- LOGIC^541
The idea of probability 0 as indicating impossibility and probability 1 as indi-
cating certainty must have had some influence on Boole's use of these symbols to
denote "nothing" and "the universe." He expressed the proposition "all X's are
Y's," for example, as xy = ÷ or i(l — y) = 0. Notice that 1 — y appears, not y — 1,
which would have made no sense. Here 1 — y corresponds to the things that are
not-y. From there, it is not far to thinking of 0 as false and 1 as true. The differ-
ence between probability and logic here is that the probability of an event may be
any number between 0 and 1, while propositions are either true or false.^2 These
analogies were brought out fully in Boole's major work, to which we now turn.
1.4. Boole's Laws of Thought. Six years later, after much reflection on the
symbolic logic that he and others had developed, Boole wrote an extended treatise,
An Investigation of the Laws of Thought, which began by recapping what he had
done earlier. The Laws of Thought began with a very general proposition that laid
out the universe of symbols to be used. These were:
1st. Literal symbols, as x, y, &c, representing things as subjects
of our conceptions.
2nd. Signs of operation, as +, —, x, 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.
3rd. The sign of identity, =.
And these symbols of Logic are in their use subject to definite
laws, partly agreeing with and partly differing from the laws of the
corresponding symbols in the science of Algebra.
Boole used + to represent disjunction (or) and juxtaposition, used in algebra
for multiplication, to represent conjunction (and). The sign — was used to stand for
"and not." In his examples, he used + only when the properties were, as we would
say, disjoint and — only when the property subtracted was, as we would say, a subset
of the property from which it was subtracted. He illustrated the equivalence of
"European men and women" (where the adjective European is intended to apply to
both nouns) with "European men and European women" as the equation z(x + y) =
zx + zy. Similarly, to express the idea that the class of men who are non-Asiatic
and white is the same as the class of white men who are not white Asiatic men, he
wrote z(x — y) = zx — zy. He attached considerable importance to what he was
later to call the index law, which expresses the fact that affirming a property twice
conveys no more information than affirming it once. That is to say, xx = x, and
he adopted the algebraic notation x^2 for xx. This piece of algebraization led him,
by analogy with the rules xO = 0 and xl = x, to conclude that "the respective
interpretations of the symbols 0 and 1 in the system of Logic are Nothing and
Universe.'" From these considerations he deduced the principle of contradiction:(^2) Classical set theory deals with propositions of the form ÷ 6 Å, which are either true or false:
Either ÷ belongs to E, or it does not, and there is no other possibility. The recently created fuzzy
set theory restores the analogy with probability, allowing an element to belong partially to a given
class and expressing the degree of membership by a function ö(÷) whose values are between 0
and 1. Thus, for example, whether a woman is pregnant or not is a classical set-theory question;
whether she is tall or not is a fuzzy set-theory question. Fuzzy-set theorists point out that their
subject is not subsumed by probability, since it deals with the properties of individuals, not those
of large sets.