INDUCTIVE REASONING.
=1. INDUCTIVE AND DEDUCTIVE REASONING DISTINGUISHED.=
It has been remarked that inference is the process of deriving a
judgment from one or two antecedent judgments, and that mediate
inference is inference by means of a middle term. But to reason by
means of a middle term necessitates two judgments; hence mediate
inference might be defined as the process of deriving a judgment from
_two_ antecedent judgments. In this treatment mediate inference and
reasoning have been used interchangeably. This, then, becomes our
definition for reasoning: _Reasoning is the process of deriving a
judgment from two antecedent judgments._
The syllogism results when the process of reasoning is formally
clothed in words. Moreover, the conclusion of the syllogism may be
_more_ general than the premises or _less_ general. This suggests the
two important kinds of reasoning; namely, inductive and deductive.
Inductive reasoning is reasoning from less general premises to a more
general conclusion. Deductive reasoning is reasoning from more general
premises to a less general conclusion.
ILLUSTRATION:
_Inductive Syllogism._ _Deductive Syllogism._
The robin, crow, sparrow, All birds have wings,
etc. have wings, The robin, crow, sparrow,
The robin, crow, sparrow, etc. are birds,
etc. are birds, ∴ The robin, crow, sparrow,
∴ All birds have wings. etc. have wings.
Iron, silver, gold, etc. are All metals are elements,
elements, Iron, silver, gold, etc. are
Iron, silver, gold, etc. are metals,
metals, ∴ Iron, silver, gold, etc. are
∴ All metals are elements. elements.
Boston, New York, Chicago, All large cities have fine
etc. have fine harbors, harbors.
Boston, New York, Chicago, Boston, New York, Chicago, etc.
etc. are large cities, are large cities,
∴ All large cities have fine ∴ Boston, New York, Chicago, etc.
harbors, have fine harbors.
The student who is sufficiently familiar with the canons of the
deductive syllogism will at once detect the fallacy of _illicit
minor_ in the foregoing inductive syllogisms; i. e., “_birds_” when
used as the predicate of the minor premise of the first syllogism
is undistributed, but as the subject of the conclusion “_birds_” is
distributed. The same might be said concerning the terms “_metals_”
and “_large cities_.” A portion of this chapter will be devoted to
answering this criticism. At this point it may be stated that _the
inductive syllogism is not supposed to conform perfectly to the canons
of the deductive syllogism_.
=2. THE INDUCTIVE HAZARD.=
Referring to the first inductive syllogism of section _one_, it is
assumed that the robin, crow and sparrow are representative birds, and
that we are thus justified in concluding that if these _type_ birds
have wings, then _all_ birds must have wings. Of course this is more
or less of a conjecture or “a hazard”; since birds without wings may
exist in some undiscovered corner of the globe. However, inasmuch
as the generalization concerns a representative quality, we deem the
assumption fairly well founded. The logical right to take this “leap
into the unknown” will be discussed later. It will profit us at this
time to realize more fully how essential the “inductive hazard” is
to the progress of the world. When the Schoolmen of mediæval time
refused to venture, they failed to progress, and thus came the dark
days. Whenever man has ignored this God given instinct which leads to
discovery, the world has stood still. _This willingness to “take a leap
into the dark” with the hope of finding, in the shadow, truth which
would enhance man’s power and increase his serviceableness, has given
to the world about all that is worth while._ It was the spirit of the
_hazard_ which pushed Columbus to the discovery of a new world; which
gave Newton the secrets of the motions of the universe; which enabled
Edison to harness a multitude of lurking forces; and Morse and Bell to
reduce distance to its lowest terms. In ordinary affairs with ordinary
men those succeed _best_ who manifest _most_ a safe, steady, persistent
spirit of discovery. _Here, then, in the “inductive hazard” have we a
most important phase of school life which, in this day of making the
work easy, is being sadly neglected._ On the other hand, an unregulated
and insane spirit of venture may result in a great waste of energy, and
in the development of low ideals of recklessness and inaccuracy. The
“inductive hazard” must be cultivated; yet it must be regulated as well,
and, as the reader already realizes, logic needs to concern itself
mainly with this _regulative_ aspect.
=3. THE COMPLEXITY OF THE PROBLEM OF INDUCTION.=
The problem of induction is much more complex than that of deduction
because of these reasons: First. Deduction as a process of reasoning
was the only kind discussed by the logicians for _two thousand years_.
Aristotle is called the father of deductive logic and this Intellectual
Giant, the greatest of ancient time and possibly of all time, so
perfected the form of deductive reasoning that, up to the time of
Francis Bacon, no scholar possessed the temerity to gainsay its
supremacy in the field of logical reasoning. For twenty centuries
Aristotle’s Deductive Logic was the _Logicians’ Bible_. On the other
hand, inductive reasoning, though it was briefly discussed by Aristotle,
received little attention till the versatile Francis Bacon placed it
upon the stage of the thinking world. This makes deduction nearly two
thousand years older than induction. Time, by eliminating the personal
equation and exposing in various ways fallacious thinking, tends to
unify and universalize truth. Hence, logicians are agreed so far as the
fundamentals of deductive logic are concerned, but are still at odds
over the true conception and use of inductive logic.
A second reason for this confused status in the field of inductive
logic is the fact of its being more closely related to the events of
every day living. Induction is the natural method of childhood; the
popular method of the school room; and the most used method of common
life. In consequence its ramifications are so varied and multitudinous,
that it will take centuries of thinking to reduce the doctrine of
induction to that uniformity and definiteness which so distinguishes
deduction.
=4. THE VARIOUS CONCEPTIONS OF INDUCTION.=
The attending quotations will give the student a fair idea of the
leading conceptions concerning induction:
(1) “Induction is the process by which we conclude that what is true
of certain individuals of a class is true of the whole class, or that
what is true at certain times will be true under similar circumstances
at all times.” “Induction, as above defined, is a process of inference;
it proceeds from the known to the unknown.” “Any process in which what
seems the conclusion is no wider than the premises from which it is
drawn, does not fall within the meaning of the term.”――J. S. Mill, A
System of Logic, 1892, p. 175.
(2) “An induction is a generalization or an inference based upon
propositions that state observed facts.” “The truth inferred may be
general or particular, but it must be one which we cannot perceive in a
single act of observation.”――Ballentine’s Inductive Logic, 1896, p. 14.
(3) “Induction is the process of inference by which we get at general
truths from particular facts or cases.”――Ryland’s Logic, 1900, p. 148.
(4) “Induction may be defined as the legitimate inference of the
general from the particular, or, of the more general from the less
general.”――Fowler, 1905, p. 10, Vol. 2.
(5) “The term induction has been used by logicians to denote this
leap of the mind from the limitations of its positive knowledge
to belief in universal laws.” “In pedagogy, however, the term
is applied to the whole process of arriving at general truths or
principles.”――Salisbury’s Theory of Teaching, p. 156.
=5. INDUCTION AND DEDUCTION CONTIGUOUS PROCESSES.=
If there is one thing above another which modern logic is emphasizing
it is the _unity of the mind_ and the _contiguity of thinking_.
Induction and deduction are _dove-tailed processes_ which characterize
all thinking worthy of the name. Where induction ceases, deduction
commences, and vice versa. It becomes the function of inductive
thinking to establish a connection between what has been experienced
and what has not been experienced. Therefore, the conclusion of an
induction must always contain more than is implied in the premises. The
premises denote facts which have been observed; whereas the conclusion
denotes the observed facts of the premises plus analogous facts which
have _not been observed_. Inductive thought ventures into the unknown,
and attempts to establish a bond of connection between it and something
already known. Induction seeks new knowledge, and does so by taking
that “leap into the dark” already referred to as the “inductive hazard.”
As soon as the mind reaches a universal truth, it sets to work to
clarify this truth. Such is accomplished by reference to other facts
which the universal is supposed to include; and this application of the
general to the particular is _deduction_. _Induction discovers_ the new
knowledge while _deduction clarifies_ it.
=6. INDUCTION AN ASSUMPTION.=
In this treatment induction as a general process has been subdivided
into induction as a mode of inference and induction as a method.
_Induction as a mode of inference is the process of reasoning from less
general premises to a more general conclusion; whereas induction as a
method is a procedure from the observation of individual facts to a
realization of a universal truth._ In either case the conclusion of an
inductive process always implies more than is contained in the premises.
This gives to the conclusion an uncertainty. No induction is absolutely
free from doubt except the so-called perfect induction, which form will
receive attention in a later section.
=7. UNIVERSAL CAUSATION.=
All inductive assumptions are made possible because of two
laws――_universal causation_ and _uniformity of nature_.
The law of universal causation may be stated in this wise:
_Nothing can occur without a cause and every cause has its
effect._ “It is a universal truth, that every fact which has
a beginning has a cause.”――Mill.
SIMPLE ILLUSTRATIONS OF UNIVERSAL CAUSATION.
The sun rises in the east. The boy throws a stone through the
window. A democratic wave sweeps the country. Prices of food
stuff are high. The bullet, shot out into space, finally falls
to the earth. Each one of these occurrences has a cause.[12]
That universal causation is a fundamental condition of all
induction may be further illustrated. The astronomer notes that
the stars in the vicinity of Vega seem to be moving outward
_from_ a common center; whereas in the opposite part of the sky
the stars seem to be moving inward _toward_ a common center.
Having observed this phenomenon, the astronomer at once looks
for a cause. Finally he decides that the phenomenon is due to
the fact that the sun, with his attending family, is moving
towards Vega. Arranged, the argument may take this form:
The stars in the vicinity of Vega seem to be moving outward
from a common center, whereas in the opposite part of the sky
the stars seem to be moving inward, When descending a mountain
the trees at the foot seem to move outward from, and those at
the top inward toward, a common center, When riding on the train
the ties in front seem to move outward while those in the rear
seem to move inward. From this we conclude that the sun with
the Earth and other planets is moving toward a spot in the sky
near Vega. Were it not for the assumption that the phenomenon
relative to the stars _had a cause_ there could have been no
induction. Moreover, any investigation concerning “democratic
waves,” “prices of food stuffs,” etc., must assume as a starting
point that these phenomena have causes.
It would appear that the mind is not satisfied with a mere
passive observation of the occurrences of the world but is
inclined to reach out for the “whys and wherefores.” Due partly
to this reason, “universal causation” is often referred to as
an _a priori_ law; meaning that it is a law which cannot be
_proved_, but must be _assumed_ in all thinking.
=8. THE LAW OF THE UNIFORMITY OF NATURE.=
Law stated: _The same antecedents are invariably followed by the
same consequents._ “That the course of nature is uniform is the
fundamental principle of induction.”――Mill. “It is not enough to
feel assured that nothing can happen without a cause (causation);
I must also feel assured that the same cause will invariably be
followed by the same effect.”――Fowler.
Referring to the observed phenomenon of the outward movement
of the stars about Vega, the astronomer might advance as an
hypothesis the fact of the solar system’s movement toward
Vega. Having done this he could then experiment with a view of
verifying this hypothesis. In this experiment he would attempt
to introduce the same cause surrounded by similar circumstances,
and then watch for the same effect. To make it concrete: suppose
the astronomer paints the side of a barn dark blue and bedecks
this with stars of white. Then taking a position as far removed
from the blue surface as his eyesight will permit, he runs
toward the barn watching the apparent movement of the artificial
stars. A similar experiment could be performed by substituting
for the _starred_ barn, the stumps on a side hill. In both
experiments he assumes that like conditions will be followed by
constant results. That is, in these particular cases, advancing
toward a group of objects is always followed by an apparent
separation of said objects.
This law of uniformity of nature not only underlies inductive
thinking but it really conditions all thinking. It implies
that the universe is a rational system functioning in a uniform
manner. Moreover, it suggests that the interpretations of the
mind are likewise uniform and whenever the mind proves a fact
to be a universal truth, this truth will _always_ remain a
truth unless the conditions change. In fact were it not for the
uniformity of nature, all activity whatsoever would be rendered
nugatory. Because of this law we have a right to assume that
grinding a knife under right conditions will always tend
to sharpen it; that surrounding a live seed with a proper
environment will result in growth; that water at the same
altitude will boil at a constant temperature, etc., etc.
The student will discern the close connection between these two
laws and the laws of thought. There is really no distinctive
mark between the law of causation and the law of sufficient
reason, while “uniformity of nature” includes identity as
one of its distinctive features. The laws differ, however,
in their application, “causation” and “uniformity of nature”
conditioning inductive thinking, while the others are concerned
with deductive thinking.
Because “uniformity of nature” expresses facts of experience, it
is regarded as an _empirical_ law, as contrasted with the law of
causation, which is supposed to be based upon an innate mental
conception or is an _a priori_ law.
=9. INDUCTIVE ASSUMPTION JUSTIFIED.=
The function of induction seems to be to universalize particulars. The
mind of man has ever been engaged in establishing connections among the
concrete experiences of daily life. This ability of his to generalize
his individual experiences has been one of the chief agencies in
elevating him to the position of “King of the animal world.” In this
disposition to generalize man has taken it for granted that nature is
honest; that what she tells him under given conditions, she will tell
him again under identical conditions. To put it in logical terms man
can depend upon the _invariability_ of nature’s activities, or _upon
the uniformity of nature_. Here, then, is one of the most fundamental
laws not only of induction but of all activity. But this law implies a
second quite as fundamental. If every cause is invariably followed by
the same effect under like conditions, then it is thereby implied that
every cause has an affect and every event is due to some cause. This,
too, is invariable. In consequence of these facts man is justified
in thinking that nature is not only honest and therefore “she
gives me _confidence_, but her every activity _means something_ and
therefore she arouses my _curiosity_.” “Uniformity of nature” engenders
confidence, “universal causation” inspires the spirit of discovery
and with these two weapons man is willing to venture into the jungle
of the unknown. _Why is man eager to undertake the “inductive hazard?”
Because, through the laws of universal causation and uniformity
of nature, his curiosity is aroused, and he is given confidence in
nature’s activities._
=10. THREE FORMS OF INDUCTIVE RESEARCH.=
Induction is a matter of universalizing less universal experiences.
In this the process may assume any one of three forms, namely:
(1) Induction by simple enumeration; (_inductio per enumerationem_);
(2) Induction by analogy; (3) Induction by analysis.
THREE FORMS ILLUSTRATED:
(1) _Simple enumeration._
Having observed a few instances the generalization is, “All birds have
wings.” The certitude of this may now be strengthened by observing more
birds and finding without exception that each has wings.
(2) _Analogy._
By noting on Mars geometric markings which resemble canals, the
generalization is vouchsafed that Mars is inhabited by human beings.
Other similarities in atmospheric conditions, existence of land and
water, etc., tend to make this generalization more plausible.
(3) _Analysis._
By analyzing water taken from a certain spring, it is found to contain
hydrogen and oxygen in the proportion of 1 to 8; in consequence a
generalization to this effect is posited. Analyses of specimens from
other sources yield similar results and thus the generalization is
given greater certitude.
As a usual thing the particular form which the induction assumes
depends on the nature of the topic under investigation and also on
the mental make-up of the investigator. The general statement that
all birds have wings could hardly be derived by means of analogy or
analysis, but is a matter of a casual observation of many instances.
Moreover, that mind given to accurate observation, but not inclined to
note resemblances or to carry on experiments, would naturally follow
the first inductive type. On the other hand, simple enumeration would
be impossible in questions like the habitability of Mars, and would
yield no results in cases requiring definite scientific experimentation
like electrolysis.
It is worthy of note that some topics lend themselves to all three
modes of procedure. To wit: (1) Enumeration. Without being taught the
rule the child is given a list of examples involving the dividing of
a decimal by a decimal and is asked to solve them. By comparing his
answers with those in the book, he somewhat accidentally discovers
what seems to be the correct rule for pointing off in the quotient.
By following this rule and each time comparing answers he establishes
the truth. (2) Analogy. If .24 ÷ .6 is the first example, the child
may resort to the well known process of dividing a common fraction by a
( 24 60 24 4 )
common fraction, ( ――― ÷ ――― = ―― = ――, ) then, because of their close
( 100 100 60 10 )
resemblance, he may reason that decimal fractions should yield the same
result. (3) Analysis. Here the child reasons that since division of
decimals is the inverse of multiplication of decimals, the rule for
pointing off might be the inverse of the multiplication rule. By trying
this out and proving his answer in each example, he becomes convinced
of the correctness of his reasoning.
=11. INDUCTION BY SIMPLE ENUMERATION.=
As its name implies this type of inductive research consists
_in observing many instances which may exemplify the particular
uniformity under consideration_. The process is quantitative rather
than qualitative, the certitude of the generalization depending on
the _mass of facts collected_ rather than on any striking resemblance
or any detailed analysis. The aim is to observe, accurately if not
scientifically, instance after instance until all doubt is removed.
The outcome of such observation may be three fold. (1) The enumeration
may be complete. This gives the so-called “perfect induction” which
will receive attention later. (2) The enumeration may be incomplete
and _without exceptions_; generalizing in this way from uncontradicted
experience gives what are termed “_empirical_” truths. (3) The
enumeration may be incomplete _with_ exceptions. It is obvious that
this type of induction could give no valid generalization; but the
result may be put in the form of a ratio between the uniformities and
the exceptions. Such a procedure is a mere “_calculation of chances_”
and the result simply an expressed probability.
THE THREE KINDS OF SIMPLE ENUMERATION ILLUSTRATED.
The subject to receive investigation is a _school examination_.
(1) Complete enumeration. Every paper is read and marked; this leads
to the generalization, “All the class have passed.”
(2) Incomplete enumeration with no exceptions. Representative
papers are read and marked in which no failures are found.
Generalization, “Probably all of the class have passed.”
(3) Incomplete enumeration with exceptions. Representative papers are
read and marked in which there are 20 failures out of the hundred
papers examined. Generalization, “Probably about 80% of the class
have passed.”
Briefly, simple enumeration may take the form of (1) _a perfect
induction_, (2) _a probable induction_, (3) _a mere calculation of
chances_. The first necessitates completed experience, the second
uncontradicted experience and the third contradicted experience.
=12. INDUCTION BY ANALOGY.=
_Induction by analogy assumes that if two (or more) things resemble
each other in certain respects, they belong to the same type, and,
therefore, any fact known of the one may be affirmed of the other._
THE TYPE.
As the definition implies, analogy involves an extensive use of types;
let us, therefore, become better acquainted with them as instruments
in analogical inductions. A type is _one_ of a group which embodies the
_essential characteristics_ of that group. How easy and natural it is
to dismiss a complex topic with the citing of an example which may be
regarded as a type; how common is the use of examples in the school
room! On second thought it becomes apparent that analogical induction
by example or type is the most common of all forms of induction either
as a method or a mode of inference. _Analogy by example (or type)
assumes that if two or more things are of the same type, they resemble
each other in every essential property._
Illustrations of analogical inductions by example or type.
(1) _Mathematics._
Example: a + b
a + b
―――――――――――――
a² + ab
+ ab + b²
―――――――――――――
a² + 2ab + b²
Inductive Inference: The square of the sum of two quantities is equal
to the square of the first, plus twice the first by the second, plus
the square of the second.
(2) _Nature._
This corn sent me as a sample produced heavy, full ears, and many of
them; hence (inductive inference), if I plant corn _like this sample_
under like conditions, I will receive in return heavy, full ears, and
many of them.
(3) _Geography._
Cities like New York, located on the coast, possess a larger foreign
element than the inland cities like Philadelphia.
(4) _Grammar._
A noun is the name of anything, as the examples, “George Washington”
and “house” would indicate.
In deriving a generalization from one or two examples the prime
essential is to select types which are _truly representative_.
Often the example used is a special type and in consequence does not
exemplify all of the essential characteristics of the group. To teach
the nature of a parallelogram by using a rectangle only, is an easy way
to commit this error; or one may affirm that the class can easily cover
the work, when the judgment is based entirely on knowledge concerning
the _brightest_ one of the grade.
Type work when judicially used is a positive time saver and a very
present help in times of perplexity. Let the skillful teacher use types
and examples extensively yet cautiously.
THE MARK OF SIMILARITY.
As opposed to analogy by type there is a second form; namely, _analogy
by one or more similar marks or qualities_. This form is best described
by the definition: _When two things resemble each other in a few marks
or qualities they resemble each other in other marks or qualities._
Illustrations of analogy by marks.
(1) Noting that two students have the same surname, I infer that they
are brothers.
(2) A man with a book under his arm rings the door bell and asks to
see “the lady of the house.” At once the conclusion is drawn that
the caller is a book agent.
(3) Two automobiles, resembling each other in shape of body, force
one to the conclusion that the machines are of the same make.
THE ERRORS OF ANALOGY BY MARKS OF SIMILARITY.
It follows that analogy by example gives generalizations of much
greater certitude than analogy by one or two marks of resemblance.
Here is a field bespattered from boundary to boundary with erroneous
thinking. The principle of resemblance being an innate tendency, this
form of error is most common with the immature. The child reasons
by analogy when he invests the poodle with the despised cognomen of
“kitty”; or honors every man who wears glasses with “papa.” In the
childhood of the race natural events were interpreted by means of
analogy. The wind blowing through the trees made sounds much like the
human voice; hence these noises were attributed to spirits. Primeval
man was led to believe by analogy that everything which moved was alive.
We may, therefore, think of our revered forbear as engaged in the
undignified task of running after his shadow, or chasing a leaf around
a stump.
THE VALUE OF ANALOGY.
Analogy being rich in its suggestions is the favored process of the
scientist and inventor. Newton reasoned by analogy when he tentatively
affirmed of the moon what he positively knew of the apple. Franklin’s
reasoning was analogical when he discovered the identity of the
electric spark and lightning. Because this form of induction so often
leads to error and at best involves a degree of probability far below
induction by analysis, some logicians are inclined to ignore its
generalizations altogether. Others deem this a mistake because of these
reasons: First. Analogy is serviceable to a high degree in suggesting
hypotheses which may be advanced either for the purpose of explanation
or verification. It has already been indicated that analogy is the
common instrument used by the inventor and discoverer. Second. The
principle of analogy, in reality, lies at the basis of classification;
because in this, things are grouped according to their resemblances.
Third. Analogical induction affords valuable training in originality
and initiative. A mind which easily and naturally discerns analogies
is “fertile in new ideas.”
REQUIREMENTS OF A TRUE ANALOGY.
It has been remarked that the certitude of an induction by simple
enumeration depends upon the number of uncontradicted instances. In
analogy the case is different as the process emphasizes the _weight_
of the points of resemblance rather than the _number_. In substance
the requirements of a logical analogy are three.
First. _The points of resemblance must be representative and not
exceptional._ For example: The argument that Mars is inhabited because
it has two moons is of little worth, since we have no proof that
moonshine is essential to life; this point of resemblance is not
representative. On the other hand, if the basis of argument is the
fact that Mars has an atmosphere, the conclusion carries some weight;
as air seems to be essential to life.
Second. _The points of resemblance must outweigh the points of
difference._ That is, the ratio of probability must always be in
favor of the resembling instances. Since it is not a matter of numbers
but of weight, a numerical proportion like this would be misleading:
Resemblances: Differences = 10:6. It is obvious that the six
differences might more than outweigh the ten resemblances. The safer
way, if it were possible, would be to attach a _value_ to each point of
resemblance or difference, and then express the proportion in terms of
the sums of these values.
Third. _There must be no difference which is absolutely incompatible
with the affirmation which we wish to prove._ For example, the fact
that the moon has no atmosphere renders nugatory any attempt to prove
the habitability of the moon.
=13. INDUCTION BY ANALYSIS.=
This, the third form of inductive research, is by far the most
important. Simple enumeration, because it depends upon the number of
observed instances, consumes much time; while we have already noted how
easy it is for analogy to lead to error. At the best, the conclusion
of these methods must be subjected to analytic investigation, if we are
seeking universal validity. Induction by analysis is superior to the
other forms because it secures a _higher degree of probability_ and is
a _positive time saver_.
Defined. We have learned that analysis is the process of separating
a whole into its related parts. We thus define induction by analysis
as _the process of separating a whole into its parts with a view of
deriving a generalization relative to the nature and causal connection
of these parts_.
ILLUSTRATIONS:
(1) Concerning the generalization that “all birds have wings,” it
becomes possible to observe in detail the nature of the wings and
advance the hypothesis that these wings are designed for aërial
navigation. This hypothesis may then be strengthened by observing that
the entire structure of the bird is adapted to flying.
(2) If it were possible to analyze the atmosphere, water, and soil of
Mars, and should such analysis reveal a composition similar to that of
the earth, it would illustrate well not only the _method_ of analysis
but also its superiority over the other methods of investigation.
(3) The physician, in diagnosing a “case,” observes that the symptoms
resemble those of typhoid; but to be positive of the truth of his
diagnosis, he takes a blood test. Noting the resemblances is induction
by analogy; but the blood test involves induction by analysis.
Induction by analysis concerns hypothesis, observation, and experiment,
including Mill’s experimental methods. These topics will receive due
attention in the chapters which follow. It will be sufficient to close
this discussion with a brief treatment of perfect induction, and
traduction.
=14. PERFECT INDUCTION.=
As has been indicated under simple enumeration, _a perfect induction
is one in which the premises enumerate all the instances denoted by the
conclusion_.
ILLUSTRATIONS:
(1) A, B, C, D, and E are all Reactionaries,
(All) The members of the committee are A, B, C, D, and E,
∴ (All) The members of the committee are Reactionaries.
(2) John, James, Albert, and Peter all have perfect eyesight,
John, James, Albert, and Peter are all the boys of my family,
∴ All the boys of my family have perfect eyesight.
(3) The first, second, and third groups are up to grade,
The first, second, and third groups include all of the children
in my room,
Hence all the children in my room are up to grade.
Because the conclusion of a perfect induction gives nothing
new――nothing but what is found in the premises, some claim that
the process is practically valueless. From the viewpoint of the
discoverer this position is well taken; yet to universalize particular
observations puts the knowledge in compact, usable form, and saves
one the trouble of returning each time to the consideration of each
particular. Thus as a process which leads to verified universals,
perfect induction is a _time saver_. In the second place it was the
method used by Socrates when he desired to lead up to a definition or
some other general truth. The Sophists were given to a careless use
of the “inductive hazard”; they were prone to generalize from one
or two particulars, or what is worse, to establish a generalization
and then attempt to fit the particular instances to it. This led to a
superficiality which the Great Pagan Educator abhorred. The fact that
perfect induction was the method used by Socrates to counteract the
teachings of the Sophists, is sufficient vindication for its use in
discouraging the indefensible assumptions of to-day, and in inspiring
warrantable generalizations based on accurate observation.
In the school room with classes addicted to careless, inaccurate work,
to accept nothing but a perfectly induced generalization, when this
is feasible, is a most valuable lesson. For example, the teacher may
not accept the generalization that all of the “first class” cities
of the U. S. are located on navigable waterways, until the pupils
have investigated the waterway conditions of every city belonging
to the class. On the other hand, there may be individual cases of
“cocksureness” which need attention. The teacher can do little for the
“know-it-all youngster” until he pricks the bubble of conceit. This may
be accomplished by allowing the youth to draw a generalization, which
seems to meet all the requirements of truth arrived at by means of
an _imperfect induction_; then without warning let the teacher give
an instance which will show the generalization to be _false_. This
involves what Socrates termed the “torpedo’s shock.” To illustrate:
Consider the “prime number” formula given by Jevons. In deriving this,
direct the class to add 2 to its square, and to this sum add 41. Give
similar directions relative to numbers 3, 4, 7 and 10. Indicating the
work as directed, would give the following:
(1) 2 + 2² + 41 = 47
(2) 3 + 3² + 41 = 53
(3) 4 + 4² + 41 = 61
(4) 7 + 7² + 41 = 97
(5) 10 + 10² + 41 = 151
A question or two will make apparent the fact that all the results
are prime numbers, and then the generalization may be drawn; namely,
X + X² + 41 = prime number. Now without warning, but under the
assumption that you desire to test deductively the general formula,
let X = 40. This gives (40 + 40² + 41) 1681, which is the square of
41 and is, therefore, not a prime number.
=15. TRADUCTION.=
It may have been noted by the student that “perfect induction” is
not induction at all according to the definition; viz.: Inductive
reasoning is reasoning from less general premises to a more general
conclusion. Referring to the first illustration of the previous section
it is apparent that the conclusion is no broader than the premises.
Ostensibly, the conclusion is a mere summary, or a generalization
of the facts mentioned in the premises. Moreover perfect induction
does not readily conform to the definition of deductive reasoning,
as in this the movement must be from the more general to the less.
We are thus forced to the conclusion that perfect induction is a form
of a third type of reasoning which is known under the cognomen of
_traduction_. This is from the Latin _trans_, and _ducere_ meaning
to lead across. Definition: _Traductive reasoning is reasoning to a
conclusion which is neither less general nor more general than the
premises._
Aside from the case of perfect induction there are other types
which well illustrate traduction. These are: First. _Reasoning from
particular_ (_or individuals_) to _particular_ (or individuals).
ILLUSTRATION:
Highland Street is the longest street in Jamaica,
Highland Street is not so long as Broadway of New York City,
∴ The longest street of Jamaica is not so long as Broadway of
New York City.
Second. _Reasoning from general to general._
ILLUSTRATION:
All growing things die,
All living things are growing things,
∴ All living things die.
It may be observed that all of the propositions in traduction are
co-extensive “A’s” or “E’s”; hence all the terms are distributed. This
eliminates any possibility of committing fallacies of distribution.
Further, the propositions may be interchanged at will, without
invalidating the particular conclusion selected. To illustrate we may
change the last argument to this:
All growing things are living things,
All things that die are growing things,
∴ All things that die are living things.
From the viewpoint of authenticity traduction is the most, and
induction the least dependable; whereas the certitude of deductive
reasoning lies somewhere between the two. On the other hand, when
looked at from the ground of serviceableness the order is reversed,
induction being the most useful form of inference and traduction the
least.
=16. OUTLINE.=
INDUCTIVE REASONING.
(1) Inductive and Deductive Reasoning Distinguished.
(2) The “Inductive Hazard.”
Essential in world’s progress.
Cultivated and regulated in school.
(3) Complexity of the Problem of Induction.
(4) Various Conceptions of Induction.
Quotations from prominent authorities.
(5) Induction and Deduction Contiguous Processes.
(6) Induction an Assumption.
A mode of inference; A method.
(7) Universal Causation.
Law stated and illustrated.
Conditions all induction.
(8) Uniformity of Nature.
Defined and illustrated.
Conditions all induction.
Empirical.
(9) Inductive Assumptions Justified.
(10) Three Forms of Inductive Research.
(1) Enumeration (2) Analogy (3) Analysis.
Illustrated.
Conditions determine form followed.
(11) Induction by Simple Enumeration.
Defined and illustrated.
Outcome threefold――these illustrated.
(12) Induction by Analogy.
Two conceptions.
Analogy by type or example. Illustrations representative.
Error of analogy.
Suggestiveness of analogy.
Value of analogy.
Requirements of a true analogy. Three.
(13) Induction by Analysis.
Importance.
Defined and illustrated.
(14) Perfect Induction.
Defined and illustrated.
Its use.
Method of Socrates.
(15) Traduction.
Defined and illustrated.
Three methods compared.
=17. SUMMARY.=
(1) Reasoning is the process of deriving a judgment from two antecedent
judgments. The syllogism is a common form of expressing the process of
reasoning.
Inductive reasoning is reasoning from less general premises to a more
general conclusion.
Deductive reasoning is reasoning from more general premises to a less
general conclusion.
The inductive syllogism is not supposed to conform to the canons of the
deductive syllogism.
(2) Positing in the conclusion more than is indicated in the premises
involves what is known as the “_inductive hazard_.”
The inductive hazard which is another expression for the spirit of
discovery, should be fostered in the school room since it has been one
of the great forces in human progress; but this venturesome spirit must
be regulated by rules, principles, and systematic procedure, or low
ideals of recklessness and inaccuracy will result.
(3) The problem of induction is more complex than that of deduction;
because the former is a comparatively new subject, and also is more
closely related to the activities of life.
(4) The opinion relative to the exact nature of induction, though
varied, may be summed up in the thought of its being the process which
leads to general truths, derived from the observation of individual
facts.
(5) Induction and deduction are contiguous processes which go to
make up the more general process of thinking. Where induction ceases,
deduction naturally commences; induction discovers new knowledge,
deduction clarifies it.
(6) Induction as a general process may be treated as a mode of
inference or as a method. In either case the conclusion comprehends
more than is contained in the premises.
Since no imperfect induction is absolutely free from doubt, on what
ground are we justified in making any inductive assumptions? The answer
follows:
(7 and 8) “Nothing can occur without a cause and every cause has
its effect,” is the law of universal causation; while the law of the
uniformity of nature is “the same antecedents are universally followed
by the same consequents.” These two laws justify inductive assumptions,
and, in a sense, condition all thinking.
(9) Uniformity of nature gives man confidence, while universal
causation arouses his curiosity. With these two weapons he is willing
to “march into the unknown.”
(10) As the process of universalizing individual experiences, induction
assumes the three forms of simple enumeration, analogy and analysis.
The form adopted is not always elective but is controlled largely by
the exigency of the case. Some topics lend themselves to all three
modes.
(11) Induction by simple enumeration consists in observing many
instances which exemplify the uniformity under consideration.
_Complete_ enumeration gives the so called perfect inductive inference;
_incomplete but uncontradicted_ enumeration leads to empirical truths;
while _incomplete_ and _contradictive_ enumeration involves a mere
calculation of chances.
(12) Induction by analogy assumes that if two (or more) things resemble
each other in certain respects, they belong to the same type, and,
therefore, any fact known of the one, may be affirmed of the other.
A most common form of analogy is reasoning by type or example. In
this it is assumed that if two or more things are of the same type,
they resemble each in every essential property. The type must be truly
representative. A second form of analogy is reasoning by marks of
resemblance. This second form often leads to egregious error.
Analogy is especially valuable in suggesting hypotheses and in giving
training in originality and initiative.
A true analogy demands that the points of resemblance be representative;
that they outweigh the points of difference, and that no disagreement
be incompatible.
(13) Induction by analysis is the process of dividing a whole into its
parts with a view of deriving a generalization relative to the nature
and causal connection of these parts.
Induction by analysis makes use of the hypothesis, of observation and
experiment, including Mill’s five methods.
(14) A perfect induction is one in which the premises enumerate all of
the instances denoted by the conclusion. It is serviceable in inspiring
care and accuracy in the establishment of generalizations.
(15) Traduction is the process of reasoning to a conclusion which is
neither less general nor more general than the premises.
Traduction includes reasoning from particular to particular or from
general to general. Perfect induction is in reality a form of
traduction.
Induction, though the most useful form of inference, is the most
untrustworthy; whereas traduction is just the reverse of this.
=18. REVIEW QUESTIONS.=
(1) Define and illustrate reasoning.
(2) Distinguish by definition and illustration between inductive and
deductive reasoning.
(3) Explain the “inductive hazard” and show its use to man.
(4) “For twenty centuries Aristotle’s Deductive Logic was the
logician’s bible.” Explain this.
(5) Show that induction and deduction are contiguous processes.
(6) Distinguish between induction as a mode of inference and
induction as a method.
(7) State and explain the law of universal causation. Illustrate
fully.
(8) Make evident that a cause may involve many antecedents.
(9) State and explain by illustration the law of uniformity of
nature.
(10) Verify by illustration the notion that the “fact of causation”
conditions all induction.
(11) Which of the two laws is empirical, “causation” or “uniformity”?
Why?
(12) Show that induction is a form of thinking.
(13) Why should the law of uniformity of nature convince man that
nature is honest? Illustrate.
(14) Show that the law of universal causation stirs the spirit of
discovery.
(15) Name and illustrate the three forms of induction.
(16) Why is it that the tendencies of the investigator often determine
the inductive form which he adopts?
(17) Explain by illustration the three-fold outcome of induction by
simple enumeration.
(18) Selecting some class room experience, illustrate analogy by
example or type.
(19) Define and exemplify types as used in logic.
(20) Remark upon the errors incident to analogy.
(21) Summarize the advantages which induction by analogy offers.
(22) State and exemplify the requirements of true analogies.
(23) Indicate the superiority of induction by analysis over the other
two forms. Illustrate.
(24) Define and illustrate perfect induction.
(25) Under what circumstances is perfect induction justified?
(26) Define and illustrate traduction.
(27) Indicate the various forms of traduction.
=19. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=
(1) Show the connection between illicit minor and the “inductive
hazard.”
(2) Show by illustration that time tends to universalize truth.
(3) “Induction and not deduction is the natural method of the child
mind.” Prove the correctness of this statement.
(4) “Induction is the process of inference by which we get at general
truths from particular facts or cases.” Prove that this is not
strictly correct according to definition.
(5) As related to establishing general truths, what are the special
functions of induction and deduction?
(6) Show that an inductive inference must of necessity be more or
less uncertain.
(7) Is there any distinction between the laws of universal causation
and sufficient reason? Hyslop’s Elements of Logic, page 329.
(8) Show that universal causation and uniformity of nature are
complementary laws. Hyslop, p. 330.
(9) Relate the “fact of causation” to the laws of thought.
(10) Distinguish between empirical and “a priori” laws.
(11) When Harvey discovered the circulation of the blood, what form
of induction did he use?
(12) What form of reasoning did Columbus follow in proving that the
earth is spherical?
(13) “It is said that the greatness of Darwin was due largely to his
habit of never ignoring an exception.” Justify by illustration
the truth of this assertion.
(14) In analogical reasoning by example, under what conditions would
_one_ illustration be as convincing as _many_?
(15) “Considering the similarities and differences, the weight of the
argument favors Mars’ habitability.” Suppose the proportion of
probability were something like this――Resemblances: Differences
= 8:7; wherein might the conclusion be erroneous?
(16) Mention a mark or characteristic which would make the
habitability of Mars incompatible?
(17) Select a topic for investigation which is peculiarly adapted to
enumeration; to analogy; to analysis.
(18) “The uniformities we expect to find in the world take two
main aspects, one of which is indicated by the term _thing_ and
the other by the term _circumstance_.” Aikin’s Principles of
Logic, 1905; p. 233. In the light of the two fundamental laws
of universal causation and uniformity of nature explain and
illustrate the quotation.
(19) Explain the principle of teleology as related to analogy. Hibben,
1908; p. 317.