Mind, Body, World- Foundations of Cognitive Science, 2013a

In this chapter, I provide an historical view of logicism and computing to introduce
these multiple vocabularies, describe their differences, and explain why all
are needed. We begin with the logicism of George Boole, which, when transformed
into modern binary logic, defined the fundamental operations of modern digital
computers.
2.2 From the Laws of Thought to Binary Logic
In 1854, with the publication of An Investigation of the Laws of Thought, George
Boole (2003) invented modern mathematical logic. Boole’s goal was to move the
study of thought from the domain of philosophy into the domain of mathematics:
There is not only a close analogy between the operations of the mind in general
reasoning and its operations in the particular science of Algebra, but there is to
a considerable extent an exact agreement in the laws by which the two classes of
operations are conducted. (Boole, 2003, p. 6)
Today we associate Boole’s name with the logic underlying digital computers
(Mendelson, 1970). However, Boole’s algebra bears little resemblance to our modern
interpretation of it. The purpose of this section is to trace the trajectory that takes
us from Boole’s nineteenth-century calculus to the twentieth-century invention of
truth tables that define logical functions over two binary inputs.
Boole did not create a binary logic; instead he developed an algebra of sets.
Boole used symbols such as x, y, and z to represent classes of entities. He then
defined “signs of operation, as +, –, ´, standing for those operations of the mind
by which the conceptions of things are combined or resolved so as to form new
conceptions involving the same elements” (Boole, 2003, p. 27). The operations of his
algebra were those of election: they selected subsets of entities from various classes
of interest (Lewis, 1918).
For example, consider two classes: x (e.g., “black things”) and y (e.g., “birds”).
Boole’s expression x + y performs an “exclusive or” of the two constituent classes,
electing the entities that were “black things,” or were “birds,” but not those that were
“black birds.”
Elements of Boole’s algebra pointed in the direction of our more modern binary
logic. For instance, Boole used multiplication to elect entities that shared properties
defined by separate classes. So, continuing our example, the set of “black birds”
would be elected by the expression xy. Boole also recognized that if one multiplied
a class with itself, the result would simply be the original set again. Boole wrote his
fundamental law of thought as xx = x, which can also be expressed as x 2 = x. He
realized that if one assigned numerical quantities to x, then this law would only be
true for the values 0 and 1. “Thus it is a consequence of the fact that the fundamental
Multiple Levels of Investigation 23