To this method it may be objected, that the claim to determine Material Truth takes for granted that the order of Nature will remain unchanged, that (for example) water not only at present is a liquid at 50° Fahrenheit, but will always be so; whereas (although we have no reason to expect such a thing) the order of Nature may alter—it is at least supposable—and in that event water may freeze at such a temperature. Any matter of fact, again, must depend on observation, either directly, or by inference—as when something is asserted about atoms or ether. But observation and material inference are subject to the limitations of our faculties; and however we may aid observation by microscopes and micrometers, it is still observation; and however we may correct our observations by repetition, comparison and refined mathematical methods of making allowances, the correction of error is only an approximation to accuracy. Outside of Formal Reasoning, suspense of judgment is your only attitude.
But such objections imply that nothing short of absolute truth has any value; that all our discussions and investigations in science or social affairs are without logical criteria; that Logic must be confined to symbols, and considered entirely as mental gymnastics. In this book prominence will be given to the character of Logic as a formal science, and it will also be shown that Induction itself may be treated formally; but it will be assumed that logical forms are valuable as representing the actual relations of natural and social phenomena.
§ 7. Symbols are often used in Logic instead of concrete terms, not only in Symbolic Logic where the science is treated algebraically (as by Dr. Venn in his Symbolic Logic), but in ordinary manuals; so that it may be well to explain the use of them before going further.
It is a common and convenient practice to illustrate logical doctrines by examples: to show what is meant by a Proposition we may give salt is soluble, or water rusts iron: the copulative exponible is exemplified by salt is savoury and wholesome; and so on. But this procedure has some disadvantages: it is often cumbrous; and it may distract the reader's attention from the point to be explained by exciting his interest in the special fact of the illustration. Clearly, too, so far as Logic is formal, no particular matter of fact can adequately illustrate any of its doctrines. Accordingly, writers on Logic employ letters of the alphabet instead of concrete terms, (say) X instead of salt or instead of iron, and (say) Y instead of soluble or instead of rusted by water; and then a proposition may be represented by X is Y. It is still more usual to represent a proposition by S is (or is not) P, S being the initial of Subject and P of Predicate; though this has the drawback that if we argue—S is P, therefore P is S, the symbols in the latter proposition no longer have the same significance, since the former subject is now the predicate.
Again, negative terms frequently occur in Logic, such as not-water, or not-iron, and then if water or iron be expressed by X, the corresponding negative may be expressed by x; or, generally, if a capital letter stand for a positive term, the corresponding small letter represents the negative. The same device may be adopted to express contradictory terms: either of them being X, the other is x (see chap. iv., §§ [7]-[8]); or the contradictory terms may be expressed by x and x̄, y and ȳ.
And as terms are often compounded, it may be convenient to express them by a combination of letters: instead of illustrating such a case by boiling water or water that is boiling, we may write XY; or since positive and negative terms may be compounded, instead of illustrating this by water that is not boiling, we may write Xy.
The convenience of this is obvious; but it is more than convenient; for, if one of the uses of Logic be to discipline the power of abstract thought, this can be done far more effectually by symbolic than by concrete examples; and if such discipline were the only use of Logic it might be best to discard concrete illustrations altogether, at least in advanced text-books, though no doubt the practice would be too severe for elementary manuals. On the other hand, to show the practical applicability of Logic to the arguments and proofs of actual life, or even of the concrete sciences, merely symbolic illustration may be not only useless but even misleading. When we speak of politics, or poetry, or species, or the weather, the terms that must be used can rarely have the distinctness and isolation of X and Y; so that the perfunctory use of symbolic illustration makes argument and proof appear to be much simpler and easier matters than they really are. Our belief in any proposition never rests on the proposition itself, nor merely upon one or two others, but upon the immense background of our general knowledge and beliefs, full of circumstances and analogies, in relation to which alone any given proposition is intelligible. Indeed, for this reason, it is impossible to illustrate Logic sufficiently: the reader who is in earnest about the cogency of arguments and the limitation of proofs, and is scrupulous as to the degrees of assent that they require, must constantly look for illustrations in his own knowledge and experience and rely at last upon his own sagacity.