DERIVED JUDGMENTS
As we have seen, a Judgment is obtained by comparing two objects of
thought according to their agreement or difference. The next higher
step, that of logical Reasoning, consists of the comparing of two ideas
through their relation to a third. This form of reasoning is called
_mediate_, because it is effected through the _medium_ of the third
idea. There is, however, a certain process of Understanding which comes
in between this mediate reasoning on the one hand, and the formation of
a plain judgment on the other. Some authorities treat it as a form of
_reasoning_, calling it _Immediate Reasoning_ or Immediate Inference,
while others treat it as a higher form of Judgment, calling it Derived
Judgment. We shall follow the latter classification, as best adapted for
the particular purposes of this book.
The fundamental principle of Derived Judgment is that ordinary Judgments
are often so related to each other that one Judgment may be derived
directly and immediately from another. The two particular forms of the
general method of Derived Judgment are known as those of (1) Opposition;
and (2) Conversion; respectively.
In order to more clearly understand the logical processes involved in
Derived Judgment, we should acquaint ourselves with the general
relations of Judgments, and with the symbolic letters used by logicians
as a means of simplifying the processes of thought. Logicians denote
each of the four classes of Judgments or Propositions by a certain
letter, the first four vowels--A, E, I and O, being used for the
purpose. It has been found very convenient to use these symbols in
denoting the various forms of Propositions and Judgments. The following
table should be memorized for this purpose:
_Universal Affirmative_, symbolized by "A."
_Universal Negative_, symbolized by "E."
_Particular Affirmative_, symbolized by "I."
_Particular Negative_, symbolized by "O."
It will be seen that these four forms of Judgments bear certain
relations to each other, from which arises what is called opposition.
This may be better understood by reference to the following table called
the Square of Opposition:
A CONTRARIES E
+------------------------+
|\ / |
| \ /S |
| C\ /E |
| O\ /I |
| N\ /R |
| T\ /O |
S| R\ /T |S
U| A\ /C |U
B| \ /I |B
A| \ /D |A
L| \ / |L
T| / \ |T
E| / D\ |E
R| / I\ |R
N| /A C\ |N
S| /R T\ |S
| /T O\ |
| /N R\ |
| /O I\ |
| /C E\ |
| / S\ |
|/ \ |
+------------------------+
I SUB-CONTRARIES O
Thus, A and E are _contraries_; I and O are _sub-contraries_; A and I,
and also E and O are _subalterns_; A and O, and also E and I are
_contradictories_.
The following will give a symbolic table of each of the four Judgments
or Propositions with the logical symbols attached:
(A) "All A is B."
(E) "No A is B."
(I) "Some A is B."
(O) "Some A is not B."
The following are the rules governing and expressing the relations above
indicated:
I. Of the Contradictories: _One must be true, and the other must be
false_. As for instance, (A) "All A is B;" and (O) "Some A is not B;"
cannot both be true at the same time. Neither can (E) "No A is B;" and
(I) "Some A is B;" both be true at the same time. They are
_contradictory_ by nature,--and if one is true, the other must be false;
if one is false, the other must be true.
II. Of the Contraries: _If one is true the other must be false; but,
both may be false_. As for instance, (A) "All A is B;" and (E) "No A is
B;" cannot both be true at the same time. If one is true the other
_must_ be false. _But_, both may be _false_, as we may see when we find
we may state that (I) "_Some_ A is B." So while these two propositions
are _contrary_, they are not _contradictory_. While, if one of them is
_true_ the other must be false, it does not follow that if one is
_false_ the other must be _true_, for both _may be false_, leaving the
truth to be found in a third proposition.
III. Of the Subcontraries: _If one is false the other must be true; but
both may be true_. As for instance, (I) "Some A is B;" and (O) "Some A
is not B;" may both be true, for they do not contradict each other. But
one or the other must be true--they can not both be false.
IV. Of the Subalterns: _If the Universal (A or E) be true the Particular
(I or O) must be true_. As for instance, if (A) "All A is B" is true,
then (I) "Some A is B" must also be true; also, if (E) "No A is B" is
true, then "Some A is not B" must also be true. The Universal carries
the particular within its truth and meaning. But; _If the Universal is
false, the particular may be true or it may be false_. As for instance
(A) "All A is B" may be false, and yet (I) "Some A is B" may be either
true or false, without being determined by the (A) proposition. And,
likewise, (E) "No A is B" may be false without determining the truth or
falsity of (O) "Some A is not B."
But: _If the Particular be false, the Universal also must be false_. As
for instance, if (I) "Some A is B" is false, then it must follow that
(A) "All A is B" must also be false; or if (O) "Some A is not B" is
false, then (E) "No A is B" must also be false. But: _The Particular may
be true, without rendering the Universal true_. As for instance: (I)
"_Some_ A is B" may be true without making true (A) "_All_ A is B;" or
(O) "Some A is not B" may be true without making true (E) "No A is B."
The above rules may be worked out not only with the symbols, as "All A
is B," but also with _any_ Judgments or Propositions, such as "All
horses are animals;" "All men are mortal;" "Some men are artists;" etc.
The principle involved is identical in each and every case. The "All A
is B" symbology is merely adopted for simplicity, and for the purpose of
rendering the logical process akin to that of mathematics. The letters
play the same part that the numerals or figures do in arithmetic or the
_a_, _b_, _c_; _x_, _y_, _z_, in algebra. Thinking in symbols tends
toward clearness of thought and reasoning.
_Exercise_: Let the student apply the principles of Opposition by using
any of the above judgments mentioned in the preceding paragraph, in the
direction of erecting a Square of Opposition of them, after having
attached the symbolic letters A, E, I and O, to the appropriate forms of
the propositions.
Then let him work out the following problems from the Tables and Square
given in this chapter.
1. If "A" is true; show what follows for E, I and O. Also what follows
if "A" be _false_.
2. If "E" is true; show what follows for A, I and O. Also what follows
if "E" be _false_.
3. If "I" is true; show what follows for A, E and O. Also what follows
if "I" be _false_.
4. If "O" is true; show what follows for A, E and I. Also what happens
if "O" be _false_.
CONVERSION OF JUDGMENTS
Judgments are capable of the process of Conversion, or _the change of
place of subject and predicate_. Hyslop says: "Conversion is the
transposition of subject and predicate, or the process of immediate
inference by which we can infer from a given preposition another having
the predicate of the original for its subject, and the subject of the
original for its predicate." The process of converting a proposition
seems simple at first thought but a little consideration will show that
there are many difficulties in the way. For instance, while it is a true
judgment that "All _horses_ are _animals_," it is not a correct Derived
Judgment or Inference that "All _animals_ are _horses_." The same is
true of the possible conversion of the judgment "All biscuit is bread"
into that of "All bread is biscuit." There are certain rules to be
observed in Conversion, as we shall see in a moment.
The Subject of a judgment is, of course, _the term of which something is
affirmed_; and the Predicate is _the term expressing that which is
affirmed of the Subject_. The Predicate is really an expression of an
_attribute_ of the Subject. Thus when we say "All horses are animals" we
express the idea that _all horses_ possess the _attribute_ of
"animality;" or when we say that "Some men are artists," we express the
idea that _some men_ possess the _attributes_ or qualities included in
the concept "artist." In Conversion, the original judgment is called the
Convertend; and the new form of judgment, resulting from the conversion,
is called the Converse. Remember these terms, please.
The two Rules of Conversion, stated in simple form, are as follows:
I. Do not change the quality of a judgment. The quality of the converse
must remain the same as that of the convertend.
II. Do not distribute an undistributed term. No term must be distributed
in the converse which is not distributed in the convertend.
The reason of these rules is that it would be contrary to truth and
logic to give to a converted judgment a higher degree of quality and
quantity than is found in the original judgment. To do so would be to
attempt to make "twice 2" more than "2 plus 2."
There are three methods or kinds of Conversion, as follows: (1) Simple
Conversion; (2) Limited Conversion; and (3) Conversion by
Contraposition.
_In Simple Conversion_, there is no change in either quality or
quantity. For instance, by Simple Conversion we may convert a
proposition by changing the places of its subject and predicate,
respectively. But as Jevons says: "It does not follow that the new one
will always be true if the old one was true. Sometimes this is the
case, and sometimes it is not. If I say, 'some churches are
wooden-buildings,' I may turn it around and get 'some wooden-buildings
are churches;' the meaning is exactly the same as before. This kind of
change is called Simple Conversion, because we need do nothing but
simply change the subjects and predicates in order to get a new
proposition. We see that the Particular Affirmative proposition can be
simply converted. Such is the case also with the Universal Negative
proposition. 'No large flowers are green things' may be converted simply
into 'no green things are large flowers.'"
_In Limited Conversion_, the quantity is changed from Universal to
Particular. Of this, Jevons continues: "But it is a more troublesome
matter, however, to convert a Universal Affirmative proposition. The
statement that 'all jelly fish are animals,' is true; but, if we convert
it, getting 'all animals are jelly fish,' the result is absurd. This is
because the predicate of a universal proposition is really particular.
We do not mean that jelly fish are 'all' the animals which exist, but
only 'some' of the animals. The proposition ought really to be 'all
jelly fish are _some_ animals,' and if we converted this simply, we
should get, 'some animals are all jelly fish.' But we almost always
leave out the little adjectives _some_ and _all_ when they would occur
in the predicate, so that the proposition, when converted, becomes
'_some_ animals are jelly fish.' This kind of change is called Limited
Conversion, and we see that a Universal Affirmative proposition, when so
converted, gives a Particular Affirmative one."
In Conversion by Contraposition, there is a change in the position of
the negative copula, which shifts the expression of the quality. As for
instance, in the Particular Negative "Some animals are not horses," we
cannot say "Some horses are not animals," for that would be a violation
of the rule that "no term must be distributed in the converse which is
not distributed in the convertend," for as we have seen in the preceding
chapter: "In Particular propositions the _subject_ is _not_
distributed." And in the original proposition, or convertend, "animals"
is the _subject_ of a Particular proposition. Avoiding this, and
proceeding by Conversion by Contraposition, we convert the Convertend
(O) into a Particular Affirmative (I), saying: "Some animals are
not-horses;" or "Some animals are things not horses;" and then
proceeding by Simple Conversion we get the converse, "Some things not
horses are animals," or "Some not-horses are animals."
The following gives the application of the appropriate form of
Conversion to each of the several four kind of Judgments or
Propositions:
(A) _Universal Affirmative_: This form of proposition is converted by
Limited Conversion. The predicate not being distributed in the
convertend, it cannot be distributed in the converse, by saying "all."
("In affirmative propositions the _predicate_ is _not_ distributed.")
Thus by this form of Conversion, we convert "All horses are animals"
into "Some animals are horses." The Universal Affirmative (A) is
converted by limitation into a Particular Affirmative (I).
(E) _Universal Negative_: This form of proposition is converted by
Simple Conversion. In a Universal Negative _both terms are distributed_.
("In universal propositions, the _subject_ is distributed;" "In
negative propositions, the _predicate_ is distributed.") So we may say
"No cows are horses," and then convert the proposition into "No horses
are cows." We simply convert one Universal Negative (E) into another
Universal Negative (E).
(I) _Particular Affirmative_: This form of proposition is converted by
Simple Conversion. For _neither term is distributed_ in a Particular
Affirmative. ("In particular propositions, the _subject_ is _not_
distributed. In affirmative propositions, the _predicate_ is _not_
distributed.") And neither term being distributed in the convertend, it
must not be distributed in the converse. So from "Some horses are males"
we may by Simple Conversion derive "Some males are horses." We simply
convert one Particular Affirmative (I), into another Particular
Affirmative (I).
(O) _Particular Negative_: This form of proposition is converted by
Contraposition or Negation. We have given examples and illustrations in
the paragraph describing Conversion by Contraposition. The Particular
Negative (I) is converted by contraposition into a Particular
Affirmative (I) which is then simply converted into another Particular
Affirmative (I).
There are several minor processes or methods of deriving judgments from
each other, or of making immediate inferences, but the above will give
the student a very fair idea of the minor or more complete methods.
_Exercise_: The following will give the student good practice and
exercise in the methods of Conversion. It affords a valuable mental
drill, and tends to develop the logical faculties, particularly that of
Judgment. The student should _convert_ the following propositions,
according to the rules and examples given in this chapter:
1. All men are reasoning beings.
2. Some men are blacksmiths.
3. No men are quadrupeds.
4. Some birds are sparrows.
5. Some horses are vicious.
6. No brute is rational.
7. Some men are not sane.
8. All biscuit is bread.
9. Some bread is biscuit.
10. Not all bread is biscuit.