A Class Room Logic / Deductive and Inductive, with Special Application to the Science and Art of Teaching

by George Hastings McNair

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 theiruniversality 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 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 maketrue 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” introducepartitive 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 asubject 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 thepredicate 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.