LOGICAL PROPOSITIONS
=1. THE NATURE OF LOGICAL PROPOSITIONS.=
Judging has been defined as the process of conjoining or disjoining
notions. This may be put in another way: “Judging is the process of
asserting or denying the agreement between two notions.” The product
of the act of judging is a judgment and when judgments are put in
word-form such expressions are called logical propositions.
Definition: _A logical proposition is a judgment expressed in words._
Just as percept and concept notions are expressed by means of logical
terms so judgment notions may be expressed by logical propositions.
To illustrate: The terms _the squirrel_ and _cracking a nut_ express
two notions, and when an agreement between them is asserted and the
product is expressed in word form, then such an expression becomes the
logical proposition, “The squirrel is cracking a nut.”
The following being _expressed_ judgments are logical propositions:
(1) All men are mortal.
(2) Some men are wise.
(3) No men are immortal.
(4) Some men are not wise.
(5) No sane person is a lover of vice.
(6) Some good orators are not good statesmen.
(7) Every man is fallible.
(8) If it rains, I shall not go.
(9) He is either sane or insane.
=2. KINDS OF LOGICAL PROPOSITIONS.=
There are three kinds of logical propositions; namely, categorical,
hypothetical and disjunctive.
_A categorical proposition_ is one in which the assertion is made
unconditionally. _An hypothetical proposition_ is one in which the
assertion depends upon a condition. _A disjunctive proposition_ is
one which asserts an alternative.
THE THREE KINDS ILLUSTRATED:
(1) “Every dog has his day.” Categorical.
(2) “If you do your best, success will reward you.” Hypothetical.
(3) “He is either stupid or indolent.” Disjunctive.
(4) “All vices are reprehensible.” Categorical.
(5) “Either you are very talented or very industrious.” Disjunctive.
(6) “If capital punishment does not aid society, it should be
abolished.” Hypothetical.
(7) “You may go provided your teacher is willing.” Hypothetical.
(8) “No intelligent man can ignore the practice of temperance.”
Categorical.
By studying the illustrations it will be observed that the categorical
propositions are direct, bold, assertive statements, whereas the
hypothetical are limited by conditions which make them less forceful.
In the second proposition, for example, “success will reward you,”
is limited by the condition, “If you do your best.” The disjunctive
may be regarded as categorical in form, but hypothetical in meaning,
because in such a proposition as, “He is either, stupid or indolent,”
a direct assertion is made which suggests the categorical, and yet it
may be implied that, if he is stupid then he is not indolent; this is
indicative of the hypothetical.
Some logicians classify propositions as categorical and conditional,
the conditional being subdivided into hypothetical and disjunctive. The
first classification seems preferable, however, as it conforms to the
three modes of reasoning.
The common word-signs of the categorical proposition are _all_,
_every_, _each_, _any_, _no_ and _some_, while those of the
hypothetical are _if_, _even if_, _unless_, _although_, _though_,
_provided that_, _when_, or any word or group of words denoting a
condition. The disjunctive symbols are _either――or_.
=3. THE FOUR ELEMENTS OF A CATEGORICAL PROPOSITION.=
Every categorical proposition should have four elements; namely,
the quantity sign, the logical subject, the copula and the logical
predicate. In the foregoing categorical propositions the quantity signs
are respectively, _every_, _all_ and _no_. In any case the quantity
sign is always attached to the subject and indicates its breadth or
extension. For example, in the two propositions, “All men are mortal”
and “Some men are wise,” the quantity sign _all_ makes the term _man_
much broader than does the quantity sign _some_.
The logical subject of a categorical proposition is the term of which
something is affirmed or denied, whereas the logical predicate of a
categorical proposition is the term which is affirmed or denied of
the subject. In the two propositions, “All men are mortal” and “No men
are immortal,” the term about which something is affirmed or denied is
_men_, while the terms which are affirmed and denied of the subject are
respectively _mortal_ and _immortal_. “Men” is, therefore, the logical
subject of each proposition, while “mortal” is the logical predicate
of the first and “immortal” the logical predicate of the second. The
copula is the connecting word between the logical subject and predicate
and denotes whether or not the latter is affirmed or denied of the
former. The copula is always some form of “_to be_” or its equivalent.
When the predicate is _denied_ of the subject, “_not_” may be used with
the copula and considered a part of it. To illustrate: in the logical
proposition, “Some men are not wise,” “_are not_” may be regarded as
the copula.
The four elements are indicated in the following categorical
propositions:
_Quantity
sign_ _Logical subject_ _Copula_ _Logical predicate_
All fixed stars are self-luminous
No wise man is going to steal
Some quadrupeds are domestic animals
Some glittering things are not gold
Some boys are not discreet
A few men are multi-millionaires
Every citizen is duty-bound to vote
The student must ever keep in mind the fact that to be absolutely
logical all categorical propositions must be expressed in terms of
the _four elements_. However, life is too short and man is too busy to
speak always in terms of the four elements. Moreover, to be logical may
often compel an awkwardness of expression and a lack of euphony which
could hardly be tolerated. For these reasons the utterances in ordinary
conversation are frequently illogical so far as the four elements are
concerned, though not necessarily illogical in meaning. When it is
desired to test the validity of any series of statements leading up to
some generalization, it may become necessary to express the statement
in terms of the four elements. The student should gain some facility
in this, otherwise he may be readily led into fallacious reasoning.
The following statements taken at random from newspapers are given in
the original and then expressed in terms of the four elements:
_The Original_ _In Terms of the Four Elements_
(1) You came too late. (1) The person is one who came too
late.
(2) I saw the swell turnout (2) The man was one who saw the swell
coming along. turnout coming along.
(3) All of the men walked. (3) All of the men were those who
walked.
(4) The robbers cut a hole (4) All the robbers were the ones who
in this floor. cut a hole in this floor.
(5) Some of these flew away. (5) Some birds were those which
flew away.
(6) The rain interfered with (6) The rain was that which interfered
the attendance. with the attendance.
(7) Our habits make or (7) All our habits are forces which
unmake us. make or unmake us.
(8) We all had a fine time. (8) All the party were those who had
a fine time.
In argumentative discourse it is often sufficient to “think the
proposition” in terms of the four elements without taking the time to
actually express it.
=4. LOGICAL AND GRAMMATICAL SUBJECT AND PREDICATE DISTINGUISHED.=
The grammatical subject is _one_ word while the logical subject is the
grammatical subject with all its modifiers except the quantity sign.
For example: in the proposition, “All white men are Caucasians,” _men_
is the grammatical subject, while _white men_ is the logical subject.
_All_ being the quantity sign simply indicates the extension of _men_
and is not a part of the logical subject.
The grammatical predicate is the verb-form together with any predicate
noun or adjective, while the logical predicate is the predicate word
or words and all its modifiers. The grammatical predicate includes
the copula, but the logical predicate never includes the copula. The
grammatical predicate does not include the object, while the logical
predicate always includes what is equivalent to the object and all
its modifiers. To illustrate: in the proposition, “Some men are wise,”
_are wise_ is the grammatical predicate, while _wise_ is the logical
predicate. And in the proposition, “He burned the red house on the
hill,” _burned_ is the grammatical predicate, while _the one who burned
the red house on the hill_ is the logical predicate.
=5. THE FOUR KINDS OF CATEGORICAL PROPOSITIONS.=
Categorical propositions are divided according to their _quantity_
into _Universal_ and _Particular_ and according to their _quality_ into
_Affirmative_ and _Negative_.
_A universal proposition is one in which the predicate refers to the
whole of the logical subject._
ILLUSTRATIONS:
(1) All men are mortal.
(2) All civilized men cook their food.
(3) No dogs are immortal.
(4) Every man was once a boy.
Considering the first proposition, “_mortal_,” the logical predicate,
refers to the whole of the logical subject “_men_.” Similarly “_cook
their food_” refers to the whole of the term “_civilized men_”;
“_immortal_” to the whole of the term “_dogs_,” and “_once a boy_”
to the whole of the term “_man_.”
In considering the definition of a universal proposition it is
necessary to keep in mind the distinction between a logical and a
grammatical subject, as in the second proposition the logical predicate,
“cook their food,” refers to only a part of the grammatical subject,
_men_, and, therefore, the proposition might fallaciously be termed a
particular proposition rather than a universal.
_A particular proposition is one in which the predicate refers to only
a part of the logical subject._
ILLUSTRATIONS:
(1) Some men are wise.
(2) Some animals are not quadrupeds.
(3) Most elements are metals.
(4) Many children are mischievous.
In the foregoing propositions _some_, _most_ and _many_ are quantity
signs and, therefore, must not be considered as a part of the logical
subjects. Considering the logical subjects and predicates in order,
the term _wise_ refers to only a part of the _men_ in the world,
_quadrupeds_ to only a part of the _animals_, _metals_ to only a part
of the _elements_ and _mischievous_ to only a part of the _children_.
_An affirmative proposition is one which expresses an agreement between
subject and predicate._
_A negative proposition is one which expresses a disagreement between
subject and predicate._
Affirmative propositions conjoin terms, negative propositions disjoin
terms. In the first the agreement is affirmed; in the second the
agreement is denied.
ILLUSTRATIONS:
None of the captives escaped. Negative.
Some teachers are just. Affirmative.
All trees grow towards heaven. Affirmative.
Some people are not companionable. Negative.
No person is above criticism. Negative.
Dividing both universal and particular propositions as to quality,
gives four kinds; namely, universal affirmative, universal negative,
particular affirmative and particular negative. No topic in logic
demands greater familiarity than these four types, as every proposition
must be reduced to one of the four before it can be used as a basis of
reasoning.
For the sake of brevity the symbols A, E, I and O are used to
designate respectively the universal affirmative, the universal
negative, the particular affirmative and the particular negative. A and
I, symbolizing the affirmative propositions, are the first and second
vowels in _Affirmo_, while E and O, symbolizing the negatives, are the
vowels in _Nego_. The common sign of the universal affirmative, or the
A proposition is _all_; of the universal negative, or E proposition
_no_; of the particular affirmative, or I proposition _some_; of the
particular negative, or O proposition _some_ with _not_ as a part of
the copula. The accompanying classification summarizes these facts,
S and P being used to symbolize the terms “_subject_” and “_predicate_.”
_Illustrations_
{ { Affirmative-A All S is P
Categorical { Universal { Negative-E No S is P
Propositions {
{ { Affirmative-I Some S is P
{ Particular { Negative-O Some S is not P
Henceforth the symbols A, E, I, O will be used to designate the four
kinds of categorical propositions. The propositions have other quantity
signs aside from the ones used above. These may be summarized:
{ A――all, every, each, any, whole.
{ E――no, none, all-not.
Quantity signs of { I――some, certain, most, a few, many, the
{ greatest part, any number.
{ O――some - - not, few.
=6. PROPOSITIONS WHICH DO NOT CONFORM TO THE LOGICAL TYPE.=
It has been observed that all expressed judgments must be reduced to
one of the four logical types A, E, I or O, before they can be used
argumentatively. Logic insists upon definiteness and clearness――there
must be no ambiguity, no opportunity for a wrong interpretation. From
this viewpoint the four types fulfill every requirement. Their meaning
cannot be misunderstood. To any one with normal intelligence their
significance may be made perfectly clear. Any argument when once put in
terms of the four types may be spelled out with mathematical precision.
In consequence it is of prime importance that the four types not only
be well understood, but that a certain facility be gained in reducing
ordinary conversation to some _one_ of these types.
(1) Indefinite and Elliptical Propositions.
It is known that every logical proposition must be expressed in terms
of the four elements――_quantity sign_, _logical subject_, _copula_ and
_logical predicate_, consequently the four types A, E, I and O which
epitomize every form of logical proposition, are expressed in terms
of these four elements. But in common conversation often the quantity
sign, as well as the copula, is omitted. See section 3.
Propositions without the quantity sign are called _indefinite_,
while those with the suppressed copula may be termed _elliptical_
propositions. Both may be made logical as the attending illustrations
will indicate:
_Illogical_ _Logical_
_Indefinite_
Men are fighting animals. _All_ men are fighting animals.
(A)
Lilies are not roses. _No_ lilies are roses. (E)
Good is the object of moral _All_ good is the object of moral
approbation. approbation. (A)
Perfect happiness is _In all cases_ perfect happiness
impossible. is impossible. (A)
_Elliptical_
Fashion rules the world. _All_ fashions _are_ ruling the
world. (A)
Trees grow. _All_ trees _are plants which_
grow. (A)
Children play. _All_ children _are_ playful. (A)
Some men cheat. Some men _are persons who_ cheat.
(I)
Here it is noted that the logical form of some propositions is not
always the most forceful. Often the logical form gives an awkward
construction and should be resorted to only for purposes of logical
argument.
The reduction of either kind to the logical form must be determined
by the meaning of the proposition. As a usual thing the indefinite is
_universal_ (either an A or an E) in meaning, while the problem of the
elliptical is to give it in terms of the _copula_, expressed with as
little awkwardness as possible.
General truths, because attended with no quantity sign, might be
classed as indefinite propositions, though their universality is so
apparent that they may be unhesitatingly classed as universals.
ILLUSTRATIONS:
“Things equal to the same thing are equal to each other.”
“Trees grow in direct opposition to gravity.”
“Honesty is the best policy.”
“A stitch in time saves nine.”
Because the indefinite proposition is so frequently of a general
nature, it is sometimes classed as _general_ rather than _indefinite_.
Sir William Hamilton would class the indefinite as an _indesignate
proposition_.
(2) Grammatical Sentences.
The grammarian divides sentences into five kinds; namely, declarative,
interrogative, imperative, optative, exclamatory. But logic recognizes
only the declarative, as it has already been seen that the four logical
types are declarative in nature. A logical proposition, then, is always
a sentence, but all sentences are not logical propositions. The four
kinds of sentences which are not logical propositions may be usually
reduced to one of the four types as the attending illustrations will
indicate:
_Illogical_ _Logical_
Interrogative. Do men have The question is asked, Do men have
the power of reason? the power of reason?[7] (A)
Imperative. “Thou shalt not All men are commanded not to
steal.” steal, or you are one who should
not steal. (E)
Optative. “I would I had a I am one who desires a million
million.” dollars. (A)
Exclamatory. “Oh, how you You are one who frightened me. (A)
frightened me!”
(3) Individual Propositions.
_An individual proposition is one which has a singular subject_; e. g.,
_Abraham Lincoln_ was an honest man. _Peter the Great_ was Russia’s
greatest ruler. _The maple tree in my yard_ is dying of old age. These
propositions, having a singular term as subject, are individual or
singular in nature. As the predicate refers to the whole of the logical
subject, individual propositions are classed as _universal_.
(4) Plurative Propositions.
Plurative propositions are those introduced by “most,” “few,” “a few,”
or equivalent quantity signs. For example, “_Most_ birds are useful to
man”; “_Few_ men know how to live”; “_A few_ of the prisoners escaped,”
are plurative propositions. “Most” means more than half, while “few”
and “a few” mean less than half. In either case the proposition is
particular. Stated logically, the illustrative propositions would take
the form of “Some birds are useful to man”; “Some men do not know how
to live”; “Some of the prisoners escaped.”
The reader will observe the difference in significance between _few_
and _a few_. The former is negative in character and when introducing
a proposition makes it a particular negative (O). The latter always
introduces a particular affirmative (I).
(5) Partitive Propositions.
Partitive propositions are particulars which imply a complementary
opposite. These arise through the ambiguous use of _all-not_, _some_
and _few_. _All-not_ may sometimes be interpreted as _not all_ and
sometimes as _no_. To illustrate: The proposition, “All men are
not mortal,” is distinctly a universal negative or an E, while the
proposition, “All that glitters is not gold,” is a particular negative
or an O. The logical form of the first is, “No men are mortal,” and
of the second, “Some glittering things are not gold.” When used in the
“not-all” sense, the proposition is partitive because if the O-meaning
is intended the I is implied. For example, “All that glitters is not
gold,” is partitive because the statement implies that some glittering
things _are_ gold (I) as well as the complement, “Some glittering
things _are not_ gold” (O). A knowledge of both the affirmative and
negative aspects is taken for granted in the statement of either the
one or the other.
“All-not,” then, is negative in any case, but universal when it
means _no_ and particular when it means _not all_. Any proposition is
partitive in nature when the quantity sign is _not all_, or _all-not_
interpreted as the equivalent of _not all_.
It may be observed here that _all_ has two distinct uses. First, it
may be used in a collective sense; second, in a distributive sense. For
example: _All_ is used in the _collective_ sense in such propositions
as, “All the members of the football team weighed exactly one ton,” or
“All the angles of the triangle are equal to two right angles.” Using
_all_ in the distributive sense would make true these: “All the members
of the football team weigh more than 140 pounds”; “All the angles of
a triangle are less than two right angles.” _All_ is used collectively
when reference is made to an aggregate, but distributively when
reference is made to each.
The quantity sign _some_ is likewise ambiguous, as it may mean (1) some
only――some, but not all, or (2) some at least――some, it may be all or
not all. When “some” is used as the quantity sign of any particular
proposition which has been accepted as logical, the second meaning,
“some at least,” is always implied. This interpretation of “_some_”
will be explained more in detail in a succeeding section.
When _some_ is used in the sense of _some only_, the partitive nature
of the proposition is apparent, as both I and O are implied. For
example, with reference to the human family, to say that “some only are
wise” necessitates an investigation, which leads to the discovery that
some are wise, while others are not wise. If the proposition be an I,
then its complementary O is implied, or if it be an O, the I is implied.
_Few_ given as a sign of a plurative proposition also serves as a sign
of the partitive. The plurative aspect is prominent when it is said
that “Few men can be millionaires” and emphasis is placed upon the
meaning that “Most men cannot be millionaires.” But when emphasis is
given to “few,” as meaning _few only_ rather than the _most are not_,
then the I and the O are both implied; e. g., _Some men become
millionaires, but the most do not_.
To put it in a word, “all-not,” “some” and “few” introduce
partitive propositions when the meaning implies both an I and an O.
When treating such in logic the meaning which seems to be given the
greater prominence must be accepted. Surely in the statement, “All that
glitters is not gold,” the O-interpretation is the one intended; namely,
“Some things which glitter are not gold.”
ILLUSTRATIONS:
(1) “_All_ men are not honest.”
(2) “_Few_ men live to be a hundred.”
(3) “_Some_ men are consistent.”
The first proposition with the emphasis placed upon _all_ suggesting
that _some men are not honest_, is the intended proposition while _some
men are honest_ is the implied. In reducing it to the logical form the
intended proposition is the one which should be used.
With the emphasis upon _few_ and _some_, the second and third
propositions may be interpreted as follows: (2) Intended proposition,
_Some men do not live to be a hundred_. Implied proposition, _Some
men do live to be a hundred_. (3) Intended proposition, _Some men are
consistent_. Implied proposition, _Some men are not consistent_.
(6) Exceptive Propositions.
These are introduced by such signs as _all except_, _all but_,
_all save_. To wit: (1) “_All except_ James and John may be excused”;
(2) “_All but_ a few of the culprits have been arrested”; (3) “_All_
birds _save_ the English sparrow are serviceable to man” are exceptive
propositions.
Exceptive propositions are universal when the exceptions are mentioned.
Universal propositions necessitate a subject more or less definite, as
the predicate of such must refer to the whole of a _definite_ subject.
It follows that in exceptive statements definiteness is secured when
the exceptions are mentioned, therefore it becomes clear how all
such propositions must be universal. Of the illustrations, the first
and third propositions are universal. Any exceptive proposition is
particular when the exceptions are referred to in general terms or
when the subject is followed by _et cetera_. The second illustrative
proposition is particular.
(7) Exclusive Propositions.
Of all propositions which vary from the logical form the exclusive
is the most misleading. Exclusives are accompanied by such words as
“only,” “alone,” “none but,” and “except.” Their peculiarity rests
in the fact that reference is made to the _whole_ of the predicate,
but only to a _part_ of the subject. For example, in the exclusive
proposition, “Only elements are metals,” _metals_ is referred to as
a whole while _elements_ is considered only in part. The true meaning
is “Some elements are all metals,” or to put it in logical form, “All
metals are elements.” _The easiest way to deal with an exclusive is
to interchange subject and predicate (convert simply) and call the
proposition an A._
PROCESS ILLUSTRATED:
_Exclusive Proposition_ _Reduced to Logical Form_
1. None but high school All who enter Training School
graduates may enter must be high school graduates.
Training School.
2. Only first-class All parlor cars are for
passengers are allowed in first-class passengers.
parlor cars.
3. Residents alone are All who are licensed to teach are
licensed to teach. residents.
4. No admittance except on All who have business may be
business. admitted.
5. Only bad men are not-wise. All who are not-wise are bad men.
6. Only some men are wise. All who are wise are men.
It is claimed by good authority that the real nature of the exclusive
is best expressed by _negating_ the subject and calling the proposition
an E; e. g., exclusive: “Only elements are metals”; logical form: “No
not-elements are metals” (E). In a succeeding chapter it is explained
how an E admits of first simple conversion and then obversion. The
following illustrate these two processes:
_Original E_: “No not-elements are metals.”
_Simple conversion_: “No metals are not-elements.”
_Obversion_: “All metals are elements.”
From this it may be seen that the statement, “The easiest way to deal
with an exclusive is to interchange subject and predicate and call the
proposition an A,” is substantially correct.
(8) Inverted Propositions.
The poet often employs the inverted proposition, illustrated by
the following: “Blessed are the merciful;” “Great is this man of
war.” An interchanging of subject and predicate makes these poetical
constructions logical; e. g., “All the merciful are blessed;” “This man
of war is great.”
NOTE.――The student should not be misled by the relative clause. Often
it may be interpreted as a part of the predicate rather than the
subject. To wit: “No man is a friend who betrays a confidence”; clearly
the logical subject is _no man who betrays a confidence_.
=7. PROPOSITIONS WHICH ARE NOT NECESSARILY ILLOGICAL.=
(1) _Analytic and Synthetic Propositions._
_An analytic proposition is one in which the predicate gives
information already implied in the subject._ Thus, “_Fire burns_,”
“_Water is wet_,” “_A triangle has three angles_” are analytic
propositions because the predicates do not give added information
to one who has any conception of the subjects. Because the attribute
mentioned by the predicate is an essential one, analytic propositions
are sometimes termed _essential_ propositions. Other names for the same
kind of proposition are _verbal_ and _explicative_.
_A synthetic proposition is one in which the predicate gives
information not necessarily implied in the subject. “Fire protects men
from the wild animal.” “A cubic foot of water weighs 62½ lbs.” “The sum
of the interior angles of a triangle is equal to two right angles.”_
These are synthetic because a common conception of the meaning of
the subject would not need to include the information given by the
predicate. Other names for synthetic propositions are _accidental_,
_real_ and _ampliative_.
The distinction between analytic and synthetic propositions is not
so clear as would on first thought appear. “_Fire burns_” might give
added information to the child or savage who knows only of the light
emitted by fire. To them, then, the proposition would be synthetic. The
distinction must be based upon the assumption that the same words mean
about the same thing to people in general.
This analytic-synthetic division of propositions finds a significance
in the domain of philosophy. To the logician the distinction is of
slight importance save in the so-called verbal disputes, viz.: disputes
which turn on the meaning of words.
(2) _Modal and Pure Propositions._
_A modal proposition states the mode or manner in which the predicate
belongs to the subject._ The signs of modal propositions are the
adverbs of time, place, degree, manner. Illustrations: “James is
walking _rapidly_.” “Honesty is _always_ the best policy.” “Aristotle
was _probably_ the greatest thinker of ancient times.”
_A pure proposition simply states that the predicate belongs, or
does not belong, to the subject._ Illustrations: “James is walking.”
“Honesty is the best policy.” “Aristotle was the greatest thinker of
ancient times.”
Some logicians refer to modal propositions as being such as indicate
_degrees_ of belief. Such words as “probably,” “certainly,” etc., would
indicate their modality.
As logic has to do with the pure proposition and not the modal, the
difference of opinion is of little import.
(3) _Truistic Propositions._
_A truistic proposition is one in which the predicate repeats the words
and the meaning of the subject._ Illustrations: “A man is a man,” “A
beast is a beast,” “A traitor is a traitor,” “What I have done I have
done.”
The truistic proposition is of little importance except in cases
where the subject is used extensionally while the predicate is used
intensionally. In the illustration, “A man is a man,” the subject
merely stands for a member of the man family, while the predicate may
indicate certain manly qualities. Against such ambiguities the logician
must be on guard.
=8. THE RELATION BETWEEN SUBJECT AND PREDICATE.=
In Chapter 5 the extension and intension of terms was explained. The
student recalls, for instance, that the term “man” may be used to
denote objects, as “white man,” “black man,” “red man,” etc. In this
sense the term “man” is used _extensionally_. But when made to stand
for the attributes “rationality,” “power of speech,” etc., the term
“man” is used intensionally.
In considering the relation between subject and predicate it is
customary to employ the terms in an extensional sense only, since such
a restriction serves the purpose of syllogistic reasoning and
conversion.
Let us, then, give attention to the _extension_ of the subject and
predicate of the categorical propositions A, E, I, O.
(1) _The Universal Affirmative or A Proposition._
_All S is P symbolizes the A proposition._ This may be interpreted as
meaning that all of the subject belongs to a part of the predicate, or
that all of the subject belongs to all of the predicate. The first
interpretation is the usual one and may be illustrated by the following
propositions:
1. “All men are mortal.”
2. “All trees grow.”
3. “All metals are elements.”
It is obvious that the subjects of these propositions include every
specimen of the particular class mentioned. For example: The subject
_all men_ includes every specimen of the human family; _all trees_
includes every object of that class; _all metals_ covers everything
which the scientist classes as such. In the three propositions, then,
reference is made to the _whole_ subject but to only a _part_ of the
predicate, as other beings beside men, such as the horse, are mortal;
and other plants aside from trees, such as the sun flower, grow; other
substances, namely oxygen, are elements.
For the sake of making the logical meaning of the four propositions
clearer, recourse may be made to Euler’s diagrams, so named because
the Swiss mathematician and logician, Leonhard Euler, first used them.
The first illustration of the A proposition, “All men are mortal,”
may be represented by two circles, a larger circle standing for the
predicate, _mortal_, and a smaller circle entirely inside the larger
representing the subject, _men_. Thus:
Illustration: FIG. 1.
It is evident that all of the smaller circle belongs to the larger.
This diagram will then fit any proposition where it may be said that
all of the subject belongs to a part of the predicate, or which may
be symbolized as “All S is some P.” (All the subject is some of the
predicate.)
The student knows that circles are plane surfaces and when such a
statement as “All men are mortal” is given, reference is made to only
that part of the “mortal” circle which is _directly underneath_ the
“men” circle. Nothing has been said relative to the remaining part of
the “mortal” circle.
“_A_” propositions which may be interpreted as meaning “All S is
all P” are called co-extensive A’s because the subject and predicate
are exactly equal in extension. Such propositions are best illustrated
by definitions; e. g.:
1. “A man is a rational biped.”
2. “A trigon is a polygon of three sides.”
3. “Teaching is the art of occasioning those activities which result
in knowledge, power and skill.”
To represent the meaning of the co-extensive A by the Euler diagram,
two circles of the same size may be drawn, one coinciding at every
point with the other. If the first circle is drawn heavily in black and
the second dotted in red, it will make clear to the eye that there are
two circles.
(2) _The Universal Negative or E Proposition._
_“No S is P” best symbolizes the E proposition_, though sometimes the
universal negative is written “All S is not P.” This latter form, as
has been explained, is ambiguous and therefore illogical.
“No S is P” surely means that no part of the subject belongs to any
part of the predicate and no part of the predicate belongs to any part
of the subject. The subject and predicate are mutually exclusive.
The following illustrate the E proposition:
1. “No man is immortal.”
2. “No true teacher works for money.”
3. “No thorough student can remain unwise.”
The E proposition may be represented by two circles, the
one entirely without the other as in Fig. 2:
Illustration: FIG. 2.
(3) _The Particular Affirmative or I Proposition._
_This may be symbolized as “Some S is P,”_ and considered as meaning
that a _part_ of the subject belongs to a _part_ of the predicate. It
has already been noted that “some” is ambiguous and that its logical
signification is “some at least_.” (It may be all or it may not be
all.) For example, the only logical interpretation which can be placed
on “Some men are wise” is, that the investigation has resulted in
finding only a _part_ of the man family wise. Whether or not all are
wise is unknown as the entire field has not received attention. In no
case can it be assumed that all the others are _not_ wise.
The I proposition illustrated:
1. “Some men are wise.”
2. “Some animals are vertebrates.”
3. “Some teachers are inspiring.”
The meaning of the I proposition may be represented by two circles
intersecting each other:
Illustration: FIG. 3.
The significant feature of the diagram is the shaded part which
represents a part of the “men” circle as belonging to a part of the
“wise” circle. The unshaded part of each circle is the unknown field.
(4) _The Particular Negative or O Proposition._
_The common symbolization of the O is “Some S is not P.”_ Put in
statement form: Some of the subject is excluded from the whole of the
predicate. Here, as in the I, the same logical import must be given to
_some_; e. g., in the proposition, “Some men are not wise,” our
knowledge is confined to the group who are not wise. Whether or not the
others are wise or not-wise is unknown.
Illustrations of the O proposition:
1. “Some men are not wise.”
2. “Some laws are not just.”
3. “Some novels are not helpful.”
The significance of the O proposition may be shown by two intersecting
circles as in Fig. 4:
Illustration: FIG. 4.
A similar diagram represents the I proposition, the only difference
being in the _part shaded_. In the O proposition the investigated field
is all of the “men” circle _outside_ of the “wise” circle, while in the
I proposition the known field is that part of the “men” circle _inside_
the “wise” circle.
In comparing the four diagrams the student will note that the
affirmative propositions are _inclusive_, while the negative
propositions are _exclusive_.
(5) _The Distribution of Subject and Predicate._
_A term is said to be distributed when it is referred to as a definite
whole._
In the proposition, “All men are mortal,” the subject _all men_ is
considered as a whole. “_All_ men” stands for every specimen of the
human race; not a single one has been left out. Again, _the whole_ is
definite; any one, if he were given the time and opportunity, could
ascertain by actual count just how many “all men” represented.
It should be observed that if the word _definite_ is not incorporated
in the definition of a distributed term, there is afforded an
opportunity for error. The attending illustrations will make this clear:
1. “All the students except John and James are dismissed.”
2. “All the students except John, James, etc., are dismissed.”
The subject of the first proposition is distributed, while the subject
of the second is undistributed. Reasons: The first subject, “All the
students except John and James,” is referred to as a whole and that
whole is definite, therefore, it is distributed; the second subject,
“All the students except John, James, etc.,” is referred to as a whole,
but as the whole is not definite, the term is not distributed. Because
_all_ is the quantity sign of the second subject the casual observer
might easily be misled in designating it as a distributed term.
Here it may be well to explain that when reference is made to subject
or predicate the _logical_ subject or predicate is meant. Unless this
is constantly kept in mind error results; e. g., in the proposition,
“All white men are Caucasians,” the logical subject is “white men,”
not “men.” If the subject were “men,” it would be undistributed, as
the whole of the man family is not considered, but the actual subject,
being “white men,” _is_ distributed because the predicate refers to
_all_ white men.
Recurring to the illustration, “All men are mortal,” we have concluded
that the subject “all men” is distributed. The predicate, “mortal,”
however, is undistributed, as reference is made to it only in part;
i. e., there are other beings aside from men that are mortal, such as
“trees,” “horses,” “dogs,” etc. _In all A propositions of the type of
“all men are mortal,” the subject is distributed while the predicate
is undistributed._ This relation is clearly shown by the diagrammatical
illustration, Fig. 1. Here _all_ of the “men” circle is identical with
only a part of the “mortal” circle. In other words, the _whole_ of the
“men” circle is considered, while reference is made to only a _part_ of
the “mortal” circle.
_In the case of the co-extensive A both subject and predicate are
distributed._ Relative to the co-extensive “All men are rational
animals,” it could likewise be said that “all rational animals are
men,” or that “all men are all of the rational animals.” Reference is
thus made to _all_ of the definite predicate as well as to all of the
definite subject.
In the E propositions, such as “No men are immortal,” the whole of the
subject is excluded from the whole of the predicate. This makes evident
the fact that _both terms are distributed_. See Fig. 2.
The I proposition, such as “Some men are wise,” concerns itself with
only a part of the subject and only a part of the predicate,
_consequently neither subject nor predicate is distributed_. This
relation is verified by the representation, Fig. 3.
_In the O proposition the subject is undistributed, while the predicate
is distributed._ For example, in the proposition, “Some men are not
wise,” “some men” would indicate that only a part of the logical
subject is under consideration. But the predicate is distributed
because “some men” is denied of the _whole_ of the predicate, “wise.”
This may become clear by studying Fig. 4. Here all of the shaded part
which stands for the subject, “some men,” is excluded from the whole
of the “wise” circle. But all of the shaded part is only a part of the
entire “men” circle, consequently the subject which the shaded part
represents (some men) is undistributed. The predicate, “wise,” however,
is distributed, as the subject is excluded from every part of it. It is
well to remember that _not_, when used with the copula, distributes the
predicate which follows it.
If the student is to succeed in testing the value of arguments, he must
ever have “at the tip of his tongue” his knowledge of the distribution
of the terms of the four logical propositions. With this in view the
following schemes are offered:
I.
_Subject_ _Predicate_
A distributed undistributed
E distributed distributed
I undistributed undistributed
O undistributed distributed
II.
A distributed undistributed
O undistributed distributed
E distributed distributed
I undistributed undistributed
III.
A All S is P
└─┘
E No S is P
└─┘ └─┘
I Some S is P
O Some S is not P
└─┘
Referring to scheme II it may be observed that A and O contradict each
other; i. e., where A is distributed O is undistributed and vice versa.
A similar relation exists between E and I.
In scheme III the bracket └─┘ under the symbol indicates the term which
is distributed.
IV. As a fourth scheme a “key word” might be adopted. Any of these
three might be used: (1) saepeo, or (2) asebinop, or (3) uaesneop.
The significance of “saepeo” is this: “_s_” stands for _subject
distributed_, “_p_” for _predicate distributed_, “_a_” “_e_” “_o_”
for the _logical propositions_ where any distribution occurs. Putting
the letters together gives this: subject distributed of propositions
A and E, predicate distributed of propositions E and O.
Similarly, “asebinop” stands for this: “_as_,” _a_ distributes its
_s_ubject; “_eb_” _e_ distributes _b_oth; “_in_,” _i_ distributes
_n_either; “_op_,” _o_ distributes the _p_redicate.
In the coined word “_uaesneop_” appear six letters which compose
“saepeo,” and the letters have the same significance. The two
additional letters, u and n, stand for universal and negative. The
interpretation of the entire word, therefore, is this: “_uaes_,”
the _u_niversals _a_ and _e_ distribute their _s_ubjects; _neop_,
the _n_egatives _e_ and _o_ distribute their _p_ redicates.
It seems to me that the last word is the most helpful as it emphasizes
the two facts which are the most used; namely, (1) Only the universals
distribute their subjects; (2) Only the negatives distribute their
predicates.
If the student will visualize “_uaesneop_” so thoroughly as never
to forget it, he will not experience difficulty in determining the
distribution of the terms of the four logical propositions.
=9. OUTLINE.=
LOGICAL PROPOSITIONS.
(1) The nature of logical propositions.
(2) Kinds of logical propositions.
Categorical
Hypothetical
Disjunctive
(3) The four elements of a categorical proposition.
(4) Logical and grammatical subject and predicate distinguished.
(5) The four kinds of categorical propositions.
Universal affirmative A
Universal negative E
Particular affirmative I
Particular negative O
(6) Propositions which do not conform to the logical type.
Indefinite and elliptical
Grammatical sentences
Individual
Plurative
Partitive
Exceptive
Exclusive
Inverted
(7) Propositions not necessarily illogical.
Analytic and synthetic
Modal and pure
Truistic
(8) The relation between subject and predicate of the four logical
propositions.
Euler’s diagrams
Distribution of subject and predicate
Uaesneop
Asebinop
Saepeo
=10. SUMMARY.=
(1) A logical proposition is a judgment expressed in words.
(2) The three kinds of logical propositions are categorical,
hypothetical, disjunctive.
A categorical proposition is one in which the assertion is made
unconditionally.
A hypothetical proposition is one in which the assertion depends
upon a condition.
A disjunctive proposition is one which asserts an alternative.
The most common word-signs of the categorical proposition are “all,”
“no,” “some” and “some-not,” of the hypothetical, “if” and of the
disjunctive, “either-or.”
(3) Every logical categorical proposition has the four elements:
quantity sign, subject, copula and predicate.
The quantity sign indicates the extension of the proposition; the
logical subject is that of which something is affirmed or denied;
the logical predicate is the term which is affirmed or denied of the
subject; the copula is always some form of “to be” and is used to
connect subject and predicate. “_Not_” is sometimes used with the
copula.
The statements of ordinary conversation are usually not expressed in
terms of the four elements, but must be, before they can be used in
testing arguments.
(4) One word usually constitutes the grammatical subject while
a word with all its modifiers goes to make up the logical subject.
The verb with any predicate word is the grammatical predicate. The
logical predicate is all which follows the copula――it may include the
predicate-word and all its modifiers as well as the modified object.
(5) Categorical propositions are divided into four kinds; universal
affirmative (A), universal negative (E), particular affirmative (I),
particular negative (O). For the sake of brevity these four are
respectively denoted by the vowels A, E, I, O.
An A proposition is one in which the predicate affirms something of
_all_ of the logical subject.
An E proposition is one in which the predicate denies something of
_all_ of the logical subject.
An I proposition is one in which the predicate affirms something of
a part of the logical subject.
An O proposition is one in which the predicate denies something of
a part of the logical subject.
Every proposition must be reduced to one of the four types before it
can be used as a basis of argumentation.
It is incumbent on the student to recognize these four types with
precision and accuracy.
(6) There are a few proposition types which are recognized as being
illogical in form. These may be defined as follows:
(1) An indefinite proposition is one without the quantity sign. It
usually may be classed as universal.
(2) An elliptical proposition is one in which the copula is
suppressed.
(3) An individual proposition is one which has a singular subject.
It is universal in content.
(4) Plurative propositions are those introduced by “most,” “a few”
or some equivalent quantity sign. These are particular in meaning.
(5) Partitive propositions are particulars which imply a
complementary opposite. These arise through the ambiguous use of
“all-not,” “some” and “few.”
“All-not” sometimes means “_no_,” while at other times it may mean
“_not-all_.” If the quantity sign means the latter, then it introduces
a partitive proposition.
“_Some_” may mean “_some only_,” or “_some at least_.” The latter is
the logical meaning. The former interpretation makes the proposition
partitive. When “few” means “few only,” it is partitive in nature.
(6) Exceptive propositions are those introduced by such signs as
“all except,” “all but,” “all save,” etc. They are universal
only when the exceptions are mentioned.
(7) Exclusive propositions are those introduced by such words as
“only,” “alone,” “none but” and “except.” In an exclusive the
predicate and not the subject is distributed. Consequently
the easiest way to make an exclusive logical is to interchange
subject and predicate and call it an A.
(8) An inverted proposition is one where the predicate precedes the
subject. Interchanging them gives the logical form.
Of the grammatical sentences only the declarative is logical.
The relative clause, though out of place, must be used with the word it
modifies.
(7) There are other propositions, though not illogical, to which the
logician usually gives some attention. These may be defined as follows:
(1) An analytical proposition is one in which the predicate gives
information already implied in the subject.
(2) A synthetic proposition is one in which the predicate gives
information not implied in the subject.
(3) A modal proposition is one which states the manner in which the
predicate belongs to the subject. The adverbs of time, place,
degree and manner are the signs of the modal proposition.
(4) A pure proposition simply states that the predicate belongs or
does not belong to the subject.
(5) A truistic or tautologous proposition is one in which the
predicate repeats the words and meaning of the subject.
(8) In considering the relation which may exist between subject and
predicate, the two terms are employed in extension only, as this use
best serves the interests of inference.
The extensional relation between subject and predicate of the four
logical propositions may be stated as follows:
Ordinary A――All of the subject belongs to a part of the predicate.
Co-extensive A――All of the subject belongs to all of the predicate.
E――None of the subject belongs to any part of the predicate.
I――Some of the subject belongs to some of the predicate.
O――Some of the subject is excluded from the whole of the predicate.
In general it may be said that the affirmative propositions are
_in_clusive while the negatives are _ex_clusive.
A term is said to be distributed when it is referred to as a definite
whole.
“A” distributes the logical subject only, “E” both logical subject and
logical predicate, “I” neither logical subject nor logical predicate,
“O” the logical predicate only. The co-extensive “A” distributes both
subject and predicate.
It is essential that the student know by heart the distribution of
the terms of the logical propositions. Some keyword like _uaesneop_
may be used as an aid to the memory. This means the _u_niversals _A_
and _E_ distribute their _s_ubjects, while the _n_egatives _E_ and _O_
distribute their _p_redicates.
=11. ILLUSTRATIVE EXERCISES.=
(1a) Examine the following list of propositions with a view to
classifying them as “A’s,” “E’s,” “I’s” or “O’s.”
_E_ 1. “None of the inmates voted.”
_A_ 2. “Benj. Franklin was the best educated American.”
_I_ 3. “Some doctors deem it right to lie to their patients.”
_A_ 4. “All earnest teachers need to observe the teaching of others.”
_I_ 5. “Some politicians are honest.”
_A_ 6. “Fools rush in where angels fear to tread.”
_O_ 7. “Some proverbs are not true to life.”
_E_ 8. “No man should infringe upon the rights of others.”
I recall that an affirmative proposition in which the predicate refers
to the whole of the subject is an A, while one where the predicate
refers to only a part of the subject is an I. Further, a negative
proposition where the predicate refers to the whole of the subject is
an E, while one in which the predicate refers to only a part of the
subject is an O. With these facts in mind, I classify the propositions
as indicated.
(1b) In a similar manner classify as to quantity and quality the
following:
(1) “All worthy workers grow to look like their work.”
(2) “Every dog has his day.”
(3) “Some of the presidents were not popular.”
(4) “No unskilled laborer can afford to own an automobile.”
(5) “Some of the ‘election prophets’ were sadly mistaken.”
(2a) Classify the following propositions and make the illogical,
logical:
(1) “Only first-class passengers may ride in parlor cars.”
(2) “Haste makes waste.”
(3) “Few men know how to act under stress.”
(4) “All which seems to ring true is not true.”
(5) “Members alone are admitted.”
(6) “None but men of integrity need apply.”
(7) “Horses trot.”
(8) “Blessed are they which are persecuted for righteousness sake.”
The first proposition is an exclusive and may be made logical by
converting and calling it an A, viz.: “All who ride in parlor cars are
first-class passengers.” (A)
The second is indefinite and elliptical and is made logical by
prefixing the universal quantity sign and expressing in terms of the
four elements. The logical form is, “All who make haste are those who
are wasteful.” (A)
The third is plurative in nature and means, “Most men do not know how
to act under stress.” It would be classed as an O.
The fourth is partitive in nature because of the ambiguous use of
“all――not.” It means, “Some who seem to ring true are not true.” (O)
The fifth is an exclusive. By converting and changing to an A the
proposition takes the logical form, “All who are admitted are members.”
The sixth is likewise an exclusive, the logical form being, “All who
apply must be men of integrity.”
The seventh is an elliptical proposition. Logical form: “All horses are
trotting animals.”
The eighth is an inverted or poetical proposition. It is made logical
by interchanging subject and predicate. Logical form: “Those who are
persecuted for righteousness sake are blessed.”
(2b) Classify the attending propositions and change to the logical form,
if necessary:
(1) “Only truthful men are honest.”
(2) “The stokers alone were saved.”
(3) “All who run do not think.”
(4) “Honesty is the best policy.”
(5) “They laugh that win.”
(6) “The good alone are happy.”
(7) “Knowledge is power.”
(8) “Only the actions of the just smell sweet and blossom in the
dust.”
=12. REVIEW QUESTIONS.=
(1) Define and illustrate logical propositions.
(2) Define and exemplify the three kinds of logical propositions.
(3) What are the usual quantity signs of the four kinds of
propositions?
(4) Name and define the four elements of a logical proposition.
(5) Select from the printed page five propositions which are not
expressed in terms of the four elements, and so express them.
(6) Distinguish between logical and grammatical subject; likewise
between logical and grammatical predicate.
(7) Define and illustrate the four kinds of categorical propositions.
(8) What makes an understanding of the four logical propositions so
important?
(9) Give the unusual quantity signs of the logical propositions.
(10) What should guide one in making an indefinite proposition logical?
(11) How are general truths usually classified?
(12) Change _birds fly_ to the logical form.
(13) How many and what kinds of grammatical sentences are logical?
(14) How would the logician deal with interrogative sentences?
(15) Give illustrations of individual propositions. How are they
usually classified?
(16) Explain the logical mode of dealing with the plurative
proposition.
(17) Exemplify the ambiguity of “all-not,” “some” and “few.”
(18) Why are propositions introduced by “all-not,” “some” and “few”
called partitive?
(19) Use “_all_” in both a partitive and collective sense. Which
signification has logic adopted?
(20) When are exceptive propositions universal and when particular?
(21) What is an exclusive proposition?
(22) Explain by circles the exclusive.
(23) Tell in full how to change an exclusive to logical form.
(24) Tell how the logician would deal with such poetical expressions
as “Blessed are the pure in heart,” “Tell me not in mournful
numbers,” “Strenuous is the man of state.”
(25) What distinction does the logician make between analytic and
synthetic propositions?
(26) Illustrate the difference between the so-called modal and pure
propositions.
(27) Explain and illustrate the truistic proposition.
(28) Show by circles the relation existing between the subject and
predicate of all the logical propositions.
(29) State in good English the relation between the subject and
predicate of all the logical propositions.
(30) Relative to the distribution of terms apply the words “uaesneop”
and “asebinop.” Which one is the more serviceable?
(31) Distinguish between the grammatical and logical subject.
(32) Explain by circles the distribution of the terms of the four
logical propositions.
(33) The statement, “A part of the subject is excluded from the whole
of the predicate,” describes which proposition? Explain how it
indicates that the predicate is distributed.
=13. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=
(1) Show that a judgment may be an individual notion as well as a
general notion.
(2) Many logicians classify logical propositions in this wise:
{ Categorical
Proposition {
{ Conditional { Hypothetical
{ Disjunctive
Give arguments for and against such a classification.
(3) “All men are bipeds” is a judgment of extension, while “Man is
wise” is a judgment of intension. Explain.
(4) “To be logical is to be pedantic.” Discuss this.
(5) Why is the proposition, “He runs,” illogical? Make it logical.
(6) Point out the reasons for calling, “White men are Caucasians,”
a particular proposition.
(7) What makes it necessary to change the propositions of ordinary
conversation to those of the four logical types?
(8) Some would call the individual proposition particular. Argue the
question.
(9) Make a list of five propositions in common speech and show how
their partitive implication may mislead.
(10) Explain by circles _some only_ and _some at least_.
(11) Explain how “_et cetera_” may change a universal to a particular
proposition.
(12) “The real nature of an exclusive is best shown by negating the
subject and calling the proposition an E.” Give arguments for
and against this statement.
(13) Show that with the immature mind all propositions must be
synthetical.
(14) Explain how a proposition may be truistic in form but not in
meaning.
(15) Show by the Euler diagram how easy it is for the careless student
to think that an “O” does not distribute its predicate.
(16) Explain by the use of two pads (a small yellow one and a large
white one) the distribution of terms.
(17) When the logician makes reference to the subject of a
proposition, show that he should exercise care in designating it
as the _logical_ subject.