The History of Mathematics: A Brief Course

540 19. LOGIC AND SET THEORY

properties of number, under another that of a geometrical problem,

and under a third that of optics.

Here Boole, like de Morgan, was arguing for the freedom to create abstract

systems and attach an interpretation to them later. This step was still something

of an innovation at the time. It was generally accepted, for example, that irrational

and imaginary numbers had a meaning in geometry but not in arithmetic. One

could not, or should not, simply conjure them into existence. Cayley raised this

objection shortly after the appearance of Boole's treatise (see Grattan-Guinness,

2000, p. 41), asking whether it made any sense to write ^x. Boole replied by

comparing the question to the existence of which he said was "a symbol (i)

which satisfies particular laws, and especially this: i^2 = —1." In other words, when

we are inventing a formal system, we are nearly omnipotent. Whatever we prescribe

will hold for the system we define. If we want a square root of —1 to exist, it will

exist (whatever "exist" may mean).

Logic and classes. Although set theory had different roots on the Continent, we

can see its basic concept—membership in a class—in Boole's work. Departing from

de Morgan's notation, he denoted a generic member of a class by an uppercase X,

and used the lowercase ÷ "operating on any subject," as he said, to denote the

class itself. Then xy was to denote the class "whose members are both X's and

Ys." This language rather blurs the distinction between a set, its members, and the

properties that determine what the members are; but we should expect that clarity

would take some time to achieve. The connection between logic and set theory is an

intimate one and one that is easy to explain. But the kind of set theory that logic

alone would have generated was different from the geometric set theory of Georg

Cantor, which is discussed in the next section.

The influence of the mathematical theory of probability on logic is both exten-

sive and interesting. The subtitle of de Morgan's Formal Logic is The Calculation

of Inference, Necessary and Probable, and, as noted above, three chapters (some

50 pages) of Formal Logic are devoted to probability and induction. Probability

deals with events, whereas logic deals with propositions. The connnection between

the two was stated by Boole in his later treatise, An Investigation of the Laws of

Thought, as follows:

[T]here is another form under which all questions in the theory

of probabilities may be viewed; and this form consists in substi-

tuting for events the propositions which assert that those events

have occurred, or will occur; and viewing the element of numerical

probability as having reference to the truth of those propositions,

not to the occurrence of the events.

Two events can combine in different ways: exactly one of Å and F may occur,
or Å and F may both occur. If the events Å and F are independent, the probability
that both Å and F occur is the product of their individual probabilities. If the two
events cannot both occur, the probability that at least one occurs is the sum of
their individual probabilities. More generally,

P(E or F) + P(£and F) = P(E) + P{F).

When these combinations of events are translated into logical terms, the result is
a logical calculus.