excerpt from The Industrialization of Intelligence: Mind and Machine in the Modern Age
Whatever one thought of its metaphysical implications, Boolean algebra was an obviously powerful analytical tool in reproducing the results of human reason, and could in the process of replicating reason significantly reduce the effort required in ratiocination. Boole demonstrated the practical utility of his method by replicating in elegant mathematics a taxing logical exercise that had been performed by Aristotle in antiquity. In a lengthy series of arguments that had been of particular interest to the philosophers of the Enlightenment, Aristotle had undertaken better to elucidate the relationship between virtue and other human characteristics such as passion, faculty and habit. Boole began by taking Aristotle’s premises:
[1] Virtue is either a passion, or a faculty, or a habit.
[2] Passions are not things according to which we are praised or blamed, or in which we exercise deliberate judgement.
[3] Faculties are not things according to which we are praised are blamed, and which are accompanied by deliberate preference.
[4] Virtue is something according to which we are praised and blamed, and which is accompanied by deliberate preference.
[5] Whatever art or science makes its works to be in a good state avoids extremes, and keeps the means in view relative to human nature.
[6] Virtue is more exact and excellent than any art or science.
He associated a symbolic variable for each of the classes involved in the premises:
v = virtue
p = passions
f = faculties
h = habits
d = things accompanied by deliberate preference
g = things causing their work to be in a good state
m = things keeping the mean in view relative to human nature
Using the symbols he thus defined, Boole restated Aristotle’s premises in algebraic terms:
[1] v = q{ p(1 – f)(1 – b) + f(1 – p)(1 – h) + h(1 – p)(1 – f) }
[2] p = q(1 – d)
[3] f = (q -1)d
[4] v = qd
[5] g = qm
[6] v = qg
Boole had now captured Aristotle’s observations in the symbols of his algebra. Having done so, he could ignore what the symbols meant, and manipulate these expressions according to the various rules of Boolean algebra. Using his own rules, he was able quickly to reproduce Aristotle’s conclusions. For example, he was able to demonstrate that, given the Boolean assumptions above, the statement:
v = h (1 – f)(1 – p)
was true, which, interpreted according to his original definitions, means ‘Virtue is a habit, and not a faculty or a passion’, or
v = hdm
which means ‘Virtue is a habit accompanied by deliberate preference and keeping in view the mean relative to human nature’. In this way was the wisdom of Aristotle made knowable through the operations reachable to anyone competent in fundamental mathematics. Boolean algebra described a way for symbols to be manipulated in a manner that was wholly unmindful of the symbols’ meaning. In theory, any logical problem could be described as Boolean statements, and a mathematician facile with the laws of Boolean algebra could proceed to derive more general or more detailed statements of truth from the original statements, new statements that could be useful to the person who had originally posed the problem. This recalls Prony’s system of employing the division of mental labor in the computation of logarithms, down to the detail that the mathematician in this hypothetical case, like the simple arithmeticians in Prony‘s case, would not need to know, and in fact would be completely unaided by knowing, the meaning that was attached to the symbols. There were opportunities for real economies along just these lines, as could be seen in the practice of large insurance companies, who used to use Boolean algebra to test the logic of their complex legal contracts. With the dizzying array of situations covered by modern insurance agreements, the only practical way to predict when the insurance company was liable and when it was not was by reducing the terms of the contract to Boolean expressions, and having teams of mathematicians discover the underlying patterns of responsibility the wayt Prony’s team discovered logarithms. Today, of course, this function would be done by electronic computers.
[back to blog post: George Boole, the Holy Trinity, and the Birth of the Computer]
[The Industrialization of Intelligence on Amazon.com]
