THE ART OF LOGICAL THINKING/PART 15
CHAPTER XV.
DEDUCTIVE REASONING
We
have seen that there are two great classes of reasoning, known respectively, as
(1) Inductive Reasoning, or the discovery of general truth from particular
truths; and (2) Deductive Reasoning, or the discovery of particular truths from
general truths.
As
we have said, Deductive Reasoning is the process of discovering particular
truths from a general truth. Thus from the general truth embodied in the
proposition "All horses are animals," when it is considered in
connection with the secondary proposition that "Dobbin is a horse,"
we are able to deduce the particular truth that: "Dobbin is an
animal." Or, in the following case we deduce a particular truth from a
general truth, as follows: "All mushrooms are good to eat; this fungus is
a mushroom; therefore, this fungus is good to eat." A deductive argument
is expressed in a deductive syllogism.
Jevons
says regarding the last stated illustration: "Here are three sentences
which state three different facts; but when we know the two first facts, we
learn or gather the third fact from the other two. When we thus learn one fact
from other facts, we infer or reason, and we do this in the mind.
Reasoning thus enables us to ascertain the nature of a thing without actual
trial. If we always needed to taste a thing before we could know whether it was
good to eat or not, cases of poisoning would be alarmingly frequent. But the
appearance and peculiarities of a mushroom may be safely learned by the eye or
the nose, and reasoning upon this information and the fact already well known,
that mushrooms are good to eat, we arrive without any danger or trouble at the
conclusion that the particular fungus before us is good to eat. To
reason, then, is to get some knowledge from other knowledge."
The
student will recognize that Deductive Reasoning is essentially an
analytic process, because it operates in the direction of analyzing a
universal or general truth into its particulars—into the particular parts which
are included within it—and asserting of them that "what is true of
the general is true of the particular." Thus in the general truth that
"All men are mortal," we see included the particular truth that
"John Smith is mortal"—John Smith having been discovered to be a man.
We deduce the particular truth about John Smith from the general truth about
"all men." We analyze "all men" and find John Smith to be
one of its particular parts. Therefore, "Deduction is an inference from
the whole to its parts; that is, an analytic process."
The
student will also recognize that Deductive Reasoning is essentially a
descending process, because it operates in the direction of a descent from
the universal to the particular; from the higher to the lower; from the broader
to the narrower. As Brooks says: "Deduction descends from higher truths to
lower truths, from laws to facts, from causes to phenomena, etc. Given the law,
we can by deduction descend to the facts that fall under the law, even if we
have never before seen the facts; and so from the cause we may pass down to
observed and even unknown phenomena."
The
general truths which are used as the basis of Deductive Reasoning are discovered
in several ways. The majority arise from Inductive Reasoning, based upon
experience, observation and experiment. For instance in the examples given
above, we could not truthfully assert our belief that: "All horses are
animals" unless we had previously studied both the horse and animals in
general. Nor without this study could we state that "Dobbin is a
horse." Nor could we, without previous study, experience and experiment
truthfully assert that: "All mushrooms are good to eat;" or that
"this fungus is a mushroom;" and that "therefore, this fungus is
good to eat." Even as it is, we must be sure that the fungus really is a
mushroom, else we run a risk of poisoning ourselves. General truths of this
kind are not intuitive, by any means, but are based upon our own
experience or the experience of others.
There
is a class of general truths which are called intuitive by
some authorities. Halleck says of these: "Some psychologists claim that we
have knowledge obtained neither through induction nor deduction; that
we recognize certain truths the moment we perceive certain objects,
without any process of inference. Under the head of intuitive knowledge are
classified such cases as the following: We perceive an object and immediately
know that it is a time relation, as existing now and then. We are said to have
an intuitive concept of time. When we are told that the whole is greater than a
part; that things equal to the same thing are equal to each other; that a
straight line cannot enclose space, we immediately, or intuitively,
recognize the truth of these statements. Attempts at proof do not make us feel
surer of their truth.... We say that it is self-evident, or that we know the
fact intuitively. The axioms of mathematics and logic are said to be intuitive."
Another
class of authorities, however, deny the nature of intuitive knowledge of truth,
or intuitive truths. They claim that all our ideas arise from sensation and
reflection, and that what we call "intuition" is merely the result of
sensation and reflection reproduced by memory or heredity. They
hold that the intuitions of animals and men are simply the
representation of experiences of the race, or individual, arising from the
impressions stored away in the subconsciousness of the individual. Halleck
states regarding this: "This school likens intuition to instinct. It
grants that the young duck knows water instinctively, plunges into it, and
swims without learning. These psychologists believe that there was a time when
this was not the case with the progenitors of the duck. They had to gain this
knowledge slowly through experience. Those that learned the proper aquatic
lesson survived and transmitted this knowledge through a modified structure, to
their progeny. Those that failed in the lesson perished in the struggle for
existence.... This school claims that the intuition of cause and effect arose
in the same way. Generations of human beings have seen the cause invariably
joined to the effect; hence, through inseparable association came the recognition
of their necessary sequence. The tendency to regard all phenomena in these
relations was with steadily increasing force transmitted by the laws of
heredity to posterity, until the recognition of the relationship has become an
intuition."
Another
class of general truths is merely hypothetical. Hypothetical means
"Founded on or including a hypothesis or supposition; assumed or taken for
granted, though not proved, for the purpose of deducing proofs of a point in
question." The hypotheses and theories of physical science are used as
general truths for deductive reasoning. Hypothetical general truths are in the
nature of premises assumed in order to proceed with the process of Deductive
Reasoning, and without which such reasoning would be impossible. They are,
however, as a rule not mere assumptions, but are rather in the nature of
assumptions rendered plausible by experience, experiment and Inductive
Reasoning. The Law of Gravitation may be considered hypothetical, and yet it is
the result of Inductive Reasoning based upon a vast multitude of facts and
phenomena.
The Primary
Basis of Deductive Reasoning may be said to rest upon the logical
axiom, which has come down to us from the ancients, and which is stated as
follows: "Whatever is true of the whole is true of its parts."
Or, as later authorities have expressed it: "Whatever is true of the
general is true of the particular." This axiom is the basis upon which we
build our Deductive Reasoning. It furnishes us with the validity of the
deductive inference or argument. If we are challenged for proof of the
statement that "This fungus is good to eat," we are able to answer
that we are justified in making the statement by the self-evident proposition,
or axiom, that "Whatever is true of the general is true of the
particular." If the general "mushroom" is good to eat, then the
particular, "this fungus" being a mushroom, must also be good to eat.
All horses (general) being animals, then according to the axiom, Dobbin
(particular horse) must also be an animal.
This
axiom has been stated in various terms other than those stated above. For
instance: "Whatever may be affirmed or denied of the whole, may be denied
or affirmed of the parts;" which form is evidently derived from that used
by Hamilton who said: "What belongs, or does not belong, to the containing
whole, belongs or does not belong, to each of the contained parts."
Aristotle formulated his celebrated Dictum as follows: "Whatever can be
predicated affirmatively or negatively of any class or term distributed, can be
predicated in like manner of all and singular the classes or individuals
contained under it."
There
is another form of Deductive Reasoning, that is a form based upon another axiom
than that of: "Whatever is true of the whole is true of the parts."
This form of reasoning is sometimes called Mathematical Reasoning, because it
is the form of reasoning employed in mathematics. Its axiom is stated as
follows: "Things which are equal to the same thing, are equal to one
another." It will be seen that this is the principle employed in
mathematics. Thus: "x equals y; and y equals 5; therefore, x equals
5." Or stated in logical terms: "A equals B; B equals C; therefore, A
equals C." Thus it is seen that this form of reasoning, as well as the
ordinary form of Deductive Reasoning, is strictly mediate, that is,
made through the medium of a third thing, or "two things being compared
through their relation to a third."
Brooks
states: "The real reason for the certainty of mathematical reasoning may
be stated as follows: First, its ideas are definite, necessary, and exact
conceptions of quantity. Second, its definitions, as the description of these
ideas are necessary, exact, and indisputable truths. Third, the axioms from
which we derive conclusions by comparison are all self-evident and necessary
truths. Comparing these exact ideas by the necessary laws of inference, the
result must be absolutely true. Or, stated in another way, using these
definitions and axioms as the premises of a syllogism, the conclusion follows inevitably.
There is no place or opportunity for error to creep in to mar or vitiate our
derived truths."
In
conclusion, we wish to call your attention to a passage from Jevons which is
worthy of consideration and recollection. Jevons says: "There is a simple
rule which will enable us to test the truth of a great many arguments, even of
many which do not come under any of the rules commonly given in books on logic.
This rule is that whatever is true of one term is true of any term
which is stated to be the same in meaning as that term. In other words, we
may always substitute one term for another if we know that they refer
to exactly the same thing. There is no doubt that a horse is some
animal, and therefore the head of a horse is the head of some animal. This
argument cannot be brought under the rules of the syllogism, because it
contains four distinct logical terms in two propositions; namely, horse, some
animal; head of horse, head of some animal. But it easily comes under the rule
which I have given, because we have simply to put 'some animal' instead of 'a
horse'. A great many arguments may be explained in this way. Gold is a metal;
therefore a piece of gold is a piece of metal. A negro is a fellow creature;
therefore, he who strikes a negro, strikes a fellow creature."
The
same eminent authority says: "When we examine carefully enough the way in
which we reason, it will be found in every case to consist in putting
one thing or term in place of another, to which we know it to have an exact
resemblance in some respect. We use the likeness as a kind of bridge, which
leads us from a knowledge of one thing to a knowledge of another; thus the
true principle of reasoning may be called the substitution of similars, or
the passing from like to like. We infer the character of one thing from the
character of something which acts as a go-between, or third term. When we are
certain there is an exact likeness, our inference is certain; when we only
believe that there probably is, or guess that there is, then our inferences are
only probable, not certain."
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