PROBABLE INFERENCE TO PARTICULARS--THE MEASUREMENT OF PROBABILITY.
Undoubtedly there are degrees of probability. Not only do we expect
some events with more confidence than others: we may do so, and our
confidence may be misplaced: but we have reason to expect some with
more confidence than others. There are different degrees of rational
expectation. Can those degrees be measured numerically?
The question has come into Logic from the mathematicians. The
calculation of Probabilities is a branch of Mathematics. We have seen
how it may be applied to guide investigation by eliminating what is
due to chance, and it has been vaguely conceived by logicians that
what is called the calculus of probabilities might be found useful
also in determining by exact numerical measurement the probability
of single events. Dr. Venn, who has written a separate treatise on the
Logic of Chance, mentions "accurate quantitative apportionment of our
belief" as one of the goals which Logic should strive to attain. The
following passage will show his drift.[1]
A man in good health would doubtless like to know whether he
will be alive this time next year. The fact will be settled
one way or the other in due time, if he can afford to wait,
but if he wants a present decision, Statistics and the Theory
of Probability can alone give him any information. He learns
that the odds are, say five to one that he will survive, and
this is an answer to his question as far as any answer can be
given. Statisticians are gradually accumulating a vast mass of
data of this general character. What they may be said to aim
at is to place us in the position of being able to say, in any
given time or place, what are the odds for or against any at
present indeterminable fact which belongs to a class admitting
of statistical treatment.
Again, outside the regions of statistics proper--which
deal, broadly speaking, with events which can be numbered or
measured, and which occur with some frequency--there is still
a large field as to which some better approach to a reasoned
intensity of belief can be acquired. What will be the issue of
a coming war? Which party will win in the next election? Will
a patient in the crisis of a given disease recover or not?
That statistics are lying here in the background, and are thus
indirectly efficient in producing and graduating our belief, I
fully hold; but there is such a large intermediate process
of estimating, and such scope for the exercise of a practised
judgment, that no direct appeal to statistics in the common
sense can directly help us. In sketching out therefore the
claims of an Ideal condition of knowledge, we ought clearly to
include a due apportionment of belief to every event of such
a class as this. It is an obvious defect that one man should
regard as almost certain what another man regards as almost
impossible. Short, therefore, of certain prevision of the
future, we want complete agreement as to the degree of
probability of every future event: and for that matter of
every past event as well.
Technically speaking, if we extend the name Modality (see p. 78) to
any qualification of the certainty of a statement of belief, what Dr.
Venn here desiderates, as he has himself suggested, is a more exact
measurement of the Modality of propositions. We speak of things as
being certain, possible, impossible, probable, extremely probable,
faintly probable, and so forth: taking certainty as the highest
degree of probability[2] shading gradually down to the zero of
the impossible, can we obtain an exact numerical measure for the
gradations of assurance?
To examine the principles of all the cases in which chances for and
against an occurrence have been calculated from real or hypothetical
data, would be to trespass into the province of Mathematics, but a few
simple cases will serve to show what it is that the calculus attempts
to measure, and what is the practical value of the measurement as
applied to the probability of a single event.
Suppose there are 100 balls in a box, 30 white and 70 black, all being
alike except in respect of colour, we say that the chances of drawing
a black ball as against a white are as 7 to 3, and the probability of
drawing black is measured by the fraction 7/10. In believing this we
proceed on the principle already explained (p. 356) of Proportional
Chances. We do not know for certain whether black or white will
emerge, but knowing the antecedent situation we expect black rather
than white with a degree of assurance corresponding to the proportions
of the two in the box. It is our degree of rational assurance that
we measure by this fraction, and the rationality of it depends on the
objective condition of the facts, and is the same for all men, however
much their actual degree of confidence may vary with individual
temperament. That black will be drawn seven times out of every ten
on an average if we go on drawing to infinity, is as certain as any
empirical law: it is the probability of a single draw that we measure
by the fraction 7/10.
When we build expectations of single events on statistics of observed
proportions of events of that kind, it is ultimately on the same
principle that rational expectation rests. That the proportion will
obtain on the average we regard as certain: the ratio of favourable
cases to the whole number of possible alternatives is the measure
of rational expectation or probability in regard to a particular
occurrence. If every year five per cent. of the children of a town
stray from their guardians, the probability of this or that child's
going astray is 1/20. The ratio is a correct measure only on the
assumption that the average is maintained from year to year.
Without going into the combination of probabilities, we are now in a
position to see the practical value of such a calculus as applied to
particular cases. There has been some misunderstanding among logicians
on the point. Mr. Jevons rebuked Mill for speaking disrespectfully
of the calculus, eulogised it as one of the noblest creations of the
human intellect, and quoted Butler's saying that "Probability is the
guide of life". But when Butler uttered this famous saying he was
probably not thinking of the mathematical calculus of probabilities
as applied to particular cases, and it was this special application to
which Mill attached comparatively little value.
The truth is that we seldom calculate or have any occasion to
calculate individual chances except as a matter of curiosity. It is
true that insurance offices calculate probabilities, but it is not the
probability of this or that man dying at a particular age. The precise
shade of probability for the individual, in so far as this depends on
vital statistics, is a matter of indifference to the company as long
as the average is maintained. Our expectations about any individual
life cannot be measured by a calculation of the chances because a
variety of other elements affect those expectations. We form beliefs
about individual cases, but we try to get surer grounds for them than
the chances as calculable from statistical data. Suppose a person were
to institute a home for lost dogs, he would doubtless try to ascertain
how many dogs were likely to go astray, and in so doing would be
guided by statistics. But in judging of the probability of the
straying of a particular dog, he would pay little heed to statistics
as determining the chances, but would proceed upon empirical knowledge
of the character of the dog and his master. Even in betting on the
field against a particular horse, the bookmaker does not calculate
from numerical data such as the number of horses entered or the
number of times the favourite has been beaten: he tries to get at
the pedigree and previous performances of the various horses in the
running. We proceed by calculation of chances only when we cannot do
better.
[Footnote 1: _Empirical Logic_, p. 556.]
[Footnote 2: Mr. Jevons held that all inference is merely
probable and that no inference is certain. But this is a
purposeless repudiation of common meaning, which he cannot
himself consistently adhere to. We find him saying that if a
penny is tossed into the air it will _certainly_ come down
on one side or the other, on which side being a matter of
probability. In common speech probability is applied to a
degree of belief short of certainty, but to say that certainty
is the highest degree of probability does no violence to the
common meaning.]