The Mathematical Analysis of Logic Being an Essay Towards a Calculus of Deductive Reasoning

by George Boole

Mathematics are essentially, as well as actually, the Science of Magnitude.

The mathematics we have to construct are the mathematics of the human intellect.

Logic considered as a Science.∗

Regarding Logic as a branch of Philosophy, and defining Philosophy as the “science of a real existence,” and “the research of causes,” and assigning as its main business the investigation of the “why, (tä dÐoti),” while Mathematics display only the “that, (tä åtÈ),” Sir W. Hamilton has contended, not simply, that the superiority rests with the study of Logic, but that the study of Mathematics is at once dangerous and useless.∗

Philosophy is then described as the science of a real existence and the research of causes.

“The philosopher seeks and knows the causes of natural things, of which the mathematician searches out and computes the numbers and the measures.”

Now we might take our stand upon the conviction of many thoughtful and reflective minds, that in the extent of the meaning above stated, Philosophy is impossible. The business of true Science, they conclude, is with laws and phenomena. The nature of Being, the mode of the operation of Cause, the why, they hold to be beyond the reach of our intelligence.

I desire here to express my conviction, that with the advance of our knowledge of all true science, an ever-increasing harmony will be found to prevail among its separate branches.

To express the class, not-X, that is, the class including all individuals that are not Xs. The class X and the class not-X together make the Universe. But the Universe is 1, and the class X is determined by the symbol x, therefore the class not-X will be determined by the symbol 1 − x. Hence the office of the symbol 1 − x attached to a given subject will be, to select from it all the not-Xs which it contains.

The hypothetical Universe, 1, shall comprehend all conceivable cases and conjunctures of circumstances.