great All except as a Unity. He had acquired the habit of thinking of each class of things as a fraction of Unity. He found that this habit simplified all study; and by strict adherence to it he had, in the very small amount of leisure at his disposal, made of himself a fairly good linguist, a learned metaphysician, and a mathematician distinguished for the originality and vigour of his methods of investigation. When, therefore, he realized what was the problem over which logicians were pondering, it naturally occurred to him to try on their behalf the experiment of using the Arithmetical Symbol of Unity for Universe of Thought. Immediately the notation fell easily into order, and all the difficulties vanished.
As simple obedience to the commands of the Pentateuch had enabled George Boole to solve the questions which the logicians of the 19th century could not solve, it seemed probable that the writers of Scripture knew more than was commonly supposed about the normal action of the human mind. He set himself the task of making a serious study of the Laws of Thought-sequence; aided by the conversation of a Jewish friend, a teacher of Hebrew at Lincoln. In 1854 he published a book to which he gave the title, The Laws of Thought.
A strange result ensued. The author had felt obliged to show that his system was not a mere fanciful outcome of religious fervour; and, in order to do this, he had interspersed his serious analysis with exhibitions of delicate and skilful modes of extracting the legitimate consequences from masses of premises. Those who were on the look-out for a rapid method of manipulating syllogisms, were at first delighted to find their problem solved; and it is impossible to speak too highly of the generosity with which several eminent logicians did honour to the man of whom they evidently thought only as a rival in their own line. But it seems never to have occurred to them that any one could care for Logic except as a method of reasoning. They proceeded to improve the method of the book called Laws of Thought, by leaving out of it all that could throw light on the Laws of Thought-Sequence!
Mr. Stanley Jevons earned a wide-spread fame as the improver of Boole's method. George Boole's own version of the matter was that Mr. Jevons and he were working for totally different objects, and that no intercourse between them could be of the least use to either. His chief anxiety seemed to be to make it impossible that Mr. Jevons should appear as in any way authorized by him. In later years Mr. Jevons was heard to say: "Boole means something that no one has understood yet; the world is not ready to understand him." The solution of Mr. Jevons' perplexity might have been found in the very title of the work on which he was commenting. The symbols which such logicians as he quite rightly discard from Boole's notation, because they are useless and cumbersome and even misleading when employed to work out logical problems, are the necessary implements which those must learn to use who wish to apply mathematical processes to analyze for themselves the Laws of Thought. Had George Boole been chiefly aiming to supply logicians with a ready-made method, his method must be confessed a very bad one; he was aiming to set the example of studying the Laws of Mental Action.
The nature of the change introduced by George Boole into logical analysis may be summed up thus:—In working a sum or equation, we seem to ourselves to be only manipulating and transforming the particular statements contained in the so-called "stating" of our question. But these very transformations are effected by means of our knowledge of certain general truths about number, such, for instance, as those registered in the multiplication-table. "Working" a sum means combining its special statements with general statements which are latent in the mind. Working a thought-sum, then, should mean transforming its premises by combining them with our general knowledge of the Laws of Mental Action. And, in fact, the Aristotelian syllogism was, so far as it went, a process of this very kind; it brought a certain thought-law into contact with the special premises of each argument; and, by means of that general law, extracted the conclusion. To make Logic a more powerful instrument than it had hitherto been, what was needed was not, as Gratry seems to suggest, to import into it a new process, the process of faith; but to extend and make freer and more vigorous our use of that process; seeing that every syllogism is, so far as it goes, an act of faith in the general Laws of the Creative Logos. The Aristotelian Logic was, in fact, an Arithmetic of reasoning, analogous to such an Arithmetic of number as might be evolved by a race whose general knowledge of the Laws of Number amounted to familiarity with the "twice" column of the multiplication-table.
We may say then that what logicians had been seeking, though without quite knowing what they sought, was a Logic made wide and vigorous by using all attainable knowledge of the Laws of Thought as freely as, in working sums, we use our knowledge of the Laws of Number. Such a Logic, it need hardly be added, is yet to be created; though George Boole did a little towards laying its foundations. He brought to light three great principles of mental action:—
First, that all sound thinking treats the Universe of Thought as a Unity; and classes of things as fractions of Unity; and that Unity itself as a fraction of a larger Unity.
Secondly, that we cannot deal logically with any statement except by comparing it impartially with the opposite statement.
Thirdly, that sound Thought is always essentially a free Pulsation between extremes. All selection, whether by an adult for himself, or by a teacher for his pupils, should consist in selection of the question to be studied. When once we have decided what we will think about, we must think with perfect impartiality on both sides. In mathematical language we state this by saying that every sound thought-process which can be carried on with respect to any element (x), depends in some way or other on considering x, first as Unity (i.e. Universal and absolute throughout the Universe of Thought), and then as Zero (i.e. non-existent in the Universe of Thought), and then combining the two conceptions. Right conduct (it is often said) is that which keeps the mean between two extremes. Now to know the mean we must know the extremes. Sound thought is related to correct conduct as the pulsation of a Light-ray is related to the onward path of the ray. Whatever hampers the free, side-ways pulsation of the wave from one extreme to the other, interferes with the pure whiteness of the Light; whatever hampers the free swinging of the mind between extremes of opinion, obstructs man's vision of what is right in moral conduct.
A country watchmaker, being asked why being dirty made a watch go fast instead of slow (as the questioner would have expected), replied: "Well, you see, when the wheels are clogged with dirt, it don't go; not to say go; it only niggles-like." This is the explanation of the hurry and fever of life. Our minds are clogged with prejudices and trammelled with superstitions; and they don't really work, they "only niggle-like." We think we are giving our children freedom by presenting to them a mass of novel ideas, the truth of which has not yet been tested, and by indoctrinating them with a mass of prejudices which differ from those of our forefathers chiefly by being expressed in new phraseology; it would be better to present to them a few subjects of study, selected according to some order which has been tested by long experience; and, when once a child's attention has been called to any topic, encourage him to think of it with absolute freedom and utter impartiality. This, and not any particular method for manipulating masses of syllogisms, is the main outcome of Boole's mathematical analysis of the Laws of Thought. It is a curious compilation; over-weighted with too great a mass and variety of material. In its attempt to bring the intellect and heart to work in unison, it fails to make its appeal to either easily intelligible. It needs a far closer study than ordinary readers can be expected to give to an author. Nevertheless, it is a very amusing book, because of its latent satire on those mental processes which religious and ethical writers call "thinking." A few quotations will make clear the author's opinion of ordinary religious controversy.
"I shall examine what are the actual premises involved; whether those premises be expressed or implied. By the actual premises I mean whatever propositions are assumed in the course of the argument, without being proved, and are employed as parts of the foundation upon which the final conclusion is built. . . . The chief practical difficulty of this inquiry will consist, not in the application of the method to the premises once determined, but in ascertaining what the premises are. In what are regarded as the most rigorous examples of reasoning applied to metaphysical questions, it will occasionally be found that different trains of thought are blended together; that particular but essential parts of the demonstration are given parenthetically, 0r out of the main course of the argument; that the meaning of a premise may be in some degree ambiguous; and, not unfrequently, that arguments, viewed by the strict laws of formal reasoning, are incorrect or inconclusive. The difficulty of determining and distinctly exhibiting the true premises of a demonstration" (in a book which we are studying and analyzing), "may, in such cases, be very considerable. But it is a difficulty which must be overcome by all who would ascertain whether a particular conclusion is proved or not, whatever form they may be prepared or disposed to give to the ulterior process of reasoning. It is a difficulty, therefore, which is not peculiar to the method of "mathematical analysis," though it manifests itself more distinctly in connection with this method than with any other. So intimate, indeed, is this connection, that it is impossible, employing the method of this treatise, to form even a conjecture as to the validity of a conclusion, without a distinct apprehension and exact statement of all the premises upon which it rests. In the more usual course of procedure" (i. e. verbal analysis), "nothing is, however, more common than to examine some of the steps of a train of argument, and thence to form a vague general impression of the scope of the whole, without any such preliminary and thorough analysis of the premises which it involves.
"The necessity of a rigorous determination of the real premises of a demonstration ought not to be regarded as an evil; especially as, when that task is accomplished, every source of doubt or ambiguity is removed. In employing the method of this treatise . . . the process of inference is conducted with a precision which might almost be termed mechanical."—Laws of Thought, ch. xiii. §§ 1,2.
"There are many special departments of science which cannot be completely surveyed from within, but require to be studied also from an external point of view, and to be regarded in connection with other and kindred subjects, in order that their full proportions may be understood." (Laws of Thought, ch. viii. § 1). The above passage is in line with George Boole's opinion that no Church can properly be reformed from within. The errors of a Church, he considered, must always be attacked from without. The ideal condition, of course, would be that Reformers within a Church should voluntarily seek the assistance of those who look on from the outside; but George Boole could not be induced to believe that such liberality was possible.
The following passage is interesting, as bearing on the question of the relation between rival systems of Theology.
"If the general truths of Logic are of such a nature that when presented to the mind they at once command assent, wherein consists the difficulty of constructing the Science of Logic ? Not in collecting the materials of knowledge, but in discriminating their nature, and determining their mutual place and relation. All sciences consist of general truths: but of those truths, some only are primary and fundamental, others are secondary and derived. The laws of elliptic motion, discovered by Kepler, are general truths in Astronomy, but they are not its fundamental truths. And it is so in the purely mathematical sciences. An almost boundless diversity of theorems which are known, and an infinite possibility of others as yet unknown, rest together upon the foundation of a few simple axioms; and yet these are all general truths. They are truths which, to an intelligence sufficiently refined, would shine forth in their own unborrowed light, without the need of those connecting links of thought, those steps of wearisome and often painful deduction, by which the knowledge of them is actually acquired. Let us define as fundamental those laws and principles from which all other general truths of science may be deduced, and into which they may all be again resolved. Shall we then err in regarding that as the true Science of Logic which, laying down certain elementary laws, confirmed by the very testimony of the mind, permits us thence to deduce, by uniform processes, the entire chain of its secondary consequences, and furnishes, for its practical applications, methods of perfect generality? Let it be considered whether in any science, viewed either as a system of truth or as the foundation of a practical art, there can properly be any other test of the completeness and the fundamental character of its laws, than the completeness of its system of derived truths, and the generality of the methods which it serves to establish. Other questions may indeed present themselves. Convenience, prescription, individual preference, may urge their claims and deserve attention. But as respects the question of what constitutes Science in its abstract integrity, I apprehend that no other considerations than the above are properly of any value."—Laws of Thought, ch. i § 5.
It is interesting to notice that the life-law of the brain, taught by Gratry, can be illustrated by means of a plain forked stick. But in order to teach geometrically Boole's complicated method of critical analysis, we must also employ the spiral wire or snake-coil. If the writer of the third Chapter of Genesis lived in our prosaic age, he would perhaps formulate his caution thus:—"Study Gratry freely. But beware of allowing yourself to be tempted by Boole's subtle and fascinating method; or it will surely lead you into mischief."
If he were in a specially honest frame of mind, he might add:—"Because Boole's method may enable you to grasp ideas quite beyond the comprehension of your Rabbi (Don, Pedagogue, or Pope, as the case may be); and therefore necessarily wicked."