truth of the class of inferences to which it belongs; and that
probability is the proportion of arguments carrying truth with them
among those of any _genus_. This is most conveniently expressed in the
nomenclature of the medieval logicians. They called the fact expressed
by a premise an _antecedent_, and that which follows from it its
_consequent_; while the leading principle, that every (or almost every)
such antecedent is followed by such a consequent, they termed the
_consequence_. Using this language, we may say that probability belongs
exclusively to _consequences_, and the probability of any consequence is
the number of times in which antecedent and consequent both occur
divided by the number of all the times in which the antecedent occurs.
From this definition are deduced the following rules for the addition
and multiplication of probabilities:
_Rule for the Addition of Probabilities._—Given the separate
probabilities of two consequences having the same antecedent and
incompatible consequents. Then the sum of these two numbers is the
probability of the consequence, that from the same antecedent one or
other of those consequents follows.
_Rule for the Multiplication of Probabilities._—Given the separate
probabilities of the two consequences, “If A then B,” and “If both A and
B, then C.” Then the product of these two numbers is the probability of
the consequence, “If A, then both B and C.”
_Special Rule for the Multiplication of Independent
Probabilities._—Given the separate probabilities of two consequences
having the same antecedents, “If A, then B,” and “If A, then C.” Suppose
that these consequences are such that the probability of the second is
equal to the probability of the consequence, “If both A and B, then C.”
Then the product of the two given numbers is equal to the probability of
the consequence, “If A, then both B and C.”
To show the working of these rules we may examine the probabilities in
regard to throwing dice. What is the probability of throwing a six with
one die? The antecedent here is the event of throwing a die; the
consequent, its turning up a six. As the die has six sides, all of which
are turned up with equal frequency, the probability of turning up any
one is 1/6. Suppose two dice are thrown, what is the probability of
throwing sixes? The probability of either coming up six is obviously the
same when both are thrown as when one is thrown—namely, 1/6. The
probability that either will come up six when the other does is also the
same as that of its coming up six whether the other does or not. The
probabilities are, therefore, independent; and, by our rule, the
probability that both events will happen together is the product of
their several probabilities, or 1/6 x 1/6. What is the probability of
throwing deuce-ace? The probability that the first die will turn up ace
and the second deuce is the same as the probability that both will turn
up sixes—namely, 1/36; the probability that the _second_ will turn up
ace and the _first_ deuce is likewise 1/36; these two events—first, ace;
second, deuce; and, second, ace; first, deuce—are incompatible. Hence
the rule for addition holds, and the probability that either will come
up ace and the other deuce is 1/36 + 1/36, or 1/18.
In this way all problems about dice, etc., may be solved. When the
number of dice thrown is supposed very large, mathematics (which may be
defined as the art of making groups to facilitate numeration) comes to
our aid with certain devices to reduce the difficulties.
II
The conception of probability as a matter of _fact_, i.e., as the
proportion of times in which an occurrence of one kind is accompanied by
an occurrence of another kind, is termed by Mr. Venn the materialistic
view of the subject. But probability has often been regarded as being
simply the degree of belief which ought to attach to a proposition, and
this mode of explaining the idea is termed by Venn the conceptualistic
view. Most writers have mixed the two conceptions together. They, first,
define the probability of an event as the reason we have to believe that
it has taken place, which is conceptualistic; but shortly after they
state that it is the ratio of the number of cases favorable to the event
to the total number of cases favorable or contrary, and all equally
possible. Except that this introduces the thoroughly unclear idea of
cases equally possible in place of cases equally frequent, this is a
tolerable statement of the materialistic view. The pure conceptualistic
theory has been best expounded by Mr. De Morgan in his _Formal Logic_:
or, the _Calculus of Inference, Necessary and Probable._
The great difference between the two analyses is, that the
conceptualists refer probability to an event, while the materialists
make it the ratio of frequency of events of a _species_ to those of a
_genus_ over that _species_, thus _giving it two terms instead of one_.
The opposition may be made to appear as follows:
Suppose that we have two rules of inference, such that, of all the
questions to the solution of which both can be applied, the first yields
correct answers to 81/100, and incorrect answers to the remaining
19/100; while the second yields correct answers to 93/100, and incorrect
answers to the remaining 7/100. Suppose, further, that the two rules are
entirely independent as to their truth, so that the second answers
correctly 93/100 of the questions which the first answers correctly, and
also 93/100 of the questions which the first answers incorrectly, and
answers incorrectly the remaining 7/100 of the questions which the first
answers correctly, and also the remaining 7/100 of the questions which
the first answers incorrectly. Then, of all the questions to the
solution of which both rules can be applied—
both answer correctly 93/100 of 81/100 or 93/100 x 81/100;
the second answers correctly and the first incorrectly 93/100 of
19/100 or 93/100 x 19/100;
the second answers incorrectly and the first correctly 7/100 of
81/100 or 7/100 x 81/100;
and both answer incorrectly 7/100 of 19/100 or 7/100 x 19/100;
Suppose, now, that, in reference to any question, both give the same
answer. Then (the questions being always such as are to be answered by
_yes_ or _no_), those in reference to which their answers agree are the
same as those which both answer correctly together with those which both
answer falsely, or 93/100 x 81/100 + 7/100 x 19/100 of all. The
proportion of those which both answer correctly out of those their
answers to which agree is, therefore—
((93 × 81)/(100 × 100))/((93 × 81)/(100 × 100)) + ((7 × 19)/(100 ×
100)) or (93 × 81)/((93 × 81) + (7 × 19)).
This is, therefore, the probability that, if both modes of inference
yield the same result, that result is correct. We may here conveniently
make use of another mode of expression. _Probability_ is the ratio of
the favorable cases to all the cases. Instead of expressing our result
in terms of this ratio, we may make use of another—the ratio of
favorable to unfavorable cases. This last ratio may be called the
_chance_ of an event. Then the chance of a true answer by the first mode
of inference is 81/19 and by the second is 93/7; and the chance of a
correct answer from both, when they agree, is—
(81 × 93)/(19 × 7) or 81/19 × 93/7,
or the product of the chances of each singly yielding a true answer.
It will be seen that a chance is a quantity which may have any
magnitude, however great. An event in whose favor there is an even
chance, or 1/1, has a probability of 1/2. An argument having an even
chance can do nothing toward re-enforcing others, since according to the
rule its combination with another would only multiply the chance of the
latter by 1.
Probability and chance undoubtedly belong primarily to consequences, and
are relative to premises; but we may, nevertheless, speak of the chance
of an event absolutely, meaning by that the chance of the combination of
all arguments in reference to it which exist for us in the given state
of our knowledge. Taken in this sense it is incontestable that the
chance of an event has an intimate connection with the degree of our
belief in it. Belief is certainly something more than a mere feeling;
yet there is a feeling of believing, and this feeling does and ought to
vary with the chance of the thing believed, as deduced from all the
arguments. Any quantity which varies with the chance might, therefore,
it would seem, serve as a thermometer for the proper intensity of
belief. Among all such quantities there is one which is peculiarly
appropriate. When there is a very great chance, the feeling of belief
ought to be very intense. Absolute certainty, or an infinite chance, can
never be attained by mortals, and this may be represented appropriately
by an infinite belief. As the chance diminishes the feeling of believing
should diminish, until an even chance is reached, where it should
completely vanish and not incline either toward or away from the
proposition. When the chance becomes less, then a contrary belief should
spring up and should increase in intensity as the chance diminishes, and
as the chance almost vanishes (which it can never quite do) the contrary
belief should tend toward an infinite intensity. Now, there is one
quantity which, more simply than any other, fulfills these conditions;
it is the _logarithm_ of the chance. But there is another consideration
which must, if admitted, fix us to this choice for our thermometer. It
is that our belief ought to be proportional to the weight of evidence,
in this sense, that two arguments which are entirely independent,
neither weakening nor strengthening each other, ought, when they concur,
to produce a belief equal to the sum of the intensities of belief which
either would produce separately. Now, we have seen that the chances of
independent concurrent arguments are to be multiplied together to get
the chance of their combination, and, therefore, the quantities which
best express the intensities of belief should be such that they are to
be _added_ when the _chances_ are multiplied in order to produce the
quantity which corresponds to the combined chance. Now, the logarithm is
the only quantity which fulfills this condition. There is a general law
of sensibility, called Fechner’s psychophysical law. It is that the
intensity of any sensation is proportional to the logarithm of the
external force which produces it. It is entirely in harmony with this
law that the feeling of belief should be as the logarithm of the chance,
this latter being the expression of the state of facts which produces
the belief.
The rule for the combination of independent concurrent arguments takes a
very simple form when expressed in terms of the intensity of belief,
measured in the proposed way. It is this: Take the sum of all the
feelings of belief which would be produced separately by all the
arguments _pro_, subtract from that the similar sum for arguments _con_,
and the remainder is the feeling of belief which we ought to have on the
whole. This is a proceeding which men often resort to, under the name of
_balancing reasons_.
These considerations constitute an argument in favor of the
conceptualistic view. The kernel of it is that the conjoint probability
of all the arguments in our possession, with reference to any fact, must
be intimately connected with the just degree of our belief in that fact;
and this point is supplemented by various others showing the consistency
of the theory with itself and with the rest of our knowledge.
But probability, to have any value at all, must express a fact. It is,
therefore, a thing to be inferred upon evidence. Let us, then, consider
for a moment the formation of a belief of probability. Suppose we have a
large bag of beans from which one has been secretly taken at random and
hidden under a thimble. We are now to form a probable judgment of the
color of that bean, by drawing others singly from the bag and looking at
them, each one to be thrown back, and the whole well mixed up after each
drawing. Suppose the first drawing is white and the next black. We
conclude that there is not an immense preponderance of either color, and
that there is something like an even chance that the bean under the
thimble is black. But this judgment may be altered by the next few
drawings. When we have drawn ten times, if 4, 5, or 6, are white, we
have more confidence that the chance is even. When we have drawn a
thousand times, if about half have been white, we have great confidence
in this result. We now feel pretty sure that, if we were to make a large
number of bets upon the color of single beans drawn from the bag, we
could approximately insure ourselves in the long run by betting each
time upon the white, a confidence which would be entirely wanting if,
instead of sampling the bag by 1,000 drawings, we had done so by only
two. Now, as the whole utility of probability is to insure us in the
long run, and as that assurance depends, not merely on the value of the
chance, but also on the accuracy of the evaluation, it follows that we
ought not to have the same feeling of belief in reference to all events
of which the chance is even. In short, to express the proper state of
our belief, not _one_ number but _two_ are requisite, the first
depending on the inferred probability, the second on the amount of
knowledge on which that probability is based.[41] It is true that when
our knowledge is very precise, when we have made many drawings from the
bag, or, as in most of the examples in the books, when the total
contents of the bag are absolutely known, the number which expresses the
uncertainty of the assumed probability and its liability to be changed
by further experience may become insignificant, or utterly vanish. But,
when our knowledge is very slight, this number may be even more
important than the probability itself; and when we have no knowledge at
all this completely overwhelms the other, so that there is no sense in
saying that the chance of the totally unknown event is even (for what
expresses absolutely no fact has absolutely no meaning), and what ought
to be said is that the chance is entirely indefinite. We thus perceive
that the conceptualistic view, though answering well enough in some
cases, is quite inadequate.
Suppose that the first bean which we drew from our bag were black. That
would constitute an argument, no matter how slender, that the bean under
the thimble was also black. If the second bean were also to turn out
black, that would be a second independent argument reënforcing the
first. If the whole of the first twenty beans drawn should prove black,
our confidence that the hidden bean was black would justly attain
considerable strength. But suppose the twenty-first bean were to be
white and that we were to go on drawing until we found that we had drawn
1,010 black beans and 990 white ones. We should conclude that our first
twenty beans being black was simply an extraordinary accident, and that
in fact the proportion of white beans to black was sensibly equal, and
that it was an even chance that the hidden bean was black. Yet according
to the rule of _balancing reasons_, since all the drawings of black
beans are so many independent arguments in favor of the one under the
thimble being black, and all the white drawings so many against it, an
excess of twenty black beans ought to produce the same degree of belief
that the hidden bean was black, whatever the total number drawn.
In the conceptualistic view of probability, complete ignorance, where
the judgment ought not to swerve either toward or away from the
hypothesis, is represented by the probability 1/2.[42]
But let us suppose that we are totally ignorant what colored hair the
inhabitants of Saturn have. Let us, then, take a color-chart in which
all possible colors are shown shading into one another by imperceptible
degrees. In such a chart the relative areas occupied by different
classes of colors are perfectly arbitrary. Let us inclose such an area
with a closed line, and ask what is the chance on conceptualistic
principles that the color of the hair of the inhabitants of Saturn falls
within that area? The answer cannot be indeterminate because we must be
in some state of belief; and, indeed, conceptualistic writers do not
admit indeterminate probabilities. As there is no certainty in the
matter, the answer lies between _zero_ and _unity_. As no numerical
value is afforded by the data, the number must be determined by the
nature of the scale of probability itself, and not by calculation from
the data. The answer can, therefore, only be one-half, since the
judgment should neither favor nor oppose the hypothesis. What is true of
this area is true of any other one; and it will equally be true of a
third area which embraces the other two. But the probability for each of
the smaller areas being one-half, that for the larger should be at least
unity, which is absurd.
III
All our reasonings are of two kinds: 1. _Explicative_, _analytic_, or
_deductive_; 2. _Amplifiative_, _synthetic_, or (loosely speaking)
_inductive_. In explicative reasoning, certain facts are first laid down
in the premises. These facts are, in every case, an inexhaustible
multitude, but they may often be summed up in one simple proposition by
means of some regularity which runs through them all. Thus, take the
proposition that Socrates was a man; this implies (to go no further)
that during every fraction of a second of his whole life (or, if you
please, during the greater part of them) he was a man. He did not at one
instant appear as a tree and at another as a dog; he did not flow into
water, or appear in two places at once; you could not put your finger
through him as if he were an optical image, etc. Now, the facts being
thus laid down, some order among some of them, not particularly made use
of for the purpose of stating them, may perhaps be discovered; and this
will enable us to throw part or all of them into a new statement, the
possibility of which might have escaped attention. Such a statement will
be the conclusion of an analytic inference. Of this sort are all
mathematical demonstrations. But synthetic reasoning is of another kind.
In this case the facts summed up in the conclusion are not among those
stated in the premises. They are different facts, as when one sees that
the tide rises _m_ times and concludes that it will rise the next time.
These are the only inferences which increase our real knowledge, however
useful the others may be.
In any problem in probabilities, we have given the relative frequency of
certain events, and we perceive that in these facts the relative
frequency of another event is given in a hidden way. This being stated
makes the solution. This is, therefore, mere explicative reasoning, and
is evidently entirely inadequate to the representation of synthetic
reasoning, which goes out beyond the facts given in the premises. There
is, therefore, a manifest impossibility in so tracing out any
probability for a synthetic conclusion.
Most treatises on probability contain a very different doctrine. They
state, for example, that if one of the ancient denizens of the shores of
the Mediterranean, who had never heard of tides, had gone to the bay of
Biscay, and had there seen the tide rise, say _m_ times, he could know
that there was a probability equal to
(m + 1)/(m + 2)
that it would rise the next time. In a well-known work by Quetelet, much
stress is laid on this, and it is made the foundation of a theory of
inductive reasoning.
But this solution betrays its origin if we apply it to the case in which
the man has never seen the tide rise at all; that is, if we put _m_ = 0.
In this case, the probability that it will rise the next time comes out
1/2, or, in other words, the solution involves the conceptualistic
principle that there is an even chance of a totally unknown event. The
manner in which it has been reached has been by considering a number of
urns all containing the same number of balls, part white and part black.
One urn contains all white balls, another one black and the rest white,
a third two black and the rest white, and so on, one urn for each
proportion, until an urn is reached containing only black balls. But the
only possible reason for drawing any analogy between such an arrangement
and that of Nature is the principle that alternatives of which we know
nothing must be considered as equally probable. But this principle is
absurd. There is an indefinite variety of ways of enumerating the
different possibilities, which, on the application of this principle,
would give different results. If there be any way of enumerating the
possibilities so as to make them all equal, it is not that from which
this solution is derived, but is the following: Suppose we had an
immense granary filled with black and white balls well mixed up; and
suppose each urn were filled by taking a fixed number of balls from this
granary quite at random. The relative number of white balls in the
granary might be anything, say one in three. Then in one-third of the
urns the first ball would be white, and in two-thirds black. In
one-third of those urns of which the first ball was white, and also in
one-third of those in which the first ball was black, the second ball
would be white. In this way, we should have a distribution like that
shown in the following table, where _w_ stands for a white ball and _b_
for a black one. The reader can, if he chooses, verify the table for
himself.
wwww.
wwwb. wwbw. wbww. bwww.
wwwb. wwbw. wbww. bwww.
wwbb. wbwb. bwwb. wbbw. bwbw. bbww.
wwbb. wbwb. bwwb. wbbw. bwbw. bbww.
wwbb. wbwb. bwwb. wbbw. bwbw. bbww.
wwbb. wbwb. bwwb. wbbw. bwbw. bbww.
wbbb. bwbb. bbwb. bbbw.
wbbb. bwbb. bbwb. bbbw.
wbbb. bwbb. bbwb. bbbw.
wbbb. bwbb. bbwb. bbbw.
wbbb. bwbb. bbwb. bbbw.
wbbb. bwbb. bbwb. bbbw.
wbbb. bwbb. bbwb. bbbw.
wbbb. bwbb. bbwb. bbbw.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
bbbb.
In the second group, where there is one b, there are two sets just
alike; in the third there are 4, in the fourth 8, and in the fifth 16,
doubling every time. This is because we have supposed twice as many
black balls in the granary as white ones; had we supposed 10 times as
many, instead of
1, 2, 4, 8, 16
sets we should have had
1, 10, 100, 1000, 10000
sets; on the other hand, had the numbers of black and white balls in the
granary been even, there would have been but one set in each group. Now
suppose two balls were drawn from one of these urns and were found to be
both white, what would be the probability of the next one being white?
If the two drawn out were the first two put into the urns, and the next
to be drawn out were the third put in, then the probability of this
third being white would be the same whatever the colors of the first
two, for it has been supposed that just the same proportion of urns has
the third ball white among those which have the first two _white-white_,
_white-black_, _black-white_, and _black-black_. Thus, in this case, the
chance of the third ball being white would be the same whatever the
first two were. But, by inspecting the table, the reader can see that in
each group all orders of the balls occur with equal frequency, so that
it makes no difference whether they are drawn out in the order they were
put in or not. Hence the colors of the balls already drawn have no
influence on the probability of any other being white or black.
Now, if there be any way of enumerating the possibilities of Nature so
as to make them equally probable, it is clearly one which should make
one arrangement or combination of the elements of Nature as probable as
another, that is, a distribution like that we have supposed, and it,
therefore, appears that the assumption that any such thing can be done,
leads simply to the conclusion that reasoning from past to future
experience is absolutely worthless. In fact, the moment that you assume
that the chances in favor of that of which we are totally ignorant are
even, the problem about the tides does not differ, in any arithmetical
particular, from the case in which a penny (known to be equally likely
to come up heads and tails) should turn up heads _m_ times successively.
In short, it would be to assume that Nature is a pure chaos, or chance
combination of independent elements, in which reasoning from one fact to
another would be impossible; and since, as we shall hereafter see, there
is no judgment of pure observation without reasoning, it would be to
suppose all human cognition illusory and no real knowledge possible. It
would be to suppose that if we have found the order of Nature more or
less regular in the past, this has been by a pure run of luck which we
may expect is now at an end. Now, it may be we have no scintilla of
proof to the contrary, but reason is unnecessary in reference to that
belief which is of all the most settled, which nobody doubts or can
doubt, and which he who should deny would stultify himself in so doing.
The relative probability of this or that arrangement of Nature is
something which we should have a right to talk about if universes were
as plenty as blackberries, if we could put a quantity of them in a bag,
shake them well up, draw out a sample, and examine them to see what
proportion of them had one arrangement and what proportion another. But,
even in that case, a higher universe would contain us, in regard to
whose arrangements the conception of probability could have no
applicability.
IV
We have examined the problem proposed by the conceptualists, which,
translated into clear language, is this: Given a synthetic conclusion;
required to know out of all possible states of things how many will
accord, to any assigned extent, with this conclusion; and we have found
that it is only an absurd attempt to reduce synthetic to analytic
reason, and that no definite solution is possible.
But there is another problem in connection with this subject. It is
this: Given a certain state of things, required to know what proportion
of all synthetic inferences relating to it will be true within a given
degree of approximation. Now, there is no difficulty about this problem
(except for its mathematical complication); it has been much studied,
and the answer is perfectly well known. And is not this, after all, what
we want to know much rather than the other? Why should we want to know
the probability that the fact will accord with our conclusion? That
implies that we are interested in all possible worlds, and not merely
the one in which we find ourselves placed. Why is it not much more to
the purpose to know the probability that our conclusion will accord with
the fact? One of these questions is the first above stated and the other
the second, and I ask the reader whether, if people, instead of using
the word probability without any clear apprehension of their own
meaning, had always spoken of relative frequency, they could have failed
to see that what they wanted was not to follow along the synthetic
procedure with an analytic one, in order to find the probability of the
conclusion; but, on the contrary, to begin with the fact at which the
synthetic inference aims, and follow back to the facts it uses for
premises in order to see the probability of their being such as will
yield the truth.
As we cannot have an urn with an infinite number of balls to represent
the inexhaustibleness of Nature, let us suppose one with a finite
number, each ball being thrown back into the urn after being drawn out,
so that there is no exhaustion of them. Suppose one ball out of three is
white and the rest black, and that four balls are drawn. Then the table
on pages 95-96 represents the relative frequency of the different ways
in which these balls might be drawn. It will be seen that if we should
judge by these four balls of the proportion in the urn, 32 times out of
81 we should find it 1/4, and 24 times out of 81 we should find it 1/2,
the truth being 1/3. To extend this table to high numbers would be great
labor, but the mathematicians have found some ingenious ways of
reckoning what the numbers would be. It is found that, if the true
proportion of white balls is _p_, and _s_ balls are drawn, then the
error of the proportion obtained by the induction will be—
half the time within 0.477 √((2p(1-p))/s)
9 times out of 10 within 1.163 √((2p(1-p))/s)
99 times out of 100 within 1.821 √((2p(1-p))/s)
999 times out of 1,000 within 2.328 √((2p(1-p))/s)
9,999 times out of 10,000 within 2.751 √((2p(1-p))/s)
9,999,999,999 times out of 10,000,000,000 within 4.77 √((2p(1-p))/s)
The use of this may be illustrated by an example. By the census of 1870,
it appears that the proportion of males among native white children
under one year old was 0.5082, while among colored children of the same
age the proportion was only 0.4977. The difference between these is
0.0105, or about one in a 100. Can this be attributed to chance, or
would the difference always exist among a great number of white and
colored children under like circumstances? Here _p_ may be taken at 1/2;
hence 2_p_(1-_p_) is also 1/2. The number of white children counted was
near 1,000,000; hence the fraction whose square-root is to be taken is
about 1/2000000. The root is about 1/1400, and this multiplied by 0.477
gives about 0.0003 as the probable error in the ratio of males among the
whites as obtained from the induction. The number of black children was
about 150,000, which gives 0.0008 for the probable error. We see that
the actual discrepancy is ten times the sum of these, and such a result
would happen, according to our table, only once out of 10,000,000,000
censuses, in the long run.
It may be remarked that when the real value of the probability sought
inductively is either very large or very small, the reasoning is more
secure. Thus, suppose there were in reality one white ball in 100 in a
certain urn, and we were to judge of the number by 100 drawings. The
probability of drawing no white ball would be 366/1000; that of drawing
one white ball would be 370/1000; that of drawing two would be 185/1000;
that of drawing three would be 61/1000; that of drawing four would be
15/1000; that of drawing five would be only 3/1000, etc. Thus we should
be tolerably certain of not being in error by more than one ball in 100.
It appears, then, that in one sense we can, and in another we cannot,
determine the probability of synthetic inference. When I reason in this
way:
Ninety-nine Cretans in a hundred are liars;
But Epimenides is a Cretan;
Therefore, Epimenides is a liar:—
I know that reasoning similar to that would carry truth 99 times in 100.
But when I reason in the opposite direction:
Minos, Sarpedon, Rhadamanthus, Deucalion, and Epimenides,
are all the Cretans I can think of;
But these were all atrocious liars,
Therefore, pretty much all Cretans must have been liars;
I do not in the least know how often such reasoning would carry me
right. On the other hand, what I do know is that some definite
proportion of Cretans must have been liars, and that this proportion can
be probably approximated to by an induction from five or six instances.
Even in the worst case for the probability of such an inference, that in
which about half the Cretans are liars, the ratio so obtained would
probably not be in error by more than 1/6. So much I know; but, then, in
the present case the inference is that pretty much all Cretans are
liars, and whether there may not be a special improbability in that I do
not know.
V
Late in the last century, Immanuel Kant asked the question, “How are
synthetical judgments _a priori_ possible?” By synthetical judgments he
meant such as assert positive fact and are not mere affairs of
arrangement; in short, judgments of the kind which synthetical reasoning
produces, and which analytic reasoning cannot yield. By _a priori_
judgments he meant such as that all outward objects are in space, every
event has a cause, etc., propositions which according to him can never
be inferred from experience. Not so much by his answer to this question
as by the mere asking of it, the current philosophy of that time was
shattered and destroyed, and a new epoch in its history was begun. But
before asking _that_ question he ought to have asked the more general
one, “How are any synthetical judgments at all possible?” How is it that
a man can observe one fact and straightway pronounce judgment concerning
another different fact not involved in the first? Such reasoning, as we
have seen, has, at least in the usual sense of the phrase, no definite
probability; how, then, can it add to our knowledge? This is a strange
paradox; the Abbé Gratry says it is a miracle, and that every true
induction is an immediate inspiration from on high.[43] I respect this
explanation far more than many a pedantic attempt to solve the question
by some juggle with probabilities, with the forms of syllogism, or what
not. I respect it because it shows an appreciation of the depth of the
problem, because it assigns an adequate cause, and because it is
intimately connected—as the true account should be—with a general
philosophy of the universe. At the same time, I do not accept this
explanation, because an explanation should tell _how_ a thing is done,
and to assert a perpetual miracle seems to be an abandonment of all hope
of doing that, without sufficient justification.
It will be interesting to see how the answer which Kant gave to his
question about synthetical judgments _a priori_ will appear if extended
to the question of synthetical judgments in general. That answer is,
that synthetical judgments _a priori_ are possible because whatever is
universally true is involved in the conditions of experience. Let us
apply this to a general synthetical reasoning. I take from a bag a
handful of beans; they are all purple, and I infer that all the beans in
the bag are purple. How can I do that? Why, upon the principle that
whatever is universally true of my experience (which is here the
appearance of these different beans) is involved in the condition of
experience. The condition of this special experience is that all these
beans were taken from that bag. According to Kant’s principle, then,
whatever is found true of all the beans drawn from the bag must find its
explanation in some peculiarity of the contents of the bag. This is a
satisfactory statement of the principle of induction.
When we draw a deductive or analytic conclusion, our rule of inference
is that facts of a certain general character are either invariably or in
a certain proportion of cases accompanied by facts of another general
character. Then our premise being a fact of the former class, we infer
with certainty or with the appropriate degree of probability the
existence of a fact of the second class. But the rule for synthetic
inference is of a different kind. When we sample a bag of beans we do
not in the least assume that the fact of some beans being purple
involves the necessity or even the probability of other beans being so.
On the contrary, the conceptualistic method of treating probabilities,
which really amounts simply to the deductive treatment of them, when
rightly carried out leads to the result that a synthetic inference has
just an even chance in its favor, or in other words is absolutely
worthless. The color of one bean is entirely independent of that of
another. But synthetic inference is founded upon a classification of
facts, not according to their characters, but according to the manner of
obtaining them. Its rule is, that a number of facts obtained in a given
way will in general more or less resemble other facts obtained in the
same way; or, _experiences whose conditions are the same will have the
same general characters_.
In the former case, we know that premises precisely similar in form to
those of the given ones will yield true conclusions, just once in a
calculable number of times. In the latter case, we only know that
premises obtained under circumstances similar to the given ones (though
perhaps themselves very different) will yield true conclusions, at least
once in a calculable number of times. We may express this by saying that
in the case of analytic inference we know the probability of our
conclusion (if the premises are true), but in the case of synthetic
inferences we only know the degree of trustworthiness of our proceeding.
As all knowledge comes from synthetic inference, we must equally infer
that all human certainty consists merely in our knowing that the
processes by which our knowledge has been derived are such as must
generally have led to true conclusions.
Though a synthetic inference cannot by any means be reduced to
deduction, yet that the rule of induction will hold good in the long run
may be deduced from the principle that reality is only the object of the
final opinion to which sufficient investigation would lead. That belief
gradually tends to fix itself under the influence of inquiry is, indeed,
one of the facts with which logic sets out.
Footnote 40:
_Popular Science Monthly_, April, 1878.
Footnote 41:
Strictly we should need an infinite series of numbers each depending
on the probable error of the last.
Footnote 42:
“Perfect indecision, belief inclining neither way, an even chance.”—DE
MORGAN, p. 182.
Footnote 43:
_Logique_. The same is true, according to him, of every performance of
a differentiation, but not of integration. He does not tell us whether
it is the supernatural assistance which makes the former process so
much the easier.
FIFTH PAPER
THE ORDER OF NATURE[44]
I
Any proposition whatever concerning the order of Nature must touch more
or less upon religion. In our day, belief, even in these matters,
depends more and more upon the observation of facts. If a remarkable and
universal orderliness be found in the universe, there must be some cause
for this regularity, and science has to consider what hypotheses might
account for the phenomenon. One way of accounting for it, certainly,
would be to suppose that the world is ordered by a superior power. But
if there is nothing in the universal subjection of phenomena to laws,
nor in the character of those laws themselves (as being benevolent,
beautiful, economical, etc.), which goes to prove the existence of a
governor of the universe, it is hardly to be anticipated that any other
sort of evidence will be found to weigh very much with minds emancipated
from the tyranny of tradition.
Nevertheless, it cannot truly be said that even an absolutely negative
decision of that question could altogether destroy religion, inasmuch as
there are faiths in which, however much they differ from our own, we
recognize those essential characters which make them worthy to be called
religions, and which, nevertheless, do not postulate an actually
existing Deity. That one, for instance, which has had the most numerous
and by no means the least intelligent following of any on earth, teaches
that the Divinity in his highest perfection is wrapped away from the
world in a state of profound and eternal sleep, which really does not
differ from non-existence, whether it be called by that name or not. No
candid mind who has followed the writings of M. Vacherot can well deny
that his religion is as earnest as can be. He worships the Perfect, the
Supreme Ideal; but he conceives that the very notion of the Ideal is
repugnant to its real existence.[45] In fact, M. Vacherot finds it
agreeable to his reason to assert that non-existence is an essential
character of the perfect, just as St. Anselm and Descartes found it
agreeable to theirs to assert the extreme opposite. I confess that there
is one respect in which either of these positions seems to me more
congruous with the religious attitude than that of a theology which
stands upon evidences; for as soon as the Deity presents himself to
either Anselm or Vacherot, and manifests his glorious attributes,
whether it be in a vision of the night or day, either of them recognizes
his adorable God, and sinks upon his knees at once; whereas the
theologian of evidences will first demand that the divine apparition
shall identify himself, and only after having scrutinized his
credentials and weighed the probabilities of his being found among the
totality of existences, will he finally render his circumspect homage,
thinking that no characters can be adorable but those which belong to a
real thing.
If we could find out any general characteristic of the universe, any
mannerism in the ways of Nature, any law everywhere applicable and
universally valid, such a discovery would be of such singular assistance
to us in all our future reasoning, that it would deserve a place almost
at the head of the principles of logic. On the other hand, if it can be
shown that there is nothing of the sort to find out, but that every
discoverable regularity is of limited range, this again will be of
logical importance. What sort of a conception we ought to have of the
universe, how to think of the _ensemble_ of things, is a fundamental
problem in the theory of reasoning.
II
It is the legitimate endeavor of scientific men now, as it was
twenty-three hundred years ago, to account for the formation of the
solar system and of the cluster of stars which forms the galaxy, by the
fortuitous concourse of atoms. The greatest expounder of this theory,
when asked how he could write an immense book on the system of the world
without one mention of its author, replied, very logically, “Je n’avais
pas besoin de cette hypothèse-là.” But, in truth, there is nothing
atheistical in the theory, any more than there was in this answer.
Matter is supposed to be composed of molecules which obey the laws of
mechanics and exert certain attractions upon one another; and it is to
these regularities (which there is no attempt to account for) that
general arrangement of the solar system would be due, and not to hazard.
If any one has ever maintained that the universe is a pure throw of the
dice, the theologians have abundantly refuted him. “How often,” says
Archbishop Tillotson, “might a man, after he had jumbled a set of
letters in a bag, fling them out upon the ground before they would fall
into an exact poem, yea, or so much as make a good discourse in prose!
And may not a little book be as easily made by chance as this great
volume of the world?” The chance world here shown to be so different
from that in which we live would be one in which there were no laws, the
characters of different things being entirely independent; so that,
should a sample of any kind of objects ever show a prevalent character,
it could only be by accident, and no general proposition could ever be
established. Whatever further conclusions we may come to in regard to
the order of the universe, thus much may be regarded as solidly
established, that the world is not a mere chance-medley.
But whether the world makes an exact poem or not, is another question.
When we look up at the heavens at night, we readily perceive that the
stars are not simply splashed on to the celestial vault; but there does
not seem to be any precise system in their arrangement either. It will
be worth our while, then, to inquire into the degree of orderliness in
the universe; and, to begin, let us ask whether the world we live in is
any more orderly than a purely chance-world would be.
Any uniformity, or law of Nature, may be stated in the form, “Every A is
B”; as, every ray of light is a non-curved line, every body is
accelerated toward the earth’s center, etc. This is the same as to say,
“There does not exist any A which is not B”; there is no curved ray;
there is no body not accelerated toward the earth; so that the
uniformity consists in the non-occurrence in Nature of a certain
combination of characters (in this case, the combination of being A with
being non-B).[46] And, conversely, every case of the non-occurrence of a
combination of characters would constitute a uniformity in Nature. Thus,
suppose the quality A is never found in combination with the quality C:
for example, suppose the quality of idiocy is never found in combination
with that of having a well-developed brain. Then nothing of the sort A
is of the sort C, or everything of the sort A is of the sort non-C (or
say, every idiot has an ill-developed brain), which, being something
universally true of the A’s, is a uniformity in the world. Thus we see
that, in a world where there were no uniformities, no logically possible
combination of characters would be excluded, but every combination would
exist in some object. But two objects not identical must differ in some
of their characters, though it be only in the character of being in
such-and-such a place. Hence, precisely the same combination of
characters could not be found in two different objects; and,
consequently, in a chance-world every combination involving either the
positive or negative of every character would belong to just one thing.
Thus, if there were but five simple characters in such a world,[47] we
might denote them by A, B, C, D, E, and their negatives by a, b, c, d,
e; and then, as there would be 2^5 or 32 different combinations of these
characters, completely determinate in reference to each of them, that
world would have just 32 objects in it, their characters being as in the
following table:
TABLE I.
ABCDE AbCDE aBCDE abCDE
ABCDe AbCDe aBCDe abCDe
ABCdE AbCdE aBCdE abCdE
ABCde AbCde aBCde abCde
ABcDE AbcDE aBcDE abcDE
ABcDe AbcDe aBcDe abcDe
ABcdE AbcdE aBcdE abcdE
ABcde Abcde aBcde abcde
For example, if the five primary characters were _hard_, _sweet_,
_fragrant_, _green_, _bright_, there would be one object which reunited
all these qualities, one which was hard, sweet, fragrant, and green, but
not bright; one which was hard, sweet, fragrant, and bright, but not
green; one which was hard, sweet, and fragrant, but neither green nor
bright; and so on through all the combinations.
This is what a thoroughly chance-world would be like, and certainly
nothing could be imagined more systematic. When a quantity of letters
are poured out of a bag, the appearance of disorder is due to the
circumstance that the phenomena are only partly fortuitous. The laws of
space are supposed, in that case, to be rigidly preserved, and there is
also a certain amount of regularity in the formation of the letters. The
result is that some elements are orderly and some are disorderly, which
is precisely what we observe in the actual world. Tillotson, in the
passage of which a part has been quoted, goes on to ask, “How long might
20,000 blind men which should be sent out from the several remote parts
of England, wander up and down before they would all meet upon Salisbury
Plains, and fall into rank and file in the exact order of an army? And
yet this is much more easy to be imagined than how the innumerable blind
parts of matter should rendezvous themselves into a world.” This is very
true, but in the actual world the _blind men_ are, as far as we can see,
_not_ drawn up in any particular order at all. And, in short, while a
certain amount of order exists in the world, it would seem that the
world is not so orderly as it might be, and, for instance, not so much
so as a world of pure chance would be.
But we can never get to the bottom of this question until we take
account of a highly-important logical principle[48] which I now proceed
to enounce. This principle is that any plurality or lot of objects
whatever have some character in common (no matter how insignificant)
which is peculiar to them and not shared by anything else. The word
“character” here is taken in such a sense as to include negative
characters, such as incivility, inequality, etc., as well as their
positives, civility, equality, etc. To prove the theorem, I will show
what character any two things, A and B, have in common, not shared by
anything else. The things, A and B, are each distinguished from all
other things by the possession of certain characters which may be named
A-ness and B-ness. Corresponding to these positive characters, are the
negative characters un-A-ness, which is possessed by everything except
A, and un-B-ness, which is possessed by everything except B. These two
characters are united in everything except A and B; and this union of
the characters un-A-ness and un-B-ness makes a compound character which
may be termed A-B-lessness. This is not possessed by either A or B, but
it is possessed by everything else. This character, like every other,
has its corresponding negative un-A-B-lessness, and this last is the
character possessed by both A and B, and by nothing else. It is obvious
that what has thus been shown true of two things is _mutatis mutandis_,
true of any number of things. Q. E. D.
In any world whatever, then, there must be a character peculiar to each
possible group of objects. If, as a matter of nomenclature, characters
peculiar to the same group be regarded as only different aspects of the
same character, then we may say that there will be precisely one
character for each possible group of objects. Thus, suppose a world to
contain five things, α, β, γ, δ, ε. Then it will have a separate
character for each of the 31 groups (with _non-existence_ making up 32
or 2^5) shown in the following table:
TABLE II.
αβ αβγ αβγδ αβγδε
α αγ αβδ αβγε
β αδ αβε αβδε
γ αε αγδ αγδε
δ βγ αγε βγδε
ε βδ αδε
βε βγδ
γδ βγε
γε βδε
δε γδε
This shows that a contradiction is involved in the very idea[49] of a
chance-world, for in a world of 32 things, instead of there being only
3^5 or 243 characters, as we have seen that the notion of a chance-world
requires, there would, in fact, be no less than 2^{32}, or 4,294,967,296
characters, which would not be all independent, but would have all
possible relations with one another.
We further see that so long as we regard characters abstractly, without
regard to their relative importance, etc., there is no possibility of a
more or less degree of orderliness in the world, the whole system of
relationship between the different characters being given by mere logic;
that is, being implied in those facts which are tacitly admitted as soon
as we admit that there is any such thing as reasoning.
In order to descend from this abstract point of view, it is requisite to
consider the characters of things as relative to the perceptions and
active powers of living beings. Instead, then, of attempting to imagine
a world in which there should be no uniformities, let us suppose one in
which none of the uniformities should have reference to characters
interesting or important to us. In the first place, there would be
nothing to puzzle us in such a world. The small number of qualities
which would directly meet the senses would be the ones which would
afford the key to everything which could possibly interest us. The whole
universe would have such an air of system and perfect regularity that
there would be nothing to ask. In the next place, no action of ours, and
no event of Nature, would have important consequences in such a world.
We should be perfectly free from all responsibility, and there would be
nothing to do but to enjoy or suffer whatever happened to come along.
Thus there would be nothing to stimulate or develop either the mind or
the will, and we consequently should neither act nor think. We should
have no memory, because that depends on a law of our organization. Even
if we had any senses, we should be situated toward such a world
precisely as inanimate objects are toward the present one, provided we
suppose that these objects have an absolutely transitory and
instantaneous consciousness without memory—a supposition which is a mere
mode of speech, for that would be no consciousness at all. We may,
therefore, say that a world of chance is simply our actual world viewed
from the standpoint of an animal at the very vanishing-point of
intelligence. The actual world is almost a chance-medley to the mind of
a polyp. The interest which the uniformities of Nature have for an
animal measures his place in the scale of intelligence.
Thus, nothing can be made out from the orderliness of Nature in regard
to the existence of a God, unless it be maintained that the existence of
a finite mind proves the existence of an infinite one.
III
In the last of these papers we examined the nature of inductive or
synthetic reasoning. We found it to be a process of sampling. A number
of specimens of a class are taken, not by selection within that class,
but at random. These specimens will agree in a great number of respects.
If, now, it were likely that a second lot would agree with the first in
the majority of these respects, we might base on this consideration an
inference in regard to any one of these characters. But such an
inference would neither be of the nature of induction, nor would it
(except in special cases) be valid, because the vast majority of points
of agreement in the first sample drawn would generally be entirely
accidental, as well as insignificant. To illustrate this, I take the
ages at death of the first five poets given in Wheeler’s _Biographical
Dictionary_. They are:
Aagard, 48.
Abeille, 70.
Abulola, 84.
Abunowas, 48.
Accords, 45.
These five ages have the following characters in common:
1. The difference of the two digits composing the number, divided by
three, leaves a remainder of _one_.
2. The first digit raised to the power indicated by the second, and
divided by three, leaves a remainder of _one_.
3. The sum of the prime factors of each age, including one, is divisible
by three.
It is easy to see that the number of accidental agreements of this sort
would be quite endless. But suppose that, instead of considering a
character because of its prevalence in the sample, we designate a
character before taking the sample, selecting it for its importance,
obviousness, or other point of interest. Then two considerable samples
drawn at random are extremely likely to agree approximately in regard to
the proportion of occurrences of a character so chosen. _The inference
that a previously designated character has nearly the same frequency of
occurrence in the whole of a class that it has in a sample drawn at
random out of that class is induction._ If the character be not
previously designated, then a sample in which it is found to be
prevalent can only serve to suggest that it _may be_ prevalent in the
whole class. We may consider this surmise as an inference if we
please—an inference of possibility; but a second sample must be drawn to
test the question of whether the character actually is prevalent.
Instead of designating beforehand a single character in reference to
which we will examine a sample, we may designate two, and use the same
sample to determine the relative frequencies of both. This will be
making two inductive inferences at once; and, of course, we are less
certain that both will yield correct conclusions than we should be that
either separately would do so. What is true of two characters is true of
any limited number. Now, the number of characters which have any
considerable interest for us in reference to any class of objects is
more moderate than might be supposed. As we shall be sure to examine any
sample with reference to these characters, they may be regarded not
exactly as predesignated, but as predetermined (which amounts to the
same thing); and we may infer that the sample represents the class in
all these respects if we please, remembering only that this is not so
secure an inference as if the particular quality to be looked for had
been fixed upon beforehand.
The demonstration of this theory of induction rests upon principles and
follows methods which are accepted by all those who display in other
matters the particular knowledge and force of mind which qualify them to
judge of this. The theory itself, however, quite unaccountably seems
never to have occurred to any of the writers who have undertaken to
explain synthetic reasoning. The most widely-spread opinion in the
matter is one which was much promoted by Mr. John Stuart Mill—namely,
that induction depends for its validity upon the uniformity of
Nature—that is, on the principle that what happens once will, under a
sufficient degree of similarity of circumstances, happen again as often
as the same circumstances recur. The application is this: The fact that
different things belong to the same class constitutes the similarity of
circumstances, and the induction is good, provided this similarity is
“sufficient.” What happens once is, that a number of these things are
found to have a certain character; what may be expected, then, to happen
again as often as the circumstances recur consists in this, that all
things belonging to the same class should have the same character.
This analysis of induction has, I venture to think, various
imperfections, to some of which it may be useful to call attention. In
the first place, when I put my hand in a bag and draw out a handful of
beans, and, finding three-quarters of them black, infer that about
three-quarters of all in the bag are black, my inference is obviously of
the same kind as if I had found any larger proportion, or the whole, of
the sample black, and had assumed that it represented in that respect
the rest of the contents of the bag. But the analysis in question hardly
seems adapted to the explanation of this _proportionate_ induction,
where the conclusion, instead of being that a certain event uniformly
happens under certain circumstances, is precisely that it does not
uniformly occur, but only happens in a certain proportion of cases. It
is true that the whole sample may be regarded as a single object, and
the inference may be brought under the formula proposed by considering
the conclusion to be that any similar sample will show a similar
proportion among its constituents. But this is to treat the induction as
if it rested on a single instance, which gives a very false idea of its
probability.
In the second place, if the uniformity of Nature were the sole warrant
of induction, we should have no right to draw one in regard to a
character whose constancy we knew nothing about. Accordingly, Mr. Mill
says that, though none but white swans were known to Europeans for
thousands of years, yet the inference that all swans were white was “not
a good induction,” because it was not known that color was a usual
generic character (it, in fact, not being so by any means). But it is
mathematically demonstrable that an inductive inference may have as high
a degree of probability as you please independent of any antecedent
knowledge of the constancy of the character inferred. Before it was
known that color is not usually a character of _genera_, there was
certainly a considerable probability that all swans were white. But the
further study of the _genera_ of animals led to the induction of their
non-uniformity in regard to color. A deductive application of this
general proposition would have gone far to overcome the probability of
the universal whiteness of swans before the black species was
discovered. When we do know anything in regard to the general constancy
or inconstancy of a character, the application of that general knowledge
to the particular class to which any induction relates, though it serves
to increase or diminish the force of the induction, is, like every
application of general knowledge to particular cases, deductive in its
nature and not inductive.
In the third place, to say that inductions are true because similar
events happen in similar circumstances—or, what is the same thing,
because objects similar in some respects are likely to be similar in
others—is to overlook those conditions which really are essential to the
validity of inductions. When we take all the characters into account,
any pair of objects resemble one another in just as many particulars as
any other pair. If we limit ourselves to such characters as have for us
any importance, interest, or obviousness, then a synthetic conclusion
may be drawn, but only on condition that the specimens by which we judge
have been taken at random from the class in regard to which we are to
form a judgment, and not selected as belonging to any sub-class. The
induction only has its full force when the character concerned has been
designated before examining the sample. These are the essentials of
induction, and they are not recognized in attributing the validity of
induction to the uniformity of Nature. The explanation of induction by
the doctrine of probabilities, given in the last of these papers, is not
a mere metaphysical formula, but is one from which all the rules of
synthetic reasoning can be deduced systematically and with mathematical
cogency. But the account of the matter by a principle of Nature, even if
it were in other respects satisfactory, presents the fatal disadvantage
of leaving us quite as much afloat as before in regard to the proper
method of induction. It does not surprise me, therefore, that those who
adopt this theory have given erroneous rules for the conduct of
reasoning, nor that the greater number of examples put forward by Mr.
Mill in his first edition, as models of what inductions should be,
proved in the light of further scientific progress so particularly
unfortunate that they had to be replaced by others in later editions.
One would have supposed that Mr. Mill might have based an induction on
_this_ circumstance, especially as it is his avowed principle that, if
the conclusion of an induction turns out false, it cannot have been a
good induction. Nevertheless, neither he nor any of his scholars seem to
have been led to suspect, in the least, the perfect solidity of the
framework which he devised for securely supporting the mind in its
passage from the known to the unknown, although at its first trial it
did not answer quite so well as had been expected.
IV
When we have drawn any statistical induction—such, for instance, as that
one-half of all births are of male children—it is always possible to
discover, by investigation sufficiently prolonged, a class of which the
same predicate may be affirmed universally; to find out, for instance,
_what sort of_ births are of male children. The truth of this principle
follows immediately from the theorem that there is a character peculiar
to every possible group of objects. The form in which the principle is
usually stated is, that _every event must have a cause_.
But, though there exists a cause for every event, and that of a kind
which is capable of being discovered, yet if there be nothing to guide
us to the discovery; if we have to hunt among all the events in the
world without any scent; if, for instance, the sex of a child might
equally be supposed to depend on the configuration of the planets, on
what was going on at the antipodes, or on anything else—then the
discovery would have no chance of ever getting made.
That we ever do discover the precise causes of things, that any
induction whatever is absolutely without exception, is what we have no
right to assume. On the contrary, it is an easy corollary, from the
theorem just referred to, that every empirical rule has an
exception.[50] But there are certain of our inductions which present an
approach to universality so extraordinary that, even if we are to
suppose that they are not strictly universal truths, we cannot possibly
think that they have been reached merely by accident. The most
remarkable laws of this kind are those of _time_ and _space_. With
reference to space, Bishop Berkeley first showed, in a very conclusive
manner, that it was not a thing _seen_, but a thing _inferred_. Berkeley
chiefly insists on the impossibility of directly seeing the third
dimension of space, since the retina of the eye is a surface. But, in
point of fact, the retina is not even a surface; it is a conglomeration
of nerve-needles directed toward the light and having only their extreme
points sensitive, these points lying at considerable distances from one
another compared with their areas. Now, of these points, certainly the
excitation of no one singly can produce the perception of a surface, and
consequently not the aggregate of all the sensations can amount to this.
But certain relations subsist between the excitations of different
nerve-points, and these constitute the premises upon which the
hypothesis of space is founded, and from which it is inferred. That
space is not immediately perceived is now universally admitted; and a
mediate cognition is what is called an inference, and is subject to the
criticism of logic. But what are we to say to the fact of every chicken
as soon as it is hatched solving a problem whose data are of a
complexity sufficient to try the greatest mathematical powers? It would
be insane to deny that the tendency to light upon the conception of
space is inborn in the mind of the chicken and of every animal. The same
thing is equally true of time. That time is not directly perceived is
evident, since no lapse of time is present, and we only perceive what is
present. That, not having the idea of time, we should never be able to
perceive the flow in our sensations without some particular aptitude for
it, will probably also be admitted. The idea of force—at least, in its
rudiments—is another conception so early arrived at, and found in
animals so low in the scale of intelligence, that it must be supposed
innate. But the innateness of an idea admits of degree, for it consists
in the tendency of that idea to present itself to the mind. Some ideas,
like that of space, do so present themselves irresistibly at the very
dawn of intelligence, and take possession of the mind on small
provocation, while of other conceptions we are prepossessed, indeed, but
not so strongly, down a scale which is greatly extended. The tendency to
personify every thing, and to attribute human characters to it, may be
said to be innate; but it is a tendency which is very soon overcome by
civilized man in regard to the greater part of the objects about him.
Take such a conception as that of gravitation varying inversely as the
square of the distance. It is a very simple law. But to say that it is
simple is merely to say that it is one which the mind is particularly
adapted to apprehend with facility. Suppose the idea of a quantity
multiplied into another had been no more easy to the mind than that of a
quantity raised to the power indicated by itself—should we ever have
discovered the law of the solar system?
It seems incontestable, therefore, that the mind of man is strongly
adapted to the comprehension of the world; at least, so far as this
goes, that certain conceptions, highly important for such a
comprehension, naturally arise in his mind; and, without such a
tendency, the mind could never have had any development at all.
How are we to explain this adaptation? The great utility and
indispensableness of the conceptions of time, space, and force, even to
the lowest intelligence, are such as to suggest that they are the
results of natural selection. Without something like geometrical,
kinetical, and mechanical conceptions, no animal could seize his food or
do anything which might be necessary for the preservation of the
species. He might, it is true, be provided with an instinct which would
generally have the same effect; that is to say, he might have
conceptions different from those of time, space, and force, but which
coincided with them in regard to the ordinary cases of the animal’s
experience. But, as that animal would have an immense advantage in the
struggle for life whose mechanical conceptions did not break down in a
novel situation (such as development must bring about), there would be a
constant selection in favor of more and more correct ideas of these
matters. Thus would be attained the knowledge of that fundamental law
upon which all science rolls; namely, that forces depend upon relations
of time, space, and mass. When this idea was once sufficiently clear, it
would require no more than a comprehensible degree of genius to discover
the exact nature of these relations. Such an hypothesis naturally
suggests itself, but it must be admitted that it does not seem
sufficient to account for the extraordinary accuracy with which these
conceptions apply to the phenomena of Nature, and it is probable that
there is some secret here which remains to be discovered.
V
Some important questions of logic depend upon whether we are to consider
the material universe as of limited extent and finite age, or quite
boundless in space and in time. In the former case, it is conceivable
that a general plan or design embracing the whole universe should be
discovered, and it would be proper to be on the alert for some traces of
such a unity. In the latter case, since the proportion of the world of
which we can have any experience is less than the smallest assignable
fraction, it follows that we never could discover any _pattern_ in the
universe except a repeating one; any design embracing the whole would be
beyond our powers to discern, and beyond the united powers of all
intellects during all time. Now, what is absolutely incapable of being
known is, as we have seen in a former paper, not real at all. An
absolutely incognizable existence is a nonsensical phrase. If,
therefore, the universe is infinite, the attempt to find in it any
design embracing it as a whole is futile, and involves a false way of
looking at the subject. If the universe never had any beginning, and if
in space world stretches beyond world without limit, there is no _whole_
of material things, and consequently no general character to the
universe, and no need or possibility of any governor for it. But if
there was a time before which absolutely no matter existed, if there are
certain absolute bounds to the region of things outside of which there
is a mere void, then we naturally seek for an explanation of it, and,
since we cannot look for it among material things, the hypothesis of a
great disembodied animal, the creator and governor of the world, is
natural enough.
The actual state of the evidence as to the limitation of the universe is
as follows: As to time, we find on our earth a constant progress of
development since the planet was a red-hot ball; the solar system seems
to have resulted from the condensation of a nebula, and the process
appears to be still going on. We sometimes see stars (presumably with
systems of worlds) destroyed and apparently resolved back into the
nebulous condition, but we have no evidence of any existence of the
world previous to the nebulous stage from which it seems to have been
evolved. All this rather favors the idea of a beginning than otherwise.
As for limits in space, we cannot be sure that we see anything outside
of the system of the Milky Way. Minds of theological predilections have
therefore no need of distorting the facts to reconcile them with their
views.
But the only scientific presumption is, that the unknown parts of space
and time are like the known parts, occupied; that, as we see cycles of
life and death in all development which we can trace out to the end, the
same holds good in regard to solar systems; that as enormous distances
lie between the different planets of our solar system, relatively to
their diameters, and as still more enormous distances lie between our
system relatively to its diameter and other systems, so it may be
supposed that other galactic clusters exist so remote from ours as not
to be recognized as such with certainty. I do not say that these are
strong inductions; I only say that they are the presumptions which, in
our ignorance of the facts, should be preferred to hypotheses which
involve conceptions of things and occurrences totally different in their
character from any of which we have had any experience, such as
disembodied spirits, the creation of matter, infringements of the laws
of mechanics, etc.
The universe ought to be presumed too vast to have any character. When
it is claimed that the arrangements of Nature are benevolent, or just,
or wise, or of any other peculiar kind, we ought to be prejudiced
against such opinions, as being the offspring of an ill-founded notion
of the finitude of the world. And examination has hitherto shown that
such beneficences, justice, etc., are of a most limited kind—limited in
degree and limited in range.
In like manner, if any one claims to have discovered a plan in the
structure of organized beings, or a scheme in their classification, or a
regular arrangement among natural objects, or a system of
proportionality in the human form, or an order of development, or a
correspondence between conjunctions of the planets and human events, or
a significance in numbers, or a key to dreams, the first thing we have
to ask is whether such relations are susceptible of explanation on
mechanical principles, and if not they should be looked upon with
disfavor as having already a strong presumption against them; and
examination has generally exploded all such theories.
There are minds to whom every prejudice, every presumption, seems
unfair. It is easy to say what minds these are. They are those who never
have known what it is to draw a well-grounded induction, and who imagine
that other people’s knowledge is as nebulous as their own. That all
science rolls upon presumption (not of a formal but of a real kind) is
no argument with them, because they cannot imagine that there is
anything solid in human knowledge. These are the people who waste their
time and money upon perpetual motions and other such rubbish.
But there are better minds who take up mystical theories (by which I
mean all those which have no possibility of being mechanically
explained). These are persons who are strongly prejudiced in favor of
such theories. We all have natural tendencies to believe in such things;
our education often strengthens this tendency; and the result is, that
to many minds nothing seems so antecedently probable as a theory of this
kind. Such persons find evidence enough in favor of their views, and in
the absence of any recognized logic of induction they cannot be driven
from their belief.
But to the mind of a physicist there ought to be a strong presumption
against every mystical theory; and, therefore, it seems to me that those
scientific men who have sought to make out that science was not hostile
to theology have not been so clear-sighted as their opponents.
It would be extravagant to say that science can at present disprove
religion; but it does seem to me that the spirit of science is hostile
to any religion except such a one as that of M. Vacherot. Our appointed
teachers inform us that Buddhism is a miserable and atheistical faith,
shorn of the most glorious and needful attributes of a religion; that
its priests can be of no use to agriculture by praying for rain, nor to
war by commanding the sun to stand still. We also hear the remonstrances
of those who warn us that to shake the general belief in the living God
would be to shake the general morals, public and private. This, too,
must be admitted; such a revolution of thought could no more be
accomplished without waste and desolation than a plantation of trees
could be transferred to new ground, however wholesome in itself, without
all of them languishing for a time, and many of them dying. Nor is it,
by-the-way, a thing to be presumed that a man would have taken part in a
movement having a possible atheistical issue without having taken
serious and adequate counsel in regard to that responsibility. But, let
the consequences of such a belief be as dire as they may, one thing is
certain: that the state of the facts, whatever it may be, will surely
get found out, and no human prudence can long arrest the triumphal car
of truth—no, not if the discovery were such as to drive every individual
of our race to suicide!
But it would be folly to suppose that any metaphysical theory in regard
to the mode of being of the perfect is to destroy that aspiration toward
the perfect which constitutes the essence of religion. It is true that,
if the priests of any particular form of religion succeed in making it
generally believed that religion cannot exist without the acceptance of
certain formulas, or if they succeed in so interweaving certain dogmas
with the popular religion that the people can see no essential analogy
between a religion which accepts these points of faith and one which
rejects them, the result may very well be to render those who cannot
believe these things irreligious. Nor can we ever hope that any body of
priests should consider themselves more teachers of religion in general
than of the particular system of theology advocated by their own party.
But no man need be excluded from participation in the common feelings,
nor from so much of the public expression of them as is open to all the
laity, by the unphilosophical narrowness of those who guard the
mysteries of worship. Am I to be prevented from joining in that common
joy at the revelation of enlightened principles of religion, which we
celebrate at Easter and Christmas, because I think that certain
scientific, logical, and metaphysical ideas which have been mixed up
with these principles are untenable? No; to do so would be to estimate
those errors as of more consequence than the truth—an opinion which few
would admit. People who do not believe what are really the fundamental
principles of Christianity are rare to find, and all but these few ought
to feel at home in the churches.
Footnote 44:
_Popular Science Monthly_, June, 1878.
Footnote 45:
[See Santayana, _Reason in Religion_.]
Footnote 46:
For the present purpose, the negative of a character is to be
considered as much a character as the positive, for a uniformity may
either be affirmative or negative. I do not say that no distinction
can be drawn between positive and negative uniformities.
Footnote 47:
There being 5 simple characters, with their negatives, they could be
compounded in various ways so as to make 241 characters in all,
without counting the characters _existence_ and _non-existence_, which
make up 243 or 3^5.
Footnote 48:
This principle was, I believe, first stated by Mr. De Morgan.
Footnote 49:
Not in every idea but only in the one so formulated.
Footnote 50:
[Note that this corollary is itself a theoretical inference and not an
empirical rule.]
SIXTH PAPER
DEDUCTION, INDUCTION, AND HYPOTHESIS[51]
I
The chief business of the logician is to classify arguments; for all
testing clearly depends on classification. The classes of the logicians
are defined by certain typical forms called syllogisms. For example, the
syllogism called _Barbara_ is as follows:
S is M; M is P:
Hence, S is P.
Or, to put words for letters—
Enoch and Elijah were men; all men die:
Hence, Enoch and Elijah must have died.
The “is P” of the logicians stands for any verb, active or neuter. It is
capable of strict proof (with which, however, I will not trouble the
reader) that all arguments whatever can be put into this form; but only
under the condition that the _is_ shall mean “_is_ for the purposes of
the argument” or “is represented by.” Thus, an induction will appear in
this form something like this:
These beans are two-thirds white;
But, the beans in this bag are (represented by) these beans;
∴ The beans in the bag are two-thirds white.
But, because all inference may be reduced in some way to _Barbara_, it
does not follow that this is the most appropriate form in which to
represent every kind of inference. On the contrary, to show the
distinctive characters of different sorts of inference, they must
clearly be exhibited in different forms peculiar to each. _Barbara_
particularly typifies deductive reasoning; and so long as the _is_ is
taken literally, no inductive reasoning can be put into this form.
_Barbara_ is, in fact, nothing but the application of a rule. The
so-called major premise lays down this rule; as, for example, _All men
are mortal._ The other or minor premise states a case under the rule;
as, _Enoch was a man._ The conclusion applies the rule to the case and
states the result: _Enoch is mortal._ All deduction is of this
character; it is merely the application of general rules to particular
cases. Sometimes this is not very evident, as in the following:
All quadrangles are figures,
But no triangle is a quadrangle;
Therefore, some figures are not triangles.
But here the reasoning is really this:
_Rule._—Every quadrangle is other than a triangle.
_Case._—Some figures are quadrangles.
_Result._—Some figures are not triangles.
Inductive or synthetic reasoning, being something more than the mere
application of a general rule to a particular case, can never be reduced
to this form.
If, from a bag of beans of which we know that 2/3 are white, we take one
at random, it is a deductive inference that this bean is probably white,
the probability being 2/3. We have, in effect, the following syllogism:
_Rule._—The beans in this bag are 2/3 white.
_Case._—This bean has been drawn in such a way that in the long run the
relative number of white beans so drawn would be equal to the relative
number in the bag.
_Result._—This bean has been drawn in such a way that in the long run it
would turn out white 2/3 of the time.
If instead of drawing one bean we draw a handful at random and conclude
that about 2/3 of the handful are probably white, the reasoning is of
the same sort. If, however, not knowing what proportion of white beans
there are in the bag, we draw a handful at random and, finding 2/3 of
the beans in the handful white, conclude that about 2/3 of those in the
bag are white, we are rowing up the current of deductive sequence, and
are concluding a rule from the observation of a result in a certain
case. This is particularly clear when all the handful turn out one
color. The induction then is:
These beans were in this bag.———————-
These beans are white.—————————
All the beans in the bag were white. | |
| | |
Which is but an inversion of the deductive | | |
syllogism. | | |
| | |
_Rule._—All the beans in the bag were white.—+ | |
_Case._—These beans were in the bag.——————+-+
_Result._—These beans are white.————————+
So that induction is the inference of the _rule_ from the _case_ and
_result_.
But this is not the only way of inverting a deductive syllogism so as to
produce a synthetic inference. Suppose I enter a room and there find a
number of bags, containing different kinds of beans. On the table there
is a handful of white beans; and, after some searching, I find one of
the bags contains white beans only. I at once infer as a probability, or
as a fair guess, that this handful was taken out of that bag. This sort
of inference is called _making an hypothesis_.[52] It is the inference
of a _case_ from a _rule_ and _result_. We have, then—
DEDUCTION.
_Rule._—All the beans from this bag are white.
_Case._—These beans are from this bag.
∴ _Result._—These beans are white.
INDUCTION.
_Case._—These beans are from this bag.
_Result._—These beans are white.
∴ _Rule._—All the beans from this bag are white.
HYPOTHESIS.
_Rule._—All the beans from this bag are white.
_Result._—These beans are white.
∴ _Case._—These beans are from this bag.
We, accordingly, classify all inference as follows:
Inference.
/———————^———————-|
Deductive or Analytic. Synthetic.
/————^—————|
Induction. Hypothesis.
Induction is where we generalize from a number of cases of which
something is true, and infer that the same thing is true of a whole
class. Or, where we find a certain thing to be true of a certain
proportion of cases and infer that it is true of the same proportion of
the whole class. Hypothesis is where we find some very curious
circumstance, which would be explained by the supposition that it was a
case of a certain general rule, and thereupon adopt that supposition.
Or, where we find that in certain respects two objects have a strong
resemblance, and infer that they resemble one another strongly in other
respects.
I once landed at a seaport in a Turkish province; and, as I was walking
up to the house which I was to visit, I met a man upon horseback,
surrounded by four horsemen holding a canopy over his head. As the
governor of the province was the only personage I could think of who
would be so greatly honored, I inferred that this was he. This was an
hypothesis.
Fossils are found; say, remains like those of fishes, but far in the
interior of the country. To explain the phenomenon, we suppose the sea
once washed over this land. This is another hypothesis.
Numberless documents and monuments refer to a conqueror called Napoleon
Bonaparte. Though we have not seen the man, yet we cannot explain what
we have seen, namely, all these documents and monuments, without
supposing that he really existed. Hypothesis again.
As a general rule, hypothesis is a weak kind of argument. It often
inclines our judgment so slightly toward its conclusion that we cannot
say that we believe the latter to be true; we only surmise that it may
be so. But there is no difference except one of degree between such an
inference and that by which we are led to believe that we remember the
occurrences of yesterday from our feeling as if we did so.
II
Besides the way just pointed out of inverting a deductive syllogism to
produce an induction or hypothesis, there is another. If from the truth
of a certain premise the truth of a certain conclusion would necessarily
follow, then from the falsity of the conclusion the falsity of the
premise would follow. Thus, take the following syllogism in _Barbara_:
_Rule._—All men are mortal.
_Case._—Enoch and Elijah were men.
∴ _Result._—Enoch and Elijah were mortal.
Now, a person who denies this result may admit the rule, and, in that
case, he must deny the case. Thus:
_Denial of Result._—Enoch and Elijah were not mortal.
_Rule._—All men are mortal.
∴ _Denial of Case._—Enoch and Elijah were not men.
This kind of syllogism is called _Baroco_, which is the typical mood of
the second figure. On the other hand, the person who denies the result
may admit the case, and in that case he must deny the rule. Thus:
_Denial of the Result._—Enoch and Elijah were not mortal.
_Case._—Enoch and Elijah were men.
∴ _Denial of the Rule._—Some men are not mortal.
This kind of syllogism is called _Bocardo_, which is the typical mood of
the third figure.
_Baroco_ and _Bocardo_ are, of course, deductive syllogisms; but of a
very peculiar kind. They are called by logicians indirect moods, because
they need some transformation to appear as the application of a rule to
a particular case. But if, instead of setting out as we have here done
with a necessary deduction in _Barbara_, we take a probable deduction of
similar form, the indirect moods which we shall obtain will be—
Corresponding to _Baroco_, an hypothesis;
and, Corresponding to _Bocardo_, an induction.
For example, let us begin with this probable deduction in _Barbara_:
_Rule._—Most of the beans in this bag are white.
_Case._—This handful of beans are from this bag.
∴ _Result._—Probably, most of this handful of beans are white.
Now, deny the result, but accept the rule:
_Denial of Result._—Few beans of this handful are white.
_Rule._—Most beans in this bag are white.
∴ _Denial of Case._—Probably, these beans were taken from another
bag.
This is an hypothetical inference. Next, deny the result, but accept the
case:
_Denial of Result._—Few beans of this handful are white.
_Case._—These beans came from this bag.
∴ _Denial of Rule._—Probably, few beans in the bag are white.
This is an induction.
The relation thus exhibited between synthetic and deductive reasoning is
not without its importance. When we adopt a certain hypothesis, it is
not alone because it will explain the observed facts, but also because
the contrary hypothesis would probably lead to results contrary to those
observed. So, when we make an induction, it is drawn not only because it
explains the distribution of characters in the sample, but also because
a different rule would probably have led to the sample being other than
it is.
But the advantage of this way of considering the subject might easily be
overrated. An induction is really the inference of a rule, and to
consider it as the denial of a rule is an artificial conception, only
admissible because, when statistical or proportional propositions are
considered as rules, the denial of a rule is itself a rule. So, an
hypothesis is really a subsumption of a case under a class and not the
denial of it, except for this, that to deny a subsumption under one
class is to admit a subsumption under another.
_Bocardo_ may be considered as an induction, so timid as to lose its
amplificative character entirely. Enoch and Elijah are specimens of a
certain kind of men. All that kind of men are shown by these instances
to be immortal. But instead of boldly concluding that all very pious
men, or all men favorites of the Almighty, etc., are immortal, we
refrain from specifying the description of men, and rest in the merely
explicative inference that _some_ men are immortal. So _Baroco_ might be
considered as a very timid hypothesis. Enoch and Elijah are not mortal.
Now, we might boldly suppose them to be gods or something of that sort,
but instead of that we limit ourselves to the inference that they are of
_some_ nature different from that of man.
But, after all, there is an immense difference between the relation of
_Baroco_ and _Bocardo_ to _Barbara_ and that of Induction and Hypothesis
to Deduction. _Baroco_ and _Bocardo_ are based upon the fact that if the
truth of a conclusion necessarily follows from the truth of a premise,
then the falsity of the premise follows from the falsity of the
conclusion. This is always true. It is different when the inference is
only probable. It by no means follows that, because the truth of a
certain premise would render the truth of a conclusion probable,
therefore the falsity of the conclusion renders the falsity of the
premise probable. At least, this is only true, as we have seen in a
former paper, when the word probable is used in one sense in the
antecedent and in another in the consequent.
III
A certain anonymous writing is upon a torn piece of paper. It is
suspected that the author is a certain person. His desk, to which only
he has had access, is searched, and in it is found a piece of paper, the
torn edge of which exactly fits, in all its irregularities, that of the
paper in question. It is a fair hypothetic inference that the suspected
man was actually the author. The ground of this inference evidently is
that two torn pieces of paper are extremely unlikely to fit together by
accident. Therefore, of a great number of inferences of this sort, but a
very small proportion would be deceptive. The analogy of hypothesis with
induction is so strong that some logicians have confounded them.
Hypothesis has been called an induction of characters. A number of
characters belonging to a certain class are found in a certain object;
whence it is inferred that all the characters of that class belong to
the object in question. This certainly involves the same principle as
induction; yet in a modified form. In the first place, characters are
not susceptible of simple enumeration like objects; in the next place,
characters run in categories. When we make an hypothesis like that about
the piece of paper, we only examine a single line of characters, or
perhaps two or three, and we take no specimen at all of others. If the
hypothesis were nothing but an induction, all that we should be
justified in concluding, in the example above, would be that the two
pieces of paper which matched in such irregularities as have been
examined would be found to match in other, say slighter, irregularities.
The inference from the shape of the paper to its ownership is precisely
what distinguishes hypothesis from induction, and makes it a bolder and
more perilous step.
The same warnings that have been given against imagining that induction
rests upon the uniformity of Nature might be repeated in regard to
hypothesis. Here, as there, such a theory not only utterly fails to
account for the validity of the inference, but it also gives rise to
methods of conducting it which are absolutely vicious. There are, no
doubt, certain uniformities in Nature, the knowledge of which will
fortify an hypothesis very much. For example, we suppose that iron,
titanium, and other metals exist in the sun, because we find in the
solar spectrum many lines coincident in position with those which these
metals would produce; and this hypothesis is greatly strengthened by our
knowledge of the remarkable distinctiveness of the particular line of
characters observed. But such a fortification of hypothesis is of a
deductive kind, and hypothesis may still be probable when such
reënforcement is wanting.
There is no greater nor more frequent mistake in practical logic than to
suppose that things which resemble one another strongly in some respects
are any the more likely for that to be alike in others. That this is
absolutely false, admits of rigid demonstration; but, inasmuch as the
reasoning is somewhat severe and complicated (requiring, like all such
reasoning, the use of A, B, C, etc., to set it forth), the reader would
probably find it distasteful, and I omit it. An example, however, may
illustrate the proposition: The comparative mythologists occupy
themselves with finding points of resemblance between solar phenomena
and the careers of the heroes of all sorts of traditional stories; and
upon the basis of such resemblances they infer that these heroes are
impersonations of the sun. If there be anything more in their
reasonings, it has never been made clear to me. An ingenious logician,
to show how futile all that is, wrote a little book, in which he
pretended to prove, in the same manner, that Napoleon Bonaparte is only
an impersonation of the sun. It was really wonderful to see how many
points of resemblance he made out. The truth is, that any two things
resemble one another just as strongly as any two others, if recondite
resemblances are admitted. But, in order that the process of making an
hypothesis should lead to a probable result, the following rules must be
followed:
1. The hypothesis should be distinctly put as a question, before making
the observations which are to test its truth. In other words, we must
try to see what the result of predictions from the hypothesis will be.
2. The respect in regard to which the resemblances are noted must be
taken at random. We must not take a particular kind of predictions for
which the hypothesis is known to be good.
3. The failures as well as the successes of the predictions must be
honestly noted. The whole proceeding must be fair and unbiased.
Some persons fancy that bias and counter-bias are favorable to the
extraction of truth—that hot and partisan debate is the way to
investigate. This is the theory of our atrocious legal procedure. But
Logic puts its heel upon this suggestion. It irrefragably demonstrates
that knowledge can only be furthered by the real desire for it, and that
the methods of obstinacy, of authority, and every mode of trying to
reach a foregone conclusion, are absolutely of no value. These things
are proved. The reader is at liberty to think so or not as long as the
proof is not set forth, or as long as he refrains from examining it.
Just so, he can preserve, if he likes, his freedom of opinion in regard
to the propositions of geometry; only, in that case, if he takes a fancy
to read Euclid, he will do well to skip whatever he finds with A, B, C,
etc., for, if he reads attentively that disagreeable matter, the freedom
of his opinion about geometry may unhappily be lost forever.
How many people there are who are incapable of putting to their own
consciences this question, “Do I want to know how the fact stands, or
not?”
The rules which have thus far been laid down for induction and
hypothesis are such as are absolutely essential. There are many other
maxims expressing particular contrivances for making synthetic
inferences strong, which are extremely valuable and should not be
neglected. Such are, for example, Mr. Mill’s four methods. Nevertheless,
in the total neglect of these, inductions and hypotheses may and
sometimes do attain the greatest force.
IV
Classifications in all cases perfectly satisfactory hardly exist. Even
in regard to the great distinction between explicative and ampliative
inferences, examples could be found which seem to lie upon the border
between the two classes, and to partake in some respects of the
characters of either. The same thing is true of the distinction between
induction and hypothesis. In the main, it is broad and decided. By
induction, we conclude that facts, similar to observed facts, are true
in cases not examined. By hypothesis, we conclude the existence of a
fact quite different from anything observed, from which, according to
known laws, something observed would necessarily result. The former, is
reasoning from particulars to the general law; the latter, from effect
to cause. The former classifies, the latter explains. It is only in some
special cases that there can be more than a momentary doubt to which
category a given inference belongs. One exception is where we observe,
not facts similar under similar circumstances, but facts different under
different circumstances—the difference of the former having, however, a
definite relation to the difference of the latter. Such inferences,
which are really inductions, sometimes present nevertheless some
indubitable resemblances to hypotheses.
Knowing that water expands by heat, we make a number of observations of
the volume of a constant mass of water at different temperatures. The
scrutiny of a few of these suggests a form of algebraical formula which
will approximately express the relation of the volume to the
temperature. It may be, for instance, that _v_ being the relative
volume, and _t_ the temperature, a few observations examined indicate a
relation of the form—
_v_ = 1 + _at_ + _bt_^2 + _ct_^3.
Upon examining observations at other temperatures taken at random, this
idea is confirmed; and we draw the inductive conclusion that all
observations within the limits of temperature from which we have drawn
our observations could equally be so satisfied. Having once ascertained
that such a formula is possible, it is a mere affair of arithmetic to
find the values of _a_, _b_, and _c_, which will make the formula
satisfy the observations best. This is what physicists call an empirical
formula, because it rests upon mere induction, and is not explained by
any hypothesis.
Such formulæ, though very useful as means of describing in general terms
the results of observations, do not take any high rank among scientific
discoveries. The induction which they embody, that expansion by heat (or
whatever other phenomenon is referred to) takes place in a perfectly
gradual manner without sudden leaps or inummerable fluctuations,
although really important, attracts no attention, because it is what we
naturally anticipate. But the defects of such expressions are very
serious. In the first place, as long as the observations are subject to
error, as all observations are, the formula cannot be expected to
satisfy the observations exactly. But the discrepancies cannot be due
solely to the errors of the observations, but must be partly owing to
the error of the formula which has been deducted from erroneous
observations. Moreover, we have no right to suppose that the real facts,
if they could be had free from error, could be expressed by such a
formula at all. They might, perhaps, be expressed by a similar formula
with an infinite number of terms; but of what use would that be to us,
since it would require an infinite number of coefficients to be written
down? When one quantity varies with another, if the corresponding values
are exactly known, it is a mere matter of mathematical ingenuity to find
some way of expressing their relation in a simple manner. If one
quantity is of one kind—say, a specific gravity—and the other of another
kind—say, a temperature—we do not desire to find an expression for their
relation which is wholly free from numerical constants, since if it were
free from them when, say, specific gravity as compared with water, and
temperature as expressed by the Centigrade thermometer, were in
question, numbers would have to be introduced when the scales of
measurement were changed. We may, however, and do desire to find
formulas expressing the relations of physical phenomena which shall
contain no more arbitrary numbers than changes in the scales of
measurement might require.
When a formula of this kind is discovered, it is no longer called an
empirical formula, but a law of Nature; and is sooner or later made the
basis of an hypothesis which is to explain it. These simple formulæ are
not usually, if ever, exactly true, but they are none the less important
for that; and the great triumph of the hypothesis comes when it explains
not only the formula, but also the deviations from the formula. In the
current language of the physicists, an hypothesis of this importance is
called a theory, while the term hypothesis is restricted to suggestions
which have little evidence in their favor. There is some justice in the
contempt which clings to the word hypothesis. To think that we can
strike out of our own minds a true preconception of how Nature acts, in
a vain fancy. As Lord Bacon well says: “The subtlety of Nature far
exceeds the subtlety of sense and intellect: so that these fine
meditations, and speculations, and reasonings of men are a sort of
insanity, only there is no one at hand to remark it.” The successful
theories are not pure guesses, but are guided by reasons.
The kinetical theory of gases is a good example of this. This theory is
intended to explain certain simple formulæ, the chief of which is called
the law of Boyle. It is, that if air or any other gas be placed in a
cylinder with a piston, and if its volume be measured under the pressure
of the atmosphere, say fifteen pounds on the square inch, and if then
another fifteen pounds per square inch be placed on the piston, the gas
will be compressed to one-half its bulk, and in similar inverse ratio
for other pressures. The hypothesis which has been adopted to account
for this law is that the molecules of a gas are small, solid particles
at great distances from each other (relatively to their dimensions), and
moving with great velocity, without sensible attractions or repulsions,
until they happen to approach one another very closely. Admit this, and
it follows that when a gas is under pressure what prevents it from
collapsing is not the incompressibility of the separate molecules, which
are under no pressure at all, since they do not touch, but the pounding
of the molecules against the piston. The more the piston falls, and the
more the gas is compressed, the nearer together the molecules will be;
the greater number there will be at any moment within a given distance
of the piston, the shorter the distance which any one will go before its
course is changed by the influence of another, the greater number of new
courses of each in a given time, and the oftener each, within a given
distance of the piston, will strike it. This explains Boyle’s law. The
law is not exact; but the hypothesis does not lead us to it exactly.
For, in the first place, if the molecules are large, they will strike
each other oftener when their mean distances are diminished, and will
consequently strike the piston oftener, and will produce more pressure
upon it. On the other hand, if the molecules have an attraction for one
another, they will remain for a sensible time within one another’s
influence, and consequently they will not strike the wall so often as
they otherwise would, and the pressure will be less increased by
compression.
When the kinetical theory of gases was first proposed by Daniel
Bernoulli, in 1738, it rested only on the law of Boyle, and was
therefore pure hypothesis. It was accordingly quite naturally and
deservedly neglected. But, at present, the theory presents quite another
aspect; for, not to speak of the considerable number of observed facts
of different kinds with which it has been brought into relation, it is
supported by the mechanical theory of heat. That bringing together
bodies which attract one another, or separating bodies which repel one
another, when sensible motion is not produced nor destroyed, is always
accompanied by the evolution of heat, is little more than an induction.
Now, it has been shown by experiment that, when a gas is allowed to
expand without doing work, a very small amount of heat disappears. This
proves that the particles of the gas attract one another slightly, and
but very slightly. It follows that, when a gas is under pressure, what
prevents it from collapsing is not any repulsion between the particles,
since there is none. Now, there are only two modes of force known to us,
force of position or attractions and repulsions, and force of motion.
Since, therefore, it is not the force of position which gives a gas its
expansive force, it must be the force of motion. In this point of view,
the kinetical theory of gases appears as a deduction from the mechanical
theory of heat. It is to be observed, however, that it supposes the same
law of mechanics (that there are only those two modes of force) which
holds in regard to bodies such as we can see and examine, to hold also
for what are very different, the molecules of bodies. Such a supposition
has but a slender support from induction. Our belief in it is greatly
strengthened by its connection with the law of Boyle, and it is,
therefore, to be considered as an hypothetical inference. Yet it must be
admitted that the kinetical theory of gases would deserve little
credence if it had not been connected with the principles of mechanics.
The great difference between induction and hypothesis is, that the
former infers the existence of phenomena such as we have observed in
cases which are similar, while hypothesis supposes something of a
different kind from what we have directly observed, and frequently
something which it would be impossible for us to observe directly.
Accordingly, when we stretch an induction quite beyond the limits of our
observation, the inference partakes of the nature of hypothesis. It
would be absurd to say that we have no inductive warrant for a
generalization extending a little beyond the limits of experience, and
there is no line to be drawn beyond which we cannot push our inference;
only it becomes weaker the further it is pushed. Yet, if an induction be
pushed very far, we cannot give it much credence unless we find that
such an extension explains some fact which we can and do observe. Here,
then, we have a kind of mixture of induction and hypothesis supporting
one another; and of this kind are most of the theories of physics.
V
That synthetic inferences may be divided into induction and hypothesis
in the manner here proposed,[53] admits of no question. The utility and
value of the distinction are to be tested by their applications.
Induction is, plainly, a much stronger kind of inference than
hypothesis; and this is the first reason for distinguishing between
them. Hypotheses are sometimes regarded as provisional resorts, which in
the progress of science are to be replaced by inductions. But this is a
false view of the subject. Hypothetic reasoning infers very frequently a
fact not capable of direct observation. It is an hypothesis that
Napoleon Bonaparte once existed. How is that hypothesis ever to be
replaced by an induction? It may be said that from the premise that such
facts as we have observed are as they would be if Napoleon existed, we
are to infer by induction that _all_ facts that are hereafter to be
observed will be of the same character. There is no doubt that every
hypothetic inference may be distorted into the appearance of an
induction in this way. But the essence of an induction is that it infers
from one set of facts another set of similar facts, whereas hypothesis
infers from facts of one kind to facts of another. Now, the facts which
serve as grounds for our belief in the historic reality of Napoleon are
not by any means necessarily the only kind of facts which are explained
by his existence. It may be that, at the time of his career, events were
being recorded in some way not now dreamed of, that some ingenious
creature on a neighboring planet was photographing the earth, and that
these pictures on a sufficiently large scale may some time come into our
possession, or that some mirror upon a distant star will, when the light
reaches it, reflect the whole story back to earth. Never mind how
improbable these suppositions are; everything which happens is
infinitely improbable. I am not saying that _these_ things are likely to
occur, but that _some_ effect of Napoleon’s existence which now seems
impossible is certain nevertheless to be brought about. The hypothesis
asserts that such facts, when they do occur, will be of a nature to
confirm, and not to refute, the existence of the man. We have, in the
impossibility of inductively inferring hypothetical conclusions, a
second reason for distinguishing between the two kinds of inference.
A third merit of the distinction is, that it is associated with an
important psychological or rather physiological difference in the mode
of apprehending facts. Induction infers a rule. Now, the belief of a
rule is a habit. That a habit is a rule active in us, is evident. That
every belief is of the nature of a habit, in so far as it is of a
general character, has been shown in the earlier papers of this series.
Induction, therefore, is the logical formula which expresses the
physiological process of formation of a habit. Hypothesis substitutes,
for a complicated tangle of predicates attached to one subject, a single
conception. Now, there is a peculiar sensation belonging to the act of
thinking that each of these predicates inheres in the subject. In
hypothetic inference this complicated feeling so produced is replaced by
a single feeling of greater intensity, that belonging to the act of
thinking the hypothetic conclusion. Now, when our nervous system is
excited in a complicated way, there being a relation between the
elements of the excitation, the result is a single harmonious
disturbance which I call an emotion. Thus, the various sounds made by
the instruments of an orchestra strike upon the ear, and the result is a
peculiar musical emotion, quite distinct from the sounds themselves.
This emotion is essentially the same thing as an hypothetic inference,
and every hypothetic inference involves the formation of such an
emotion. We may say, therefore, that hypothesis produces the _sensuous_
element of thought, and induction the _habitual_ element. As for
deduction, which adds nothing to the premises, but only out of the
various facts represented in the premises selects one and brings the
attention down to it, this may be considered as the logical formula for
paying attention, which is the _volitional_ element of thought, and
corresponds to nervous discharge in the sphere of physiology.
Another merit of the distinction between induction and hypothesis is,
that it leads to a very natural classification of the sciences and of
the minds which prosecute them. What must separate different kinds of
scientific men more than anything else are the differences of their
_techniques_. We cannot expect men who work with books chiefly to have
much in common with men whose lives are passed in laboratories. But,
after differences of this kind, the next most important are differences
in the modes of reasoning. Of the natural sciences, we have, first, the
classificatory sciences, which are purely inductive—systematic botany
and zoölogy, mineralogy, and chemistry. Then, we have the sciences of
theory, as above explained—astronomy, pure physics, etc. Then, we have
sciences of hypothesis—geology, biology, etc.
There are many other advantages of the distinction in question which I
shall leave the reader to find out by experience. If he will only take
the custom of considering whether a given inference belongs to one or
other of the two forms of synthetic inference given on page 134, I can
promise him that he will find his advantage in it, in various ways.
Footnote 51:
_Popular Science Monthly_, August, 1878.
Footnote 52:
[Later Pierce called it _presumptive inference_. See Baldwin’s
_Dictionary_ art. _Probable Inference_.]
Footnote 53:
This division was first made in a course of lectures by the author
before the Lowell Institute, Boston, in 1866, and was printed in the
_Proceedings of the American Academy of Arts and Sciences_, for April
9, 1867.