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Henry Craik, ed.  English Prose.  1916.
Vol. V. Nineteenth Century
 
The Necessary Laws of Thought
By Sir William Hamilton (1788–1856)
 
From Lectures

THE HIGHEST of all logical laws, in other words, the supreme law of thought, is what is called the principle of Contradiction, or more properly the principle of Non-Contradiction. It is this:—A thing cannot be and not be at the same time—Alpha est, Alpha non est, are propositions which cannot both be true at once. A second fundamental law of thought, or rather the principle of Contradiction viewed in a certain aspect, is called the principle of Excluded Middle, or more fully, the principle of Excluded Middle between two Contradictories. A thing either is or it is not—Aut est Alpha aut non est; there is no medium; one must be true, both cannot. These principles require, indeed admit of, no proof. They prove everything, but are proved by nothing. When I, therefore, have occasion to speak of these laws by name, you will know to what principle I refer.
  1
  Now, then, I lay it down as a law which, though not generalised by philosophers, can be easily proved to be true by its application to the phenomena—That all that is conceivable in thought, lies between two extremes, which, as contradictory of each other, cannot both be true, but of which, as mutual contradictories, one must. For example, we conceive space—we cannot but conceive space. I admit, therefore, that Space, indefinitely, is a positive and necessary form of thought. But when philosophers convert the fact, that we cannot but think space—or, to express it differently, that we are unable to imagine anything out of space,—when philosophers, I say, convert this fact with the assertion, that we have a notion,—a positive notion, of absolute or of infinite space, they assume, not only what is not contained in the phenomenon, nay, they assume what is the very reverse of what the phenomenon manifests. It is plain, that space must either be bounded or not bounded. These are contradictory alternatives; on the principle of Contradiction, they cannot both be true, and, on the principle of Excluded Middle, one must be true. This cannot be denied, without denying the primary laws of intelligence. But, though space must be admitted to be necessarily either finite or infinite, we are able to conceive the possibility, neither of its finitude, nor of its infinity.  2
  We are altogether unable to conceive space as bounded,—as finite; that is, as a whole beyond which there is no further space. Every one is conscious that this is impossible. It contradicts also the supposition of space as a necessary notion; for if we could imagine space as a terminated sphere, and that sphere not itself enclosed in a surrounding space, we should not be obliged to think everything; in space; and, on the contrary, if we did imagine this terminated sphere as itself in space, in that case we should not have actually conceived all space as a bounded whole. The one contradictory is thus found inconceivable; we cannot conceive space as positively limited.  3
  On the other hand, we are equally powerless to realise in thought the possibility of the opposite contradictory; we cannot conceive space as infinite, as without limits. You may launch out in thought beyond the solar walk, you may transcend in fancy even the universe of matter, and rise from sphere to sphere in the region of empty space, until imagination sinks exhausted;—with all this what have you done? You have never gone beyond the finite, you have attained at best only to the indefinite, and the indefinite, however expanded, is still always the finite. As Pascal energetically says, “Inflate our conceptions as we may, with all the finite possible we cannot make one atom of the infinite.” “The infinite is infinitely incomprehensible.” Now then, both contradictories are equally inconceivable, and could we limit our attention to one alone, we should deem it at once impossible and absurd, and suppose its unknown opposite as necessarily true. But as we not only can, but are constrained to consider both, we find that both are equally incomprehensible; and yet though unable to view either as possible, we are forced by a higher law to admit that one, but one only is necessary.  4
  That the conceivable lies always between two inconceivable extremes, is illustrated by every other relation of thought. We have found the maximum of space incomprehensible, can we comprehend its minimum? This is equally impossible. Here, likewise, we recoil from one inconceivable contradictory only to infringe upon another. Let us take a portion of space however small, we can never conceive it as the smallest. It is necessarily extended, and may, consequently be divided into a half or quarters, and each of these halves or quarters may again be divided into other halves or quarters, and this ad infinitum. But if we are unable to construe to our mind the possibility of an absolute minimum of space, we can as little represent to ourselves the possibility of an infinite divisibility of any extended entity.  5
  In like manner Time: this is a notion even more universal than space, for while we exempt from occupying space the energies of mind, we are unable to conceive these as not occupying time. Thus, we think everything, mental and material, as in time, and out of time we can think nothing. But, if we attempt to comprehend time, either in whole or in part, we find that thought is hedged in between two incomprehensibles. Let us try the whole. And here let us look back—let us consider time a parte ante. And here we may surely flatter ourselves that we shall be able to conceive time as a whole, for here we have the past period bounded by the present; the past cannot, therefore, be infinite or eternal, for a bounded infinite is a contradiction. But we shall deceive ourselves. We are altogether unable to conceive time as commencing; we can easily represent to ourselves time under any relative limitation of commencement and termination, but we are conscious to ourselves of nothing more clearly, than that it would be equally possible to think without thought, as to construe to the mind an absolute commencement, or an absolute termination, of time, that is, a beginning and an end beyond which time is conceived as non-existent. Goad imagination to the utmost, it still sinks paralysed within the bounds of time, and time survives as the condition of the thought itself in which we annihilate the universe. On the other hand, the concept of past time as without limit—without commencement, is equally impossible. We cannot conceive the infinite regress of time; for such a notion could only be realised by the infinite addition in thought of finite times, and such an addition would itself require an eternity for its accomplishment. If we dream of effecting this, we only deceive ourselves by substituting the indefinite for the infinite, than which no two notions can be more opposed. The negation of a commencement of time involves, likewise, the affirmation, that an infinite time has, at every moment, already run; that is, it implies the contradiction, that an infinite has been completed. For the same reasons, we are unable to conceive an infinite progress of time; while the infinite regress and the infinite progress taken together, involve the triple contradiction of an infinite concluded, of an infinite commencing, and of two infinities, not exclusive of each other.  6
  Now take the parts of time—a moment, for instance; this we must conceive, as either divisible to infinity, or that it is made up of certain absolutely smallest parts. One or other of the contradictories must be the case. But each is, to us, equally unconceivable. Time is a protensive quantity, and, consequently, any part of it, however small, cannot, without a contradiction, be imagined as not divisible into parts, and these parts into others ad infinitum. But the opposite alternative is equally impossible; we cannot think this infinite division. One is necessarily true; but neither can be conceived possible. It is on the inability of the mind to conceive either the ultimate indivisibility, or the endless divisibility of space and time, that the arguments of the Eleatic Zeno against the possibility of motion are founded,—arguments which at least show, that the possibility of motion, however certain as a fact, cannot be conceived possible, as it involves a contradiction.  7
  The same principle could be shown in various other relations, but what I have now said is, I presume, sufficient to make you understand its import. Now the law of mind, that the conceivable is in every relation bounded by the inconceivable, I call the Law of the Conditioned. You will find many philosophers who hold an opinion the reverse of this,—maintaining that the absolute is a native or necessary notion of intelligence. This, I conceive, is an opinion founded on vagueness and confusion. They tell us we have a notion of absolute or infinite space, of absolute or infinite time. But they do not tell us in which of the opposite contradictories this notion is realised. Though they are exclusive of each other, and though both are only negations of the conceivable on its opposite poles, they confound together these exclusive inconceivables into a single notion; suppose it positive; and baptise it with the name of absolute. The sum, therefore, of what I have now stated is, that the Conditioned is that which is alone conceivable or cogitable; the Unconditioned, that which is inconceivable or incogitable. The conditioned or the thinkable lies between two extremes or poles; and these extremes or poles are each of them unconditioned, each of them inconceivable, each of them exclusive or contradictory of the other. Of these two repugnant opposites, the one is that of Unconditional or Absolute Limitation; the other that of Unconditional or Infinite Illimitation. The one we may, therefore, in general call the Absolutely Unconditioned, the other the Infinitely Unconditioned; or, more simply, the Absolute and the Infinite; the term absolute expressing that which is finished or complete, the term infinite that which cannot be terminated or concluded. These terms, which, like the Absolute and Infinite themselves, philosophers have confounded, ought not only to be distinguished, but opposed as contradictory. The notion of either unconditioned is negative:—the absolute and the infinite can each only be conceived as a negation of the thinkable. In other words, of the absolute and infinite we have no conception at all. On the subject of the unconditioned,—the absolute and infinite, it is not necessary for me at present further to dilate.  8
  I shall only add in conclusion, that, as this is the one true, it is the only orthodox, inference. We must believe in the infinity of God; but the infinite God cannot by us, in the present limitation of our faculties, be comprehended or conceived. A Deity understood, would be no Deity at all; and it is blasphemy to say that God only is as we are able to think Him to be. We know God, according to the finitude of our faculties; but we believe much that we are incompetent properly to know. The Infinite, the infinite God, is what, to use the words of Pascal, is infinitely inconceivable. Faith—belief—is the organ by which we apprehend what is beyond our knowledge. In this, all divines and philosophers, worthy of the name, are found to coincide; and the few who assert to man a knowledge of the infinite, do this on the daring, the extravagant, the paradoxical supposition, either that Human Reason is identical with the Divine, or that Man and the Absolute are one.  9
  The assertion has, however, sometimes been hazarded, through a mere mistake of the object of knowledge or conception; as if that could be an object of knowledge, which was not known; as if that could be an object of conception, which was not conceived.  10
  It has been held that the infinite is known or conceived, though only a part of it (and every part, be it observed, is ipso facto finite,) can be apprehended; and Aristotle’s definition of the infinite has been adopted by those who disregard his declaration, that the infinite, qua infinite, is beyond the reach of human understanding. To say that the infinite can be thought, but only inadequately thought, is a contradiction in adjecto; it is the same as saying, that the infinite can be known, but only known as finite.  11
  The Scriptures explicitly declare that the infinite is for us incognisable;—they declare that the finite, and the finite alone, is within our reach. It is said, to cite one text out of many, that “now I know in part” (i.e., the finite); “but then” (i.e., in the life to come), “shall I know even as I am known” (i.e., without limitation).  12
 
 
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