In the preceding chapters we have considered (_i_) what thinking is; (_ii_) the importance of its special training; (_iii_) the natural tendencies that lend themselves to its training; and (_iv_) some of the special obstacles in the way of its training under school conditions. We come now to the relation of _logic_ to the purpose of mental training. [Sidenote: Three senses of term _logical_] [Sidenote: The practical is the important meaning of _logical_] In its broadest sense, any thinking that ends in a conclusion is logical--whether the conclusion reached be justified or fallacious; that is, the term _logical_ covers both the logically good and the illogical or the logically bad. In its narrowest sense, the term _logical_ refers only to what is demonstrated to follow necessarily from premises that are definite in meaning and that are either self-evidently true, or that have been previously proved to be true. Stringency of proof is here the equivalent of the logical. In this sense mathematics and formal logic (perhaps as a branch of mathematics) alone are strictly logical. Logical, however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions. If only the word _artificial_ were associated with the idea of _art_, or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that logical refers to artificial thought. [Sidenote: Care, thoroughness, and exactness the marks of the logical]