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

equation of thought is of the second degree, that we perform the operation of analysis
and classification, by division into pairs of opposites, or, as it is technically said,
by dichotomy” (Boole, 2003, pp. 50–51). Still, this dichotomy was not to be exclusively
interpreted in terms of truth or falsehood, though Boole exploited this representation
in his treatment of secondary propositions. Boole typically used 0 to
represent the empty set and 1 to represent the universal set; the expression 1 – x
elected those entities that did not belong to x.
Boole’s operations on symbols were purely formal. That is, the actions of his
logical rules were independent of any semantic interpretation of the logical terms
being manipulated.
We may in fact lay aside the logical interpretation of the symbols in the given equation;
convert them into quantitative symbols, susceptible only of the values 0 and
1; perform upon them as such all the requisite processes of solution; and finally
restore to them their logical interpretation. (Boole, 2003, p. 70)
This formal approach is evident in Boole’s analysis of his fundamental law. Beginning
with x 2 = x, Boole applied basic algebra to convert this expression into x – x 2 = 0.
He then simplified this expression to x(1 – x) = 0. Note that none of these steps
are logical in nature; Boole would not be able to provide a logical justification for
his derivation. However, he did triumphantly provide a logical interpretation of his
result: 0 is the empty set, 1 the universal set, x is some set of interest, and 1 – x is the
negation of this set. Boole’s algebraic derivation thus shows that the intersection of
x with its negation is the empty set. Boole noted that, in terms of logic, the equation
x(1 – x) = 0 expressed,
that it is impossible for a being to possess a quality and not to possess that quality at
the same time. But this is identically that ‘principle of contradiction’ which Aristotle
has described as the fundamental axiom of all philosophy. (Boole, 2003, p. 49)
It was important for Boole to link his calculus to Aristotle, because Boole not only
held Aristotelian logic in high regard, but also hoped that his new mathematical
methods would both support Aristotle’s key logical achievements as well as extend
Aristotle’s work in new directions. To further link his formalism to Aristotle’s logic,
Boole applied his methods to what he called secondary propositions. A secondary
proposition was a statement about a proposition that could be either true or false.
As a result, Boole’s analysis of secondary propositions provides another glimpse of
how his work is related to our modern binary interpretation of it.
Boole applied his algebra of sets to secondary propositions by adopting a temporal
interpretation of election. That is, Boole considered that a secondary proposition
could be true or false for some duration of interest. The expression xy would
now be interpreted as electing a temporal period during which both propositions
x and y are true. The symbols 0 and 1 were also given temporal interpretations,
24 Chapter 2