Dealing mainly with the subjects and materials of reasoning, with thoughts rather than with thinking, these writers, with the tendency of specialists, have not appreciated the labors of another school of logicians, who have made the investigation of the process of thinking itself their especial province. This is abstract logic, or pure logic, sometimes called, inasmuch as it deals with forms only, “formal logic,” or because it deals with names and not things, “the logic of names.” It dates its rise as an independent science from the discovery of what is known as “the quantification of the predicate,” claimed by Sir William Hamilton. Of writers upon it may be mentioned Professor De Morgan, W. Stanley Jevons, and especially Professor George Boole of Belfast. The latter, one of the subtlest thinkers of this age, and eminent as a mathematician, succeeded in making an ultimate analysis of the laws of thinking, and in giving them a symbolic notation, by which not only the truth of a simple proposition but the relative degree of truth in complex propositions may be accurately estimated.[24-1] This he did by showing that the laws of correct thinking can be expressed in algebraic notation, and, thus expressed, will be subject to all the mathematical laws of an algebra whose symbols bear the uniform value of unity or nought (1 or 0)--a limitation required by the fact that pure logic deals in notions of quality only, not of quantity. This mathematical form of logic was foreseen by Kant when he declared that all mathematical reasoning derives its validity from the logical laws; but no one before Professor Boole had succeeded in reaching the notation which subordinated these two divisions of abstract thought to the same formal types. His labors have not yet borne fruit in proportion to their value, and they are, I believe, comparatively little known. But in the future they will be regarded as epochal in the science of mind. They make us to see the same law governing mind and matter, thought and extension. Not the least important result thus achieved was in emphasizing the contrast between the natural laws of mental association, and the laws of thinking which are the foundation of the syllogism. By attending to this distinction we are enabled to keep the form and the matter of thought well apart--a neglect to do which, or rather a studied attempt to ignore which, is the radical error of the logic devised by Hegel, as I shall show more fully a little later.