542 19. LOGIC AND SET THEORYx^2 = ÷ ÷(1 — ÷) = 0, that is, no object can have a property and simultaneously
not have that property.^3
Boole was carried away by his algebraic analogies. Although he remained within
the confines of his initial principles for a considerable distance, when he got to
Chapter 5 he introduced the concept of developing a function. That is, for each
algebraic expression f(x), no matter how complicated, to find an equivalent linear
expression ax + 6(1 — x), one that would have the same values as f(x) for ÷ = 0
and i = l. That expression would obviously be f(l)x + /(0)(1 - x). Boole gave a
convoluted footnote to explain this simple fact by deriving it from Taylor's theorem
and the idempotence property.
Like De Morgan, after discussing his 0-1 logic, Boole then turned to philosophy,
metaphysics, and probability, placing himself in the philosophical camp of Poisson
and de Morgan. He gave the now-familiar rule for the conditional probability of A
given  as the probability of both A and  divided by the probability of B. He also
gave a formal definition of independence, saying that two events are independent if
"the probability of the happening of either of them is unaffected by our expectation
of the occurrence or failure of the other." All of this was done in words, but could
have been done symbolically, as he surely realized.
Application to jurisprudence. Nearly all of the early writers on probability, statis-
tics, and logic had certain applications in mind, to insurance in the case of statistics,
especially to the decisions of courts. The question of the believability of witnesses
and the probability that a jury has been deceived interested Laplace, Quetelet,
Poisson, de Morgan, and Boole, among others. Boole, for example, gave as an
example, the following problem:The probability that a witness A speaks the truth is p, the probabil-
ity that another witness  speaks the truth is q, and the probability
that they disagree in a statement is r. What is the probability that
if they agree, their statement is true?Boole gave the answer as (p + q — r)/(2(l — r)). He claimed to prove as a
theorem the following proposition:From the records of the decisions of a court or deliberative assem-
bly, it is not possible to deduce any definite conclusion respecting
the correctness of the individual judgments of its members.1.5. Venn. It is interesting to compare the mathematization of logic with the
mathematization of probability. Both have ultimately been successful, but both
were resisted to some extent as an intrusion of mathematics into areas of philosophy
where it had no legitimate business. The case for removing mathematics from
philosophy was made by John Venn, whose name is associated with a common
tool of set theory: Venn diagrams, so-called, although the idea really goes back
to Euler. In his book The Logic of Chance, which was first published in 1867,
then revised a decade later and revised once again after another decade, Venn
(^3) Nowadays, a ring in which every element is idempotent, that is, the law ÷ (^2) = ÷ holds, is called
a Boolean ring. It is an interesting exercise to show that such a ring is always commutative and
of characteristic 2, that is, ÷ + ÷ = 0 for all x. The subsets of a given set form a Boolean ring
when addition is interpreted as symmetric difference, that is, A+B means "either A or  but
not both."