Logica e realta' secondo george boole

A century and a quarter ago, a man died who, according to Bertrand Russell, discovered pure mathematics. Better known perhaps for the algebras named after him, or as the creator of modem logic, George Boole was one of the most influential thinkers of the 19th century. His celebrated treatise, The Laws of Thought, published in 1854, opened up new possibilities, not just for logic or mathematics, but also for the yet-to-be-born science of electronic computing. What has been the contribution of Boole's legacy -his ideas, methods and discoveries- to contemporary science and technology? Let us begin with a quick tour of his masterpiece, whose full title is An Investigation of the Laws of Thought on which are founded the Mathematical Theories of Logic and Probabilities.

From the opening paragraph, Boole states the ambitious design of his work: "to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method.'. In the course of these inquiries, he also intends to collect "some probable intimations concerning the nature and constitution of the human mind".

For a subject that is essentially mathematical, Boole employs remarkably little technical jargon. He seems to write for the cultivated intellect, not just the experts, quoting poets and philosophers, often in the original Greek or Latin, and occasionally shifting from logic to psychology, sociology or metaphysics.

According to Boole, the operations of the mind engaged in reasoning are ruled by certain "algebraic" laws, analogous to the laws of the familiar operations on numerical quantities (addition, multiplication and so on). From these fundamental laws, which he expresses using mathematical symbols, he constructs a method for solving problems in logic. First, the hypotheses, or initial assumptions, are put in the form of equations. Then, algebraic manipulations of the symbols replace the usual process of logical deduction. Thus, reasoning is performed by 'calculating' or, to put it bluntly, logic is reduced to algebra.

To be sure, what Boole understood by "logic" is only what we would call today the algebra of classes and propositional logic. But he must be credited with a radically new approach, from which modem mathematical logic emerged. In his algebra of classes, collections of things are represented by single letters. For example, a may represent the class of all persons "ill with AIDS"; h the class of "healthy" people; v the class of those carrying the AIDS virus (or HIV-positive); and b those persons who came in contact with blood. The universal class, or class of all things under discussion, is denoted 1 (in our case, 1 = "people"). The symbol 0 stands for the empty class or class with no members. Classes obtained out of other classes by certain operations are represented by algebraic expressions. Thus, ab is the class of all things that are members of both classes a and b (those ill with AIDS who have been in contact with blood) and 1-v is the class of things under discussion that are not members of v (the HIV-negative class). In case two classes have no members in common, you can form their "sum" h+a, which stands for the class of all persons who are either healthy or ill with AIDS. Relations among classes are then written as equations.

A logical view of AIDS

To illustrate this, assume that the AIDS virus (HIV) may be transmitted only through sexual intercourse or by coming in contact with blood. Also, let us say that someone is "ill with AIDS" if, in addition to carrying the my, the person presents what might be called the "AIDS symptoms". These relations are translated into the two equations:

v(1-(x+(1-x)b)=0

a=vs

where x and s are the classes of those who had sexual intercourse and those who have the AIDS symptoms, respectively. From the assumed relations among the classes, other relations follow "logically", in other words, are implied by them. Boole's method provides a series of rules for "solving" the system of equations and interpreting the solution so as to yield the implied relations. For example, solving the above system for the class (1-s)+sx, would result in the final equation:

(1-s)+sx=a(1-b)+(1-a)v+(0/0)ab+(0/0)(1-v)

whose interpretation is: the class of all those individuals who either do not have the AIDS symptoms or did not refrain from sex (or both) consists of all those ill with AIDS who did not come in contact with blood; all HIV-positives who do not have AIDS; an indefinite part -some, none or all- of those ill with AIDS who have been in contact with blood and an indefinite part of the HIV-negative class. This constitutes the complete relation among the given classes that is implicit in the assumptions.