IMMEDIATE INFERENCE (CONTINUED)――OBVERSION,
CONVERSION, CONTRAVERSION AND INVERSION.
(2) _IMMEDIATE INFERENCE BY OBVERSION._
_Obversion is the process of changing a proposition from the
affirmative form to its equivalent negative or from the negative form
to its equivalent affirmative._
Some authorities refer to this process as “Inference by Privitive
Conception,” but Obversion seems to be a better term.
Obversion is based upon the principle that _two negatives are
equivalent to one affirmative_. With this double negative principle in
mind let us experiment with the four logical propositions, A, E, I, O.
_The A Proposition._
Example: “All thoughtful men are wise.” Insert the double negative and
the proposition reads: “All thoughtful men are not not-wise.” Changed
to the logical form this becomes: “No thoughtful men are not-wise.”
Simplified and we have, finally: “No thoughtful men are unwise.”
Thus by the process of obversion we have passed from the original
proposition, “All thoughtful men are wise,” to “No thoughtful men
are unwise.” In the first proposition the subject “_thoughtful men_”
is denied of the predicate “_unwise_.” Assuming that “unwise” is the
contradictory of “wise,” then: “What is affirmed of a predicate may
be denied of its contradictory.” Recourse to circles will make this
clearer. In the previous chapter it has been suggested that _not_
bisects the world. For example: What can _not_ be included in the
_wise_ class may be placed under the not-wise or _unwise_ class.
Likewise a circle bisects space――there is the space inside the circle
and the space outside the circle. Let the space inside the circle
represent all wise beings, then the space outside the circle would
represent all not-wise or unwise beings; e. g.,
Illustration: FIG. 5.
Now representing _thoughtful men_ by a smaller circle and placing it
inside the larger we have,
Illustration: FIG. 6.
Referring to Fig. 6 we note that all of the smaller circle belongs
to the larger or that none of the smaller circle belongs to the space
outside of the larger. Hence the two propositions: “All thoughtful men
are wise” (A), and “No thoughtful men are unwise” (E) have virtually
the same meaning though the same subject is related to different
predicates.
The use of the positive or negative form depends upon circumstances.
Often the negative puts the thought in a more forceful way.
In passing from, “All thoughtful men are wise,” to “No thoughtful
men are unwise,” it was necessary to prefix _not_ to the predicate
_wise_ and substitute for _not_ its equivalent _un_. If the original
predicate were unwise or not-wise, then the reverse order of dropping
the _un_ or _not_ could be followed. This process of prefixing the
_not_ to an affirmative predicate or of dropping the _not_ from a
negative predicate is referred to as _negating the predicate_. Before
substituting _in_, _im_, _un_, etc., for _not_, one must make sure
that the substitution really gives the contradictory; there are some
logicians who claim that unwise, for instance, is not the contradictory
of wise.
In comparing the first proposition with the second it is observed that
the first is an A, while the second is an E, also that the predicate of
the first was _negated_ to form the predicate of the second. Thus the
rule: Negate the predicate and change A to E.
_To sum up_:
The obversion of an A proposition.
1. Principle:
Two negatives are equivalent to one affirmative.
2. Rule:
Negate the predicate and change the A to an E by using the sign
_no_ instead of _all_.
3. Process illustrated.
_The Original Proposition (A)_ _The Obverse (E)_
All men are mortal. No men are immortal.
All maples are trees. No maples are not-trees.
All teachers should be No teacher should be
sympathetic. un-sympathetic.
All pain is unpleasant. No pain is pleasant.
All men are imperfect. No men are perfect.
All birds are feathered No birds are non-feathered
animals. animals.
All men are not-trees. No men are trees.
All scalene triangles are No scalene triangles are
non-equilateral. equilateral.
_The E Proposition._
It is obvious that the process of obverting an E is simply the reverse
of obverting an A. Consequently, the same principle obtains; whereas
the _process_ may be illustrated by reading the foregoing illustrations
reversely.
The rule for obverting E is: _Negate the predicate and change the E to
an A by changing the sign no to all._
_The I Proposition._
Let us note the result when the double negative principle is applied to
the I proposition.
Original: “Some men are wise.”
Adding two negatives: “Some men are not not-wise.”
The foregoing simplified: “Some men are not unwise.”
In comparing the first proposition with the last it is observed that
the first is an I while the last is an O; it is also observed that the
predicate of the first was _negated_ in order to form the predicate
of the last. Thus the rule: “Negate the predicate and change the I to
an O.”
The use of circles may make this clearer:
Illustration: FIG. 7.
The significant part of Fig. 7 is that which is inked. Here we have
represented the part of the “men” circle which is common to the “wise”
circle. Thus the inked part represents “Some men are wise.” If the
inked part is entirely inside of the “wise” circle, no part of it can
belong to the “unwise” space without. Thus the obverse, “Some men are
not unwise.”
_Summary._
The obversion of an I proposition.
1. Principle:
Same as with A.
2. Rule:
Negate the predicate and change the I to an O.
3. Process illustrated.
_The Original Proposition (I)_ _The Obverse (O)_
Some water is pure. Some water is not impure.
Some curves are perfect. Some curves are not imperfect.
Some friends are loyal. Some friends are not disloyal.
Some men are true. Some men are not not-true.
Some precious stones are Some precious stones are not
imperfect. perfect.
Some plants are not-trees. Some plants are not trees.
Some boys are not-honest. Some boys are not honest.
It must be borne in mind that when “_not_” is used without the hyphen
it makes the proposition negative, because when “_unhyphened_,” “_not_”
must be thought of in connection with the copula and not in connection
with the predicate; while “_not_” attached to the predicate with
a hyphen simply makes the predicate negative without affecting the
quality of the proposition; e. g., “Some plants are not trees” is
a negative proposition, while “Some plants are not-trees” is an
affirmative proposition with a negative predicate.
It may not be clearly seen how it is possible, by following the rule
given, to pass from such a proposition as “Some plants are _not-trees_,”
to “Some plants are _not trees_.” Let us illustrate the steps:
1. The original: “Some plants are not-trees.”
2. Negating predicate: “Some plants are trees.”
3. Changing to an O: “Some plants are not trees.”
Dropping the not from “1” and then adding it again to “2” is simply
putting into operation the double negative idea, so that there is no
violation of the principle.
_The O Proposition._
O bears the same relation to I that E bears to A. The principle
involved is the same. The process is illustrated by reading reversely
the scheme of illustrations under I. The rule is as follows: _To obvert
an O negate the predicate and change the O to an I by eliminating the
not._
_Summary of Obverting the Four Logical Propositions._
1. Principle:
Two negatives are equivalent to one affirmative.
2. Rules:
{ (1) A to E
Negate the predicate { (2) E to A
and change { (3) I to O
{ (4) O to I
(3) _IMMEDIATE INFERENCE BY CONVERSION._
_Conversion is the process of inferring from a given proposition
another which has, as its subject, the predicate of the given
proposition, and, as its predicate, the subject of the given
proposition._ It is simply a matter of transposing subject and
predicate. The original proposition is called the _convertend_ while
the derived proposition is named the _converse_.
The process of conversion is limited by two rules. First rule. No term
must be distributed in the converse which is not distributed in the
convertend. Second rule. The quality of the converse must be the same
as that of the convertend. More briefly: (1) _Do not distribute an
undistributed term._ (2) _Do not change the quality._
We recall that a term is distributed when it is referred to as a
_definite whole_. An undistributed term is referred to only _in part_.
The principle underlying rule “1,” therefore, is the one which forms
the basis of inference by opposition; namely, “_Whatever may be said of
the entire class may be said of a part of that class_.” The converse of
this is _not true_, that is, “What is said of part of a class cannot be
said of the whole of that class.” When we distribute an undistributed
term we are saying of the _whole_ class what was said only of a _part_
of that class. This is fallacious. On the other hand, we may say of a
part what was said of the whole, or “undistribute” a distributed term.
We recall that the conclusion of the whole matter of inference by
opposition was, that only an I could be inferred from an A and only an
O from an E, or to put it in another way: Only an affirmative from an
affirmative and only a negative from a negative. This establishes the
truth of the second rule in conversion: “Do not change the quality.”
Let us apply the two rules to the four logical propositions.
_Converting an A proposition._
Take as a type, “All horses are quadrupeds.” Here the subject
“_horses_” is distributed, but the predicate “_quadrupeds_” is
undistributed. In transposing subject and predicate we cannot
distribute the term “quadrupeds,” according to the rule which says, “Do
not distribute an undistributed term.” Hence in interchanging subject
and predicate we cannot say, “All quadrupeds are horses,” but must
limit the assertion to, “_Some_ quadrupeds are horses.” Logicians call
this process _Conversion_ by _Limitation_.
_Conversion by Limitation Exemplified Further._
_Convertend._ _Converse._
All metals are elements. Some elements are metals.
All bees buzz. Some buzzing insects are bees.
All men are fallible. Some fallible beings are men.
All good teachers are Some sympathetic persons are good
sympathetic. teachers.
The conclusions from the foregoing are these: First, the usual mode of
converting an A is to interchange subject and predicate, limiting the
latter by the word “_some_” or a word of similar significance. Second,
this mode is called _conversion by limitation_. Third, the converse of
an _A_ is an _I._
_The Co-extensive A._
In the conversion of A propositions there is the one exception of
“co-extensive A’s,” such as truisms and definitions. It will be
remembered that with these both subject and predicate are distributed;
hence, they may be interchanged without limiting the predicate by
“some.” To illustrate: The converse of the truism, “A man is a _man_.”
is “A _man_ is a man,” while the converse of the definition, “A man
is a rational animal,” is “A rational animal is a man.” This mode
of interchanging subject and predicate without limiting the latter
is called _Simple Conversion_. The ordinary A proposition is thus
_converted_ by _limitation_, while the co-extensive A is _converted
simply_.
_Converting an E proposition._
As both terms of the E proposition are distributed it is not possible
to violate the rule of distribution. It is to be remembered that no
fallacy is committed by “undistributing” a term which is already
distributed.
_Illustrations._
_Convertend._ _Converse._
No men are immortal. No immortals are men. Simply.
No birds are quadrupeds. No quadrupeds are birds. Simply.
No metals are compounds. No compounds are metals. Simply.
No men are immortal. Some immortals (at least) are not
men. Limitation.
No birds are quadrupeds. Some quadrupeds are not birds.
Limitation.
No metals are compounds. Some compounds are not metals.
Limitation.
Three facts are evident relative to the converting of an E. First: An E
proposition may be converted either _simply_ or by _limitation_. Second:
E may be converted into either _E_ or _O_. Third: If the converse is
an O then is the inference a _weakened_ one, being _particular_ when it
could just as well be _universal_.
_Converting an I proposition._
With an I proposition neither term is distributed. Thus care must be
used lest an undistributed term in the convertend be distributed in the
converse. _Illustrations_:
_Convertend._ _Converse._
Some men are wise. Some wise beings are men.
Some teachers scold. Some who scold are teachers.
Some high school graduates Some who enter college are high
enter college. school graduates.
Some Americans live simply. Some who live simply are Americans.
From the foregoing we conclude first, that I is _converted simply_;
second, that I is converted _into I_.
_The O Proposition._
With an O proposition the subject is _undistributed_ while the
predicate is distributed. This condition presents a peculiar
difficulty. Consider, for example, the O proposition, “Some men are
not wise.” Convert this into, “Some wise beings are not men,” and the
_undistributed_ subject of the convertend, which is “men,” becomes the
_distributed_ predicate of the converse. _Thus the O proposition cannot
be converted without violating the rule for distribution._
_A Summary of How the Four Logical Propositions May be Converted._
1. _A._ The ordinary A proposition may be converted by _limitation_
only. The co-extensive A may be converted _simply_.
2. _E._ The E proposition is converted _simply_. The E may also be
converted by _limitation_, but the inference thus obtained is
_weakened_.
3. _I._ The I proposition may be converted _simply_ only.
4. _O._ The O proposition _cannot be converted_.
(4) _INFERENCE BY CONTRAVERSION._ (Contraposition).
This mode of inference is usually referred to as inference by
_contraposition_, but contraversion, indicating more definitely the
nature of the process, is a better term. Contraversion involves _two_
steps: First, _obversion_; second, _conversion_. The same principles
and rules evident in these _two_ processes obtain in inference by
contraversion. The following scheme, therefore, ought to be sufficient
to make the matter clear:
_Inference by Contraversion._
1. _The Given Proposition._ 2. _Obverted._
A. All men are mortal. No men are immortal.
All trees are plants. No trees are not-plants.
E. No men are infallible. All men are fallible.
No men are trees. All men are not-trees.
I. Some men are wise. Some men are not not-wise.
O. Some water is not pure. Some water is impure.
Some houses are not white. Some houses are not-white.
3. _Converted; giving the contraverse of the original proposition._
No immortals are men.
No not-plants are trees.
Some fallible beings are men.
Some not-trees are men.
An O cannot be converted, consequently the contraversion of an
I is impossible.
Some impure liquids are water.
Some not-white buildings are houses.
It is indicated in the foregoing scheme that “I” cannot be contraverted.
This is due to the fact that the obverse of an I is an O, and it will
be remembered that “O” cannot be converted. All the other propositions
admit of contraversion.
=4. EPITOME OF THE FOUR PROCESSES OF IMMEDIATE INFERENCE IN CONNECTION
WITH THE FOUR LOGICAL PROPOSITIONS.=
[*]│Proposition│Name of │Inference │Principle
│symbolized │Process │symbolized │involved
───┼───────────┼─────────────┼───────────┼─────────────────────────
A │All S is │Opposition │Some S is │What is said of _all_ may
│ P[†] │ │ P (I) │ be said of _some_.
│ │ │ │
│ │Obversion │No S is │Two negatives are
│ │ │ not-P (E)│ equivalent to one
│ │ │ │ affirmative.
│ │ │ │
│ │Conversion by│Some P is │An undistributed term
│ │ Limitation │ S (I) │ cannot be distributed.
│ │ │ │
│ │Contraversion│No not-P │Same principles which
│ │ │ is S (E) │ obtain in obverting
│ │ │ │ A and converting E.
───┼───────────┼─────────────┼───────────┼─────────────────────────
E │No S is P │Opposition │Some S is │What is said of _all_ may
│ │ │ not P (O)│ be said of _some_.
│ │ │ │
│ │Obversion │All S is │Two negatives are
│ │ │ not-P (A)│ equivalent to one
│ │ │ │ affirmative.
│ │ │ │
│ │Simple │No P is S │Distribution not
│ │ Conversion │ (E) │ affected.
│ │ │ │
│ │Contraversion│Some not-P │An undistributed term
│ │ │ is S (I) │ cannot be distributed.
───┼───────────┼─────────────┼───────────┼─────────────────────────
I │Some S is P│Opposition │Doubtful │None.
│ │ │ │
│ │Obversion │Some S │Two negatives are
│ │ │ is not │ equivalent to one
│ │ │ not-P (O)│ affirmative.
│ │ │ │
│ │Conversion │Some P is │Distribution not
│ │ │ S (I) │ affected.
│ │ │ │
│ │Contraversion│Impossible │None.
───┼───────────┼─────────────┼───────────┼─────────────────────────
O │Some S is │Opposition │Doubtful │None.
│ not P │ │ │
│ │ │ │
│ │Obversion │Some S is │Two negatives are
│ │ │ not-P (I)│ equivalent to one
│ │ │ │ affirmative.
│ │ │ │
│ │Conversion │Impossible │None.
│ │ │ │
│ │Contraversion│Some not-P │Same as in obversion of
│ │ │ is S (I) │ O and conversion of I.
* – Name of proposition
† – “S” represents any subject and “P” any predicate.
INFERENCE BY INVERSION.
Some logicians treat of a form of immediate inference known as
inversion though it is of small importance and of little practical
value.
The process can be applied only to propositions A and E. In the one
case the _contradictory subject_ is limited by “_some_” and then denied
of the predicate, whereas, in the other case, the contradictory subject
is merely affirmed of the predicate.
_Illustrations._
_The Given Proposition._ _The Inverse._
I. All S is P. (A) Some not-S is not P. (O)
All planets rotate. Some not-planets do not rotate.
II. No S is P. (E) Some not-S is P. (I)
No men are immortal. Some not-men are immortal.
From the foregoing we are able to conclude _that the inverse of “A” is
found by negating the subject and changing to an “O”; while the inverse
of “E” is found by negating the subject and changing to an “I.”_
=5. OUTLINE.=
IMMEDIATE INFERENCE――OPPOSITION――OBVERSION, CONVERSION,
CONTRAVERSION AND INVERSION.
1. The Nature of Inference.
2. Immediate and Mediate Inference.
3. The Forms of Immediate Inference.
(1) Opposition.
(a) Scheme of Opposition.
(b) Square of Opposition.
(2) Obversion.
(3) Conversion.
(a) Simply.
(b) By Limitation.
(4) Contraversion.
Inversion.
=6. SUMMARY.=
1. Inference is the thought process of deriving a judgment from one or
two antecedent judgments.
2. Immediate inference is inference without the use of a middle term.
Mediate inference is inference by means of a middle term.
3. The four common forms of immediate inference are (1) opposition,
(2) obversion, (3) conversion, (4) contraversion.
(1) The name _opposition_ stands for certain definite relations which
exist between the logical propositions when they are given the same
subject and predicate. The one principle underlying opposition is:
Whatever is said of the entire class may be said of a part of that
class. The two statements which sum up opposition are first, an I may
be derived from an A; and second, an O may be derived from an E.
The crucial fact made obvious by the square of opposition is that A and
O are mutually contradictory; likewise E and I.
(2) Obversion is the process of passing from an affirmative to its
equivalent negative or from a negative to its equivalent affirmative.
“Two negatives are equivalent to one affirmative,” is the basic
principle of obversion.
The proposition A may be obverted by negating the predicate and
changing to an E. “E” is obverted by negating the predicate and
changing to an A. “I” is obverted by negating the predicate and
changing to an O. “O” is obverted by negating the predicate and
changing to an I.
(3) Conversion is the process of inferring from a given proposition
another which has as its subject the _predicate of the given
proposition_ and as its predicate the _subject of the given
proposition_.
Conversion is limited by the two rules, (1) do not distribute an
undistributed term; (2) do not change the quality.
To convert an A interchange subject and predicate, limiting the latter
by _some_, or a word of like significance. This is called conversion by
limitation.
The co-extensive A may be converted without limiting the predicate.
This is called simple conversion.
An E proposition may be converted either simply or by limitation. When
converted by limitation the inference is a _weakened_ one.
An I proposition is converted simply only.
The O proposition does not admit of conversion.
(4) Immediate inference by contraversion is a process involving first
_obversion_ and then _conversion_.
“A,” “E” and “O” may be contraverted; “I” cannot be contraverted.
=7. ILLUSTRATIVE EXERCISES.=
(1a) From the antecedent judgment, “All weeds are plants,” I am able
to derive by immediate inference these judgments: (1) “All weeds are
not not-plants,” or “No weeds are not plants.” (2) “No not-plants are
weeds.” (3) “Some plants are weeds.” (4) “Some weeds are plants.”
(1b) “All vertebrates have a backbone.” From the foregoing judgment
derive immediately five different conclusions.
(2a) “All good citizens try to vote,”
“Albert White is a _good citizen_,”
Hence, “Albert White will try to vote.”
I know that the above is an example of mediate inference because the
two antecedent judgments make use of the middle term, “_good citizen_.”
(2b) Why is the following illustrative of mediate inference?
“All wise men are close observers,”
“All wise men are thoughtful,”
Hence, “Some thoughtful men are close observers.”
(3a) Derive immediate inferences by opposition from the following:
(1) “Good men are wise.”
(2) “No teacher can afford to be unjust.”
(3) “All birds fly.”
(4) “None of the inner planets are as large as the earth.”
I first determine that “1” and “3” are A propositions, while “2” and
“4” are E’s. Then I recall that by opposition an I may be derived from
an A and an O from an E. Hence, the inferences are:
(1) “Some good men are wise.”
(2) “Some teachers cannot afford to be unjust.”
(3) “Some birds fly.”
(4) “Some of the inner planets are not so large as the earth.”
(3b) Derive by opposition inferences from the following:
(1) “No true woman will neglect her home for society.”
(2) “All patriotic men love the flag.”
(3) “Fools rush in where angels fear to tread.”
(4a) Obvert the following:
(1) “All earnest teachers are diligent students.”
(2) “No self-respecting man can afford to be careless in his personal
appearance.”
(3) “Some of the great teachers of the past did not practice what
they preached.”
(4) “Some weeds are beautiful.”
I determine first the logical character of each proposition, finding
the first to be an A, the second an E, the third an O and the fourth an
I. Then I recall that in obversion the predicate must always be
_negated_ and an A must be changed to an E or an E to an A; also an I
must be changed to an O or an O to an I. Hence, the obverse of each
proposition is:
(1) “No earnest teacher is a not-diligent student.”
(2) “All self-respecting men can afford to be not-careless (careful)
in their personal appearance.”
(3) “Some of the great teachers of the past did not-practice (failed
to practice) what they preached.”
(4) “Some weeds are not not-beautiful.”
(4b) Infer by obversion from the following:
(1) “All roses are beautiful.”
(2) “None of the members of the stock exchange are dishonest.”
(3) “Some pupils are not industrious.”
(4) “Some teachers are tactful.”
(5a) Convert the following:
(1) “All that glitters is not gold.”
(2) “All good men are wise.”
(3) “Some books are to be chewed and digested.”
(4) “No man is perfectly happy.”
It is first necessary to determine the logical character of each
proposition. Carelessness might lead one to call the first proposition
an A because it is introduced by the quantity sign “_all_.” But on
second thought we note that the meaning is to the effect that _some
glittering things are not gold_; this is an O. It is clear that
the second is an A, the third an I and the fourth an E. It is now
expedient to recall the rules regarding conversion. These are, (1) do
not distribute an undistributed term; (2) do not change the quality.
We may now attempt to interchange the subject and predicate of each
proposition, with the following results:
(1) Conversion impossible.
(2) “Some wise men are good men.”
(3) “Some things to be chewed and digested are books.”
(4) “No perfectly happy being is a man.”
When attempting to convert proposition (1), I find that the subject
which is undistributed becomes distributed, hence the rule pertaining
to distribution is violated. This conclusion is verified by recalling
the fact that an O proposition cannot be converted. The second
proposition, being an A, is converted by limitation; while the third
and fourth are converted simply, as is the natural procedure with all
I’s and E’s.
(5b) Convert these propositions:
(1) “Blessed are the meek.” (All the meek are blessed.)
(2) “None but material bodies gravitate.” (All gravitating bodies are
material.)
(3) “Gold is not a compound substance.”
(4) “Usually cruel men are cowards.”
NOTE.――_The first proposition is poetical while the second is an
exclusive._
(6a) Contravert the following propositions:
(1) “All virtue is praiseworthy.”
(2) “Some teachers are not tactful.”
(3) “A man who lies is not to be trusted.”
Contraversion consists in obverting first, and then converting;
consequently, the contraverse of the three propositions is as follows:
(1) “No unpraiseworthy deed is virtue.”
(2) “Some not-tactful persons are teachers.”
(3) “Some untrustworthy men are those who lie.”
(6b) Write the contraverse of the following:
(1) “All honest men pay their debts.”
(2) “All men are rational.”
(3) “Nearly all the troops have left the town.”
(4) “Some teachers are not patient.”
(7a) The attending scheme indicates the logical process and rule
involved in passing from one proposition to another:
A. “All men are imperfect.”
Process: Obversion.
Rule: Negate predicate and change to E.
E. “No men are perfect.”
Process: Simple Conversion.
Rule: Interchange subject and predicate.
E. “No perfect beings are men.”
Process: Contraversion.
Rule: Obvert and then convert.
I. “Some not-men are perfect beings.”
(7b) Treat in a manner similar to the above the proposition, “All
horses are quadrupeds.”
=8. REVIEW QUESTIONS.=
(1) What is inference?
(2) What is the meaning of antecedent?
(3) Define (1) judging, (2) a judgment.
(4) All roses are beautiful,
This flower is a rose,
This flower is beautiful.
Write this example of mediate inference in equation form. Name
the middle term.
(5) Define immediate inference. Illustrate.
(6) Define mediate inference. Illustrate.
(7) Name the five forms of immediate inference.
(8) What principle is involved in inference by opposition?
(9) Draw the scheme of opposition.
(10) Make use of this scheme in deriving inferences from the following
propositions:
(a) “Good men are wise.”
(b) “No king is infallible.”
(c) “Cattle are ruminants.”
(d) “All who cheat the railroads are not honest.”
(11) What are contradictory propositions? Illustrate.
(12) What would be the simplest way of disproving the statement that
“No great religious teacher has been consistent?”
(13) Why are A and E said to be contrary propositions?
(14) Define obversion.
(15) By what other name is obversion known?
(16) State the basic principle of obversion.
(17) Illustrate the process known as _negating the predicate_.
(18) State the rule for obverting an A proposition.
(19) Obvert the following:
(1) “All the boys in my room are industrious.”
(2) “Honesty is the best policy.”
(3) “Only the industrious are truly successful.”
(20) First state the rule and then obvert the following:
(1) “Some plants are biennial.”
(2) “Planets are not suns.”
(3) “Blessed are the merciful.”
(4) “These samples are not perfect.”
(21) Define conversion.
(22) State and illustrate the rules which condition the process of
conversion.
(23) Convert, if possible, the following:
(1) “Some men practice sophistry.”
(2) “Few men know how to live.”
(3) “Some of the inhabitants are not civilized.”
(4) “All the world is a stage.”
(5) “None of my pupils failed.”
(6) “Experience is a hard taskmaster.”
(24) Why may co-extensive propositions be converted simply?
(25) Describe the process of inference by contraversion.
=9. PROBLEMS FOR ORIGINAL THOUGHT AND INVESTIGATION.=
(1) What ground is there for the belief that immediate inference, so
called, is merely a matter of the interpretation of propositions?
(2) Is there any difference between reasoning and inference?
(3) When the conclusion is reached that two rooms are of the same
width, because each is five yards wide, what is the middle term?
(4) Put in equation form:
All teachers instruct,
John Jones is a teacher,
John Jones instructs.
Show that the equations are not _absolutely true_.
(5) Indicate the true relation between the subjects and predicates
of the foregoing by using the algebraic signs > and <.
(6) Why cannot an A be derived from an I?
(7) Why cannot an O be derived from an A?
(8) The basic principle of obversion is “Two negatives are equivalent
to one affirmative.” Show by means of circles that this is not
absolutely true; take as an illustrative proposition, “No men are
not mortal.”
(9) Show that agreeable and disagreeable are not contradictory terms.
(10) Why should the logician class individual propositions as
universal?
(11) Show by circles that there is a difference in signification
between, “Some men are _not wise_” and “Some men are _not-wise_.”
(12) Show by circles that the O proposition cannot be converted.
(13) “The I proposition cannot be contraverted.” Make this clear.
(14) Is there any difference in meaning between, “All illogical work
is unscholarly” and “No illogical work is scholarly?” Explain by
circles.
(15) State the logical process involved in passing from each
proposition to its succeeding one:
(1) “All men are imperfect.”
(2) “No men are perfect.”
(3) “No perfect beings are men.”
(4) “Some not-men are perfect beings.”
(5) “Some perfect beings are not-men.”
(6) “Some perfect beings are not men.”
(16) It is sometimes said that in sub-contraries there is really no
opposition. Do you agree? Give arguments.