La percée due à Boole

George Peacock

[Peacock 1833, p. 194]
A : Whatever form is algebraically equivalent to another when expressed in general symbol, must continue to be equivalent whatever those symbols denote. B : Converse Proposition : Whatever equivalent form, is discoverable in arithmetical algebra considered as the science of suggestion, when the symbols are general in their form, though specifics in their value, will continue to be an equivalent form when the symbols are general in their nature as well as their form.

[Peacock1833, p. 196-197]
The operations called addition and subtraction are denoted by the signs + and - ; They are the inverse of each other. (. . . ) The operations called multiplication and division are denoted by the signs x and ÷, or more frequently by a conventional position of the quantities or symbols which respect to each other.(. . . ) The operations called multiplication and division are the inverse of each other.

Dunkan F. Gregory

[Gregory 1839, p. 34]
Whatever is proved of the latter symbols, from the known laws of their combination, must be equally true of all other symbols which are subjects to the same laws of combination.

Richard Whately

[Whately 1832, p. xvii]
The truth is, that a very small proportion, even of distinguish students, ever become proficient in logic.; and that by far the greater part pass through the University without knowing anything at all of the subject. I do not mean that they have not learned by rote a string of technical terms; but that they understand absolutely nothing whatever of the principles of the science.

George Bentham

[Bentham 1827, p. 149]
( & ) consists of two substantive terms with their respective signs of extend, and of a copula, expressive of the identity or diversity of those terms.

[Bentham 1827, p. 165]
Thus, in the proposition "Plants are not animals", not must be considered as attached to the copula; for, if we were, by what is called conversio per accidens, to attach the not to animals, the term not-animals becomes adjective (. . . ) and the proposition thus reduced becomes : "Every plant is a being-not-animal". All negative propositions might thus be reduced to affirmatives; but it appears to be a useless complication.

George Boole

[Boole 1847, p. 3] Who are acquainted with the present state of the theory of Symbolical Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relation supposed, is equally admissible.

[Boole 1847, p. 4]
It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical analysis.

[Boole 1854, p. 12]
That Logic, as a science, is susceptible of very wide applications is admitted; but it is equally certain that its ultimate forms and processes are mathematical.

[Boole 1854, p. 49]
That axiom of Metaphysicians which is termed the principle of contradiction and which affirms that it is impossible for anything to possess a quality, and in the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x²=x.

[Boole 1854, p. 37-38]
Let us conceive, then, of an algebra in which the symbols x, y z etc. admit indifferently of the values 0 and 1, and of these values alone The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extend with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.

[Boole 1854, p. 165]
Let x represent an act of the mind by which we fix our regard upon that portion of time for which the proposition X is true ; and let this meaning be understood when it is asserted that x denote the time for which the proposition X is true. (. . .) We shall term x the representative symbol of the proposition X.

[Boole 1854, p. 248]
Thus, instead of considering the numerical fraction p as expressing the probability of the occurrence of an event E, let it be viewed as representing the probability of the truth of the proposition X, which asserts that the event E will occur.

[Boole 1854, p. 12]
It is not of the essence of mathematics to be conversant with the ideas of numbers and quantity.

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