OF THE PROBABILITY OF CHANCES.
But in order to bestow on this system its full force and evidence, we
must carry our eye from it a moment to consider its consequences, and
explain, from the same principles, some other species of reasoning
which are derived from the same origin.
Those philosophers who have divided human reason into _knowledge and
probability_, and have defined the first to be _that evidence which
arises from the comparison of ideas_, are obliged to comprehend all our
arguments from causes or effects under the general term of probability.
But though every one be free to use his terms in what sense he pleases;
and accordingly, in the precedent part of this discourse, I have
followed this method of expression; 'tis however certain, that in
common discourse we readily affirm, that many arguments from causation
exceed probability, and may be received as a superior kind of evidence.
One would appear ridiculous who would say, that 'tis only probable the
sun will rise to-morrow, or that all men must die; though 'tis plain we
have no further assurance of these facts than what experience affords
us. For this reason t'would perhaps be more convenient, in order at
once to preserve the common signification of words, and mark the
several degrees of evidence, to distinguish human reason into three
kinds, viz. _that from knowledge, from proofs, and from probabilities_.
By knowledge, I mean the assurance arising from the comparison of
ideas. By proofs, those arguments which are derived from the relation
of cause and effect, and which are entirely free from doubt and
uncertainty. By probability, that evidence which is still attended with
uncertainty. 'Tis this last species of reasoning I proceed to examine.
Probability or reasoning from conjecture may be divided into two kinds,
viz. that which is founded on _chance_, and that which arises from
_causes_. We shall consider each of these in order.
The idea of cause and effect is derived from experience, which,
presenting us with certain objects constantly conjoined with each
other, produces such a habit of surveying them in that relation, that
we cannot, without a sensible violence, survey them in any other. On
the other hand, as chance is nothing real in itself, and, properly
speaking, is merely the negation of a cause, its influence on the mind
is contrary to that of causation; and 'tis essential to it to leave the
imagination perfectly indifferent, either to consider the existence or
non-existence of that object which is regarded as contingent. A cause
traces the way to our thought, and in a manner forces us to survey such
certain objects in such certain relations. Chance can only destroy
this determination of the thought, and leave the mind in its native
situation of indifference; in which, upon the absence of a cause, 'tis
instantly reinstated.
Since, therefore, an entire indifference is essential to chance, no
one chance can possibly be superior to another, otherwise than as it
is composed of a superior number of equal chances. For if we affirm
that one chance can, after any other manner, be superior to another,
we must at the same time affirm, that there is something which gives
it the superiority, and determines the event rather to that side than
the other; that is, in other words, we must allow of a cause, and
destroy the supposition of chance, which we had before established. A
perfect and total indifference is essential to chance, and one total
indifference can never in itself be either superior or inferior to
another. This truth is not peculiar to my system, but is acknowledged
by every one that forms calculations concerning chances.
And here 'tis remarkable, that though chance and causation be directly
contrary, yet 'tis impossible for us to conceive this combination of
chances, which is requisite to render one hazard superior to another,
without supposing a mixture of causes among the chances, and a
conjunction of necessity in some particulars, with a total indifference
in others. Where nothing limits the chances, every notion that the
most extravagant fancy can form is upon a footing of equality; nor can
there be any circumstance to give one the advantage above another.
Thus, unless we allow that there are some causes to make the dice fall,
and preserve their form in their fall, and lie upon some one of their
sides, we can form no calculation concerning the laws of hazard. But
supposing these causes to operate, and supposing likewise all the rest
to be indifferent and to be determined by chance, 'tis easy to arrive
at a notion of a superior combination of chances. A dye that has four
sides marked with a certain number of spots, and only two with another,
affords us an obvious and easy instance of this superiority. The mind
is here limited by the causes to such a precise number and quality of
the events; and, at the same time, is undetermined in its choice of any
particular event.
Proceeding, then, in that reasoning, wherein we have advanced three
steps; _that_ chance is merely the negation of a cause, and produces a
total indifference in the mind; _that_ one negation of a cause and one
total indifference can never be superior or inferior to another; and
_that_ there must always be a mixture of causes among the chances, in
order to be the foundation of any reasoning. We are next to consider
what effect a superior combination of chances can have upon the mind,
and after what manner it influences our judgment and opinion. Here
we may repeat all the same arguments we employed in examining that
belief which arises from causes; and may prove, after the same manner,
that a superior number of chances produces our assent neither by
_demonstration_ nor _probability_. 'Tis indeed evident, that we can
never, by the comparison of mere ideas, make any discovery which can
be of consequence in this affair, and that 'tis impossible to prove
with certainty that any event must fall on that side were there is a
superior number of chances. To suppose in this case any certainty, were
to overthrow what we have established concerning the opposition of
chances, and their perfect equality and indifference.
Should it be said, that though in an opposition of chances, 'tis
impossible to determine with _certainty_ on which side the event will
fall, yet we can pronounce with certainty, that 'tis more likely and
probable 'twill be on that side where there is a superior number of
chances, than where there is an inferior: should this be said, I
would ask, what is here meant by _likelihood and probability_? The
likelihood and probability of chances is a superior number of equal
chances; and consequently, when we say 'tis likely the event will fall
on the side which is superior, rather than on the inferior, we do no
more than affirm, that where there is a superior number of chances
there is actually a superior, and where there is an inferior there is
an inferior, which are identical propositions, and of no consequence.
The question is, by what means a superior number of equal chances
operates upon the mind, and produces belief or assent, since it appears
that 'tis neither by arguments derived from demonstration, nor from
probability.
In order to clear up this difficulty, we shall suppose a person to take
a dye, formed after such a manner as that four of its sides are marked
with one figure, or one number of spots, and two with another; and to
put this dye into the box with an intention of throwing it: 'tis plain,
he must conclude the one figure to be more probable than the other,
and give the preference to that which is inscribed on the greatest
number of sides. He in a manner believes that this will lie uppermost;
though still with hesitation and doubt, in proportion to the number of
chances which are contrary: and according as these contrary chances
diminish, and the superiority increases on the other side, his belief
acquires new degrees of stability and assurance. This belief arises
from an operation of the mind upon the simple and limited object before
us; and therefore its nature will be the more easily discovered and
explained. We have nothing but one single dye to contemplate, in order
to comprehend one of the most curious operations of the understanding.
This dye formed as above, contains three circumstances worthy of our
attention. First, certain causes, such as gravity, solidity, a cubical
figure, &c. which determine it to fall, to preserve its form in its
fall, and to turn up one of its sides. Secondly, a certain number
of sides, which are supposed indifferent. Thirdly, a certain figure
inscribed on each side. These three particulars, form the whole nature
of the dye, so far as relates to our present purpose; and consequently
are the only circumstances regarded by the mind in its forming a
judgment concerning the result of such a throw. Let us therefore
consider gradually and carefully what must be the influence of these
circumstances on the thought and imagination.
First, we have already observed, that the mind is determined by custom
to pass from any cause to its effect, and that upon the appearance
of the one, 'tis almost impossible for it not to form an idea of the
other. Their constant conjunction in past instances has produced such
a habit in the mind, that it always conjoins them in its thought, and
infers the existence of the one from that of its usual attendant. When
it considers the dye as no longer supported by the box, it cannot
without violence regard it as suspended in the air; but naturally
places it on the table, and views it as turning up one of its sides.
This is the effect of the intermingled causes, which are requisite to
our forming any calculation concerning chances.
Secondly, 'tis supposed, that though the dye be necessarily determined
to fall, and turn up one of its sides, yet there is nothing to fix the
particular side, but that this is determined entirely by chance. The
very nature and essence of chance is a negation of causes, and the
leaving the mind in a perfect indifference among those events which
are supposed contingent. When, therefore, the thought is determined
by the causes to consider the dye as falling and turning up one of
its sides, the chances present all these sides as equal, and make us
consider every one of them, one after another, as alike probable and
possible. The imagination passes from the cause, viz. the throwing of
the dye, to the effect, viz. the turning up one of the six sides; and
feels a kind of impossibility both of stopping short in the way, and of
forming any other idea. But as all these six sides are incompatible,
and the dye cannot turn up above one at once, this principle directs us
not to consider all of them at once as lying uppermost, which we look
upon as impossible: neither does it direct us with its entire force to
any particular side; for in that case this side would be considered
as certain and inevitable; but it directs us to the whole six sides
after such a manner as to divide its force equally among them. We
conclude in general, that some one of them must result from the throw:
we run all of them over in our minds: the determination of the thought
is common to all; but no more of its force falls to the share of any
one, than what is suitable to its proportion with the rest. 'Tis after
this manner the original impulse, and consequently the vivacity of
thought arising from the causes, is divided and split in pieces by the
intermingled chances.
We have already seen the influence of the two first qualities of the
dye, viz. the _causes_, and the _number_, and _indifference_ of the
sides, and have learned how they give an impulse to the thought, and
divide that impulse into as many parts as there are units in the number
of sides. We must now consider the effects of the third particular,
viz. the _figures_ inscribed on each side. 'Tis evident, that where
several sides have the same figure inscribed on them, they must concur
in their influence on the mind, and must unite upon one image or idea
of a figure all those divided impulses, that were dispersed over the
several sides, upon which that figure is inscribed. Were the question
only what side will be turned up, these are all perfectly equal, and no
one could ever have any advantage above another. But as the question
is concerning the figure, and as the same figure is presented by more
than one side, 'tis evident that the impulses belonging to all these
sides must re-unite in that one figure, and become stronger and more
forcible by the union. Four sides are supposed in the present case
to have the same figure inscribed on them, and two to have another
figure. The impulses of the former are therefore superior to those of
the latter. But as the events are contrary, and 'tis impossible both
these figures can be turned up; the impulses likewise become contrary,
and the inferior destroys the superior, as far as its strength goes.
The vivacity of the idea is always proportionable to the degrees of the
impulse or tendency to the transition; and belief is the same with the
vivacity of the idea, according to the precedent doctrine.