The educational theory that geometry is exceptionally good training for the reason—apart from its practical utility in mechanics—is thus evidently a mistake. Abstract geometry may induce habits of minute observation and exact definition, but reason nowhere enters into the study. As a rational gymnastic there is nothing better than the game of chess.
XXXV—INDUCTION
Those who contend that there is a kind of argument called Inductive different from the Deductive, illustrate their view by some such example as the following:—'This, that, and the other magnet' [that is, all the magnets we know] 'attract iron; therefore all possible magnets attract iron.' They say there is an irresistible compulsion in the mind to draw such a conclusion from information of the kind exemplified, and they contrast that type of thought with a deductive argument like—'All magnets attract iron; this object is a magnet; therefore it attracts iron.' They figure the former as a progress upwards, the latter as a regress downwards.
That is Induction as understood by J. S. Mill and Sir William Hamilton; on this point these philosophers happen to agree.
The first of those arguments is a deduction with the precedent omitted. Expressed in full it amounts to this—'Any relation observed several times to subsist between two classes of objects, and concerning which no exception has ever been observed, may be taken as universal; there is such a relation between known magnets and known iron; therefore it may be regarded as universal.' The precedent is not a mental compulsion, but a result of experience. Induction as above defined is therefore only a species of deductive conclusions.
Most logicians take the word Induction in its etymological sense, as meaning systematic observation carried on with a view to obtaining a general idea of some class of objects; or of establishing a categorical relation between one object or class and another, by eliminating all the alternative correlatives. In neither operation would Induction be argument.
In science a 'perfect induction' is one in which all existing objects of a class, or all objects related in a certain manner, have been perceived, so that there is no other object concerning which a conclusion can be drawn. In such cases, says Mill, there is no induction—only a summary of experience. He evidently regarded the conclusion with respect to unknown cases as the essence of induction, whereas in the scientific sense the induction is the positive content of the idea, or the abstract relation—the unknown cases are ignored, or there may be none. In scientific writings induction sometimes means the method of observation rather than the result—the method of correcting inferences by perception, wherever possible.
XXXVI—ARISTOTLE'S DICTUM
This is usually put into English thus—'Whatever is affirmed or denied of a class, may be affirmed or denied of any part of that class,' and such an affirmation or denial is supposed to be an act of reason. Archbishop Whately expounds the Dictum in analysing the following theorem—Whatever exhibits marks of design had an intelligent author; the world exhibits marks of design; therefore the world had an intelligent author.