THE ART OF LOGICAL THINKING/PART 10
CHAPTER X.
IMMEDIATE REASONING
In
the process of Judgment we must compare two concepts and ascertain their
agreement of disagreement. In the process of Reasoning we follow a similar
method and compare two judgments, the result of such comparison being the
deduction of a third judgment.
The
simplest form of reasoning is that known as Immediate Reasoning, by which is
meant the deduction of one proposition from another which implies it.
Some have defined it as: "reasoning without a middle term." In
this form of reasoning only one proposition is required for the premise,
and from that premise the conclusion is deduced directly and without the
necessity of comparison with any other term of proposition.
The
two principal methods employed in this form of Reasoning are; (1) Opposition;
(2) Conversion.
Opposition exists
between propositions having the same subject and predicate, but differing in
quality or quantity, or both. The Laws of Opposition are as follows:
I.
(1) If the universal is true, the particular is true. (2) If the particular is
false, the universal is false. (3) If the universal is false, nothing follows.
(4) If the particular is true, nothing follows.
II.
(1) If one of two contraries is true, the other is false. (2) If one of two
contraries is false, nothing can be inferred. (3) Contraries are never both
true, but both may be false.
III.
(1) If one of two sub-contraries is false, the other is true. (2) If one of two
sub-contraries is true, nothing can be inferred concerning the other. (3)
Sub-contraries can never be both false, but both may be true.
IV.
(1) If one of two contradictories is true, the other is false. (2) If one of
two contradictories is false, the other is true. (3) Contradictories can never
be both true or both false, but always one is true and the other is false.
In
order to comprehend the above laws, the student should familiarize himself with
the following arrangement, adopted by logicians as a convenience:
|
|
Universal |
|
Affirmative |
(A) |
Propositions |
|||||
|
Particular |
|
Affirmative |
(I) |
Examples
of the above: Universal Affirmative (A): "All men are mortal;"
Universal Negative (E): "No man is mortal;" Particular Affirmative
(I): "Some men are mortal;" Particular Negative (O): "Some men
are not mortal."
The
following examples of abstract propositions are often used by logicians as
tending toward a clearer conception than examples such as given above:
(A)
"All A is B."
(I)
"Some A is B."
(E)
"No A is B."
(O)
"Some A is not B."
These
four forms of propositions bear certain logical relations to each other, as
follows:
A
and E are styled contraries. I and O are sub-contraries;
A and I and also E and O are called subalterns; A and O and
also I and E are styled contradictories.
A
close study of these relations, and the symbols expressing them, is necessary
for a clear comprehension of the Laws of Opposition stated a little further
back, as well as the principles of Conversion which we shall mention a little
further on. The following chart, called the Square of Opposition, is also
employed by logicians to illustrate the relations between the four classes of
propositions:
Conversion is
the process of immediate reasoning by which we infer from a given proposition
another proposition having the predicate of the original for its subject
and the subject of the original for its predicate; or stated in a few
words: Conversion is the transposition of the subject and predicate of
a proposition. As Brooks states it: "Propositions or judgments
are converted when the subject and predicate change places in
such a manner that the resulting judgment is an inference from the given
judgment." The new proposition, resulting from the operation or
Conversion, is called the Converse; the original proposition is called the
Convertend.
The
Law of Conversion is that: "No term must be distributed in the Converse
that is not distributed in the Convertend." This arises from the obvious
fact that nothing should be affirmed in the derived proposition than there is
in the original proposition.
There
are three kinds of Conversion; viz: (1) Simple Conversion; (2)
Conversion by Limitation; (3) Conversion by Contraposition.
In Simple
Conversion there is no change in either quality or quantity. In Conversion
by Limitation the quality is changed from universal to particular. In
Conversion by Negation the quality is changed but not the quantity. Referring
to the classification tables and symbols given in the preceding pages of this
chapter, we may now proceed to consider the application of these methods of
Conversion to each of the four kinds of propositions; as follows:
The
Universal Affirmative (symbol A) proposition is converted by Limitation, or by a
change of quality from universal to particular. The predicate not being
"distributed" in the convertend, we must not distribute it in the
converse by saying "all." Thus in this case we must convert
the proposition, "all men are mortal" (A), into "some mortals
are men" (I).
The
Universal Negative (symbol E) is converted by Simple Conversion, in which there
is no change in either quality or quantity. For since both terms of
"E" are distributed, they may both be distributed in the converse
without violating the law of conversion. Thus "No man is mortal" is
converted into: "No mortals are men." "E" is converted into
"E."
The
Particular Affirmative (symbol I) is also converted
by Simple Conversion in which there is no change in either quality or quantity.
For since neither term is distributed in "I," neither term may be
distributed in the converse, and the latter must remain "I." For
instance; the proposition: "Some men are mortal" is converted into
the proposition, "Some mortals are men."
The
Particular Negative (symbol O) is converted by Conversion by Negation, in which
the quality is changed but not the quantity. Thus in converting the
proposition: "Some men are not mortal," we must not say
"some mortals are not men," for in so doing we would distribute men in
the predicate, where it is not distributed in the convertend. Avoiding
this, we transfer the negative particle from the copula to the
predicate so that the convertend becomes "I" which is
converted by Simple Conversion. Thus we transfer "Some men are not
mortal" into "Some men are not-mortal" from which we easily
convert (by simple Conversion) the proposition: "Some not-mortals are
men."
It
will be well for students, at this point, to consider the three following
Fundamental Laws of Thought as laid down by the authorities, which are as
follows:
The
Law of Identity, which states that: "The same quality or thing is always the
same quality or thing, no matter how different the conditions in which it
occurs."
The
Law of Contradiction, which states that: "No thing can at the same time and place
both be and not be."
The
Law of Excluded Middle, which states that: "Everything
must either be or not be; there is no other alternative or middle course."
Of
these laws, Prof. Jevons, a noted authority, says: "Students are seldom
able to see at first their full meaning and importance. All arguments may be
explained when these self-evident laws are granted; and it is not too much to
say that the whole of logic will be plain to those who will constantly use
these laws as the key."
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