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."