Personality Profile #1 | George Boole

Boole and the Algebra of Logic:

George Boole (1815–1864) was a British mathematician and he’s considered to be the inventor of The algebra of logic, which is a system showing the underlying mathematical structure of logic.

Boole proposed that logical propositions should be expressed as algebraic equations. The algebraic manipulation of the symbols in the equations provides a fail-safe method of logical deduction, i.e. logic can be reduced to algebra. He replaced the operation of multiplication by the word AND and addition by the word OR. The symbols in the equations can stand for collections of objects (sets) or statements in logic. For example: if x is the set of all green cars and y is the set of all big cars, then x+y is the set of all cars that are green or big, and xy is the set of all cars that are green and big.

In 1847, boole put his ideas into an essay, called Mathematical Analysis of Logic. It was the ground-breaking work that laid the foundations for what is known today as Boolean algebra and the propositional calculus. It not only expanded on Gottfried Leibniz's earlier theory on the correlation between logic and math, but rather suggested that logic was principally a discipline of mathematics, rather than philosophy.

Boole’s view of logic is clearly stated in the opening section of his Mathematical Analysis of Logic:

"Those who are acquainted with the present state of the theory of Symbolical Algebra, are aware that the validity of the processes of anaIysis 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 relations supposed, is equally admissible, and it is thus that the same processes may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics.... We might justly assign it as the definitive character of a true Calculus, that it is a method resting upon the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation. That to the existing forms of Analysis a quantitative interpretation is assigned, is the result of circumstances by which those forms were determined, and is not to be construed into a universal condition of Analysis. 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, regardless that in its objects and in its instruments it must at present stand alone.'

In the Following year (1848), boole published a short account of his Calculus of Logic. It Doesn't add much to the

theory of his Mathematical Analysis of Logic, but it is somewhat easier to follow, and it ends with a paragraph which clearly summarizes Boole's position:

"The view which these enquiries present of the nature of language is a very interesting one. They exhibit it not as a mere collection of signs, but as a system of expressions, the elements of which are subject to the laws of the thought which they represent. That these laws are as rigorously mathematical as the laws which govern the purely quantitative conceptions of space and time, of number and magnitude, is a conclusion which I do not hesitate to submit to the exactest scrutiny”.