538 19. LOGIC AND SET THEORY
De Morgan's notation in this work was not the best, and very little of it has
caught on. He used a parenthesis in roughly the same way as the modern notation
for implication. For example, X) Y denoted the proposition "Every X is a Y."
Nowadays we would write X D Y (read "X horseshoe V") for "X implies Y."
The rest of his notation--X : Y for "Some X's are not Ya," X.Y for "No X's
are Ys" and × Y for "Some X's are Ys"—is no longer used. For the negation of
these properties he used lowercase letters, so that ÷ denoted not-X. De Morgan
introduced the useful "necessary" and "sufficient" language into implications: X) Y
meant that Y was necessary for X and X was sufficient for Y. He gave a table
of the relations between I or é and Y or y for the relations X) Y, X.Y, Õ) X,
and ÷. y. For example, given that X implies Y, he noted that this relation made
Y necessary for X, y an impossible condition for X, y a sufficient condition for x,
and Y a contingent (not necessary, not sufficient, not impossible) condition for x.
For compound propositions, he wrote PQ for conjunction (his word), meaning
both Ñ and Q are asserted, and Ñ, Q for disjunction (again, his word), meaning
either Ñ or Q. He then noted what are still known as De Morgan's laws:
The contrary of PQ is ñ, q. Not both is either not one or not the
other, or not either. Not either Ñ nor Q (which we might denote
by : Ñ, Q or .P, Q) is logically 'not Ñ and not Q' or pq: and this
is then the contrary of Ñ, Q.
De Morgan's theory of probability. De Morgan devoted three chapters (Chapters 9
through 11) to probability and induction, starting off with a very Cartesian princi-
ple:
That which we know, of which we are certain, of which we are well
assured nothing could persuade us to the contrary, is the existence
of our own minds, thoughts, and perceptions.
He then took the classical example of a certain proposition, namely that 2+2 =
4 and showed by analyzing the meaning of 2, 4, and + that "It is true, no doubt,
that 'two and two' is four, in amount, value, &c. but not in form, construction,
definition, &C."^1 He continued:
There is no further use in drawing distinction between the knowlege
which we have of our own existence, and that of two and two
amounting to four. This absolute and inassailable feeling we shall
call certainty. We have lower grades of knowledge, which we usually
call degrees of belief, but they are really degrees of knowledge... It
may seem a strange thing to treat knowledge as a magnitude, in
the same manner as length, or weight, or surface. This is what all
writers do who treat of probability... But it is not customary to
make the statement so openly as I now do.
As this passage shows, for de Morgan probability was a subjective entity. He
said that degree of probability meant degree of belief. In this way he placed himself
firmly against the frequentist position, saying, "I throw away objective probability
(^1) There is some intellectual sleight-of-hand here. The effectiveness of this argument depends on
the reader's not knowing—and de Morgan's not stating—what is meant by addition of integers
and by equality of integers, so that 2 + 2 cannot be broken down into any terms simpler than
itself. It really can be proved that 2 + 2 = 4.