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:

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 coming along.	(2) The man was one who saw the swell turnout coming along.
(3) All of the men walked.	(3) All of the men were those who walked.
(4) The robbers cut a hole in this floor.	(4) All the robbers were the ones who 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 the attendance.	(6) The rain was that which interfered with the attendance.
(7) Our habits make or unmake us.	(7) All our habits are forces which 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:

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:

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:

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
Categorical Propositions		Universal		Affirmative-A	All S is P
				Negative-E	No S is P
		Particular		Affirmative-I	Some S is P
				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:

Quantity signs of		A—all, every, each, any, whole.
		E—no, none, all-not.
		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 approbation.	All good is the object of moral approbation. (A)
Perfect happiness is impossible.	In all cases perfect happiness 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:

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 power of reason?	The question is asked, Do men have the power of reason?7 (A)
Imperative. “Thou shalt not steal.”	All men are commanded not to steal, or you are one who should not steal. (E)
Optative. “I would I had a million.”	I am one who desires a million dollars. (A)
Exclamatory. “Oh, how you frightened me!”	You are one who frightened me. (A)

(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:

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 graduates may enter Training School.	All who enter Training School must be high school graduates.
2. Only first-class passengers are allowed in parlor cars.	All parlor cars are for first-class passengers.
3. Residents alone are licensed to teach.	All who are licensed to teach are residents.
4. No admittance except on business.	All who have business may be 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 62¹⁄₂ 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.