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pletely adequate, our knowledge ought to admit of analysis after analysis ad infinitum, so that adequate knowledge would be impossible. But, as Dr Thomson remarks, we may consider any knowledge adequate which carries the analysis sufficiently far for the purpose in view. A mechanist, for instance, has adequate knowledge of a machine, if he not only know its several wheels and parts, but the purposes, materials, forms, and actions of those parts; provided again that he knows all the mechanical properties of the materials, and the geometrical properties of the forms which may influence the working of the machine. But he is not expected to go on still further and explain why iron or wood of a particular quality is strong or brittle, why oil acts as a lubricator, or on what axioms the principles of mechanical forces are founded.
Lastly, we must notice the very important distinction of symbolical and intuitive knowledge. From the original meaning of the word, intuitive would denote that which we gain by seeing (Latin, intueor, to look at), and any knowledge which we have directly through the senses, or by immediate communication to the mind, is called intuitive. Thus we may learn intuitively what a square or a hexagon is, but hardly what a chiliagon, or figure of 1000 sides, is.
We could not tell the difference by sight of a figure of 1000 sides and a figure of 1001 sides. Nor can we imagine any such figure completely before the mind. It is known to us only by name or symbolically. All large numbers, such as those which state the velocity of light (186,000 miles per second), the distance of the sun (91,000,000 miles), and the like, are known to us only by symbols, and they are beyond our powers of imagination.
Infinity is known in a similar way, so that we can in an intellectual manner become acquainted with that of which our senses could never inform us. We speak also
of nothing, of zero, of that which is self-contradictory, of the non-existent, or even of the unthinkable or inconceivable, although the words denote what can never be realized in the mind and still less be perceived through the senses intuitively, but can only be treated in a merely symbolical way.
In arithmetic and algebra we are chiefly occupied with symbolical knowledge only, since it is not necessary in working a long arithmetical question or an algebraical problem that we should realise to ourselves at each step the meaning of the numbers and symbols. We learn from algebra that if we multiply together the sum and difference of two quantities we get the difference of the squares; as, in symbols
(a + b)(a - b) = 02-82; which is readily seen to be true, as follows:
a + 6
In the above we act darkly or symbolically, using the letters a and b according to certain fixed rules, without knowing or caring what they mean; and whatever meaning we afterwards give to a and b we may be sure the process holds good, and that the conclusion is true without going over the steps again.
But in geometry, we argue by intuitive perception of the truth of each step, because we actually employ a representation in the mind of the figures in question, and satisfy ourselves that the requisite properties are really possessed by the figures. Thus the algebraical truth shown above in symbols may be easily proved to hold true
of lines and rectangles contained under those lines, as a corollary of the 5th Prop. of Euclid's Second Book.
Much might be said concerning the comparative advantages of the intuitive and symbolical methods. The latter is usually much the less laborious, and gives the most widely applicable answers; but the symbolical seldom or never gives the same command and comprehension of the subject as the intuitive method. Hence the study of geometry is always indispensable in education, although the same truths are often more readily proved by algebra. It is the peculiar glory of Newton that he was able to explain the motions of the heavenly bodies by the geometric or intuitive method; whereas the greatest of his successors, such as Lagrange or Laplace, have treated these motions by the aid of symbols.
What is true of mathematical subjects may be applied to all kinds of reasoning; for words are symbols as much as A, B, C, or x, y, z, and it is possible to argue with words without any consciousness of their meaning. Thus if I.say that “selenium is a dyad element, and a dyad element is one capable of replacing two equivalents of hydrogen,” no one ignorant of chemistry will be able to attach any meaning to these terms, and yet any one will be able to conclude that “selenium is capable of replacing two equivalents of hydrogen.” Such a person argues in a purely symbolical manner. Similarly, whenever in common life we use words, without having in mind at the moment the full and precise meaning of the words, we possess symbolical knowledge only.
There is no worse habit for a student or reader to acquire than that of accepting words instead of a knowledge of things. It is perhaps worse than useless to read a work on natural history about Infusoria, Foraminifera, Rotifera and the like, if these names do not convey clear images to the mind. Nor can a student who has not
witnessed experiments, and examined the substances with
Hamilton's Lectures on Logic. Lect. IX.
A TERM standing alone is not capable of expressing truth; it merely refers the mind to some object or class of objects, about which something inay be affirmed or denied, but about which the term itself does not affirm or deny anything. “Sun," "air,” “table,” suggest to every mind objects of thought, but we cannot say
sun is true," or “air is mistaken,” or “table is false.” We must join words or terms into sentences or propositions before they can express those reasoning actions of the mind to which
truth or falsity may be attributed. “The sun is bright," “the air is fresh,” “the table is unsteady,” are statements which may be true or may be false, but we can certainly entertain the question of their truth in any circumstances. Now as the logical term was defined to be any combination of words expressing an act of simple apprehension, so a logical proposition is any combination of words expressing an act of judgment. The proposition is in short the result of an act of judgment reduced to the form of language.
What the logician calls a proposition the grammarian calls a sentence. But though every proposition is a sentence, it is not to be supposed that every sentence is a proposition. There are in fact several kinds of sentences. more or less distinct from a proposition, such as a Sentence Interrogative or Question, a Sentence Imperative or a Command, a Sentence Optative, which expresses a wish, and an Exclamatory Sentence, which expresses an emotion of wonder or surprise. These kinds of sentence may possibly be reduced, by a more or less indirect mode of expression, to the form of a Sentence Indicative, which is the grammatical name for a proposition; but until this be done they have no proper place in Logic, or at least no place which logicians have hitherto sufficiently explained.
The name proposition is derived from the Latin words pro, before, and pono, I place, and means the laying or placing before any person the result of an act of judgment. Now every act of judgment or comparison must involve the two things brought into comparison, and every proposition will naturally consist of three partsthe two terms or names denoting the things compared, and the copula or verb indicating the connection between them, as it was ascertained in the act of judgment. Thus the proposition, "Gold is a yellow substance," expresses