2. METHOD OF AGREEMENT.

(1) Principle stated. As stated by Mill the principle of the Method of Agreement is this: “If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstances in which alone all the instances agree is the cause (or effect) of the given phenomenon.”

This notion is given in clearer terms by Jevons and Creighton. Viz.: “The sole invariable antecedent of a phenomenon is probably its cause”; and “The sole invariable consequent of a phenomenon is probably its effect.”

It is known that an antecedent is anything which precedes; while a consequent is anything which follows. To be regarded as a cause, an antecedent must be invariable, and to be regarded as an effect, a consequent must likewise be invariable. Antecedents and consequents which are in no way constant could hardly have any causal connection.

(2) Method symbolized. Let P₁, P₂, P₃, P₄, etc., represent the phenomenon as it may appear the first, second, third, fourth, etc., times, and let A, B, C, etc., stand for the various antecedents, or the various consequents as the case may demand. These two forms may now be used to illustrate the two statements which summarize Agreement:

First statement.

Second statement.

In the first case, the sole invariable antecedent is A, and, therefore, we infer that A is probably the cause of P. In the second case, the invariable consequent being A, is probably the effect of P.

(3) Concrete examples illustrating first statement.

The Problem: Cause of John’s tardiness.

On investigation the various antecedents are these: (1) John has his breakfast at seven; (2) after breakfast he carries his father’s dinner to him and (3) feeds the hens; and then (4) goes to school by the path through the woods and around the mill pond.

Phenomenon as a consequent. John is tardy. Determining to do away with the tardiness, the teacher brings about a variation in the antecedents, varying one at a time taken in the order indicated above.

The teacher is now quite certain that the tardiness is due to the route through the woods and around the pond.

Using, as symbols, the initial letters of the italicized “key-words” of the antecedents as stated above, the case of tardiness may be symbolized as follows:

“w” standing for route through the woods, is seen to be the invariable antecedent.

(4) Concrete example illustrating the second statement.

The Problem: To determine the effect of direct primaries.

First trial.
Antecedent		Consequents
Direct primary		1. Greater expense to candidate,
		2. Greater interest shown,
		3. Better men nominated,
		4. “Bumper” crops.
Second trial.
Direct primary		1. Greater expense to candidate,
		2. Greater interest shown,
		3. Better men nominated,
		4. Crops below average.
Third trial.
Direct primary		1. No greater expense,
		2. Greater interest shown,
		3. Better men nominated,
		4. Crops average.
Fourth trial.
Direct primary		1. No greater expense,
		2. No greater interest,
		3. Better men nominated,
		4. Crops average.

It is seen that the invariable consequent is, “Better men nominated.” We may, therefore, conclude that this is a probable effect of “Direct primaries.”

(5) Distinguishing features of method of agreement. The essential characteristics of the method of agreement are three:

Giving attention to the attending symbolized illustrations it may be noted that “P,” the phenomenon, always happens; while in the case of the first symbolization, “D” is the invariable antecedent and “A, B, C, E, G, L, M, F, I” are the variable antecedents. “K” is the invariable antecedent of the second and “H, I, L, T, M, W, X, Y, Z, S” are the variable antecedents.

(6) A Matter of Observation and Experiment.

On studying the problem relative to the tardiness of John, it appears that in obtaining the various antecedents the work would be largely a matter of observation. Carrying the father’s dinner, the route through the woods, etc., are facts which observation would make evident. However, when it becomes necessary to vary these antecedents with a view to finding the invariable one, the procedure is experimental as well as a matter of casual observation. Moreover, in connection with the direct primary problem the question would be largely a matter of experiment; though observation would obtain as a subsidiary condition. We may conclude from this that the method of agreement involves both observation and experiment; and since the student will discover that the other methods impose similar demands, we are justified in designating these five special methods of induction as those of observation as well as of experiment.

(7) Advantages and Disadvantages of the Method of Agreement.

The concrete cases given to illustrate the method of agreement present a simple combination of antecedents and consequents. In life, however, such simplicity does not usually obtain and in consequence the method of agreement gives rise to a few serious difficulties. These may be summarized as (a) Plurality of causes; (b) Immaterial antecedents; (c) Complexity of phenomena; (d) Uncertainty of conclusion.

(a) Plurality of causes is mentioned by Mill as constituting the “characteristic imperfection” of the method of agreement. As the term signifies, plurality of causes represents a condition where a given phenomenon has more than one cause, or where different causes produce the same effect. For example, “A poor crop” may be due to drought, neglect, pests, etc.; heat may be caused by friction, electricity, combustion. Unfavorable home conditions; ill health; dislike for teacher—any one of these might be followed by irregular attendance.

(b) Immaterial antecedents are those which precede a given phenomenon and yet, under the most favorable situations, have no causal connection with said phenomenon. For example, the various antecedents of the heavy rain may have been a south wind, forgetting to take an umbrella, missing the car and having to walk, etc. Clearly these antecedents, with the exception of the first, are immaterial.

(c) The law of agreement demands that all the material antecedents receive consideration, but often the situation is too complex to make this possible; a fair illustration of such would be an attempt to ascertain all of the antecedents of “the high cost of living.”

(d) The law of agreement never precludes the possibility of error; as it is quite impossible to carry the analysis to the point of absolute certainty. Of all the methods, “agreement” is the least reliable. Despite the foregoing objections, however, the method is of positive value because of its suggestiveness; opening the door to plausible hypotheses it gives the investigators a working basis.

3. METHOD OF DIFFERENCE.

(1) Principle stated.

Says Mill, “If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ is the effect or the cause of an indispensable part of the cause, of the phenomenon.”

To put this in simple terms: Whatever is invariably present when the phenomenon occurs and invariably absent when the phenomenon does not occur, other circumstances remaining the same, is probably the cause or the effect of the phenomenon.

(2) Method symbolized.

Using the same symbols as were used in “Agreement.”

In the first instance A is probably the cause of the phenomenon, since it is present when the phenomenon occurs and absent when it does not occur. For a similar reason, A is the effect in the second case.

(3) Concrete illustrations.

(A) A wise teacher in ascertaining the cause of John’s tardiness would have suggested at once a change of route. Using as symbols the initial letters of the key-words of the antecedents in the case, the following results:

(B) First trial.

Problem: Unprepared home work.

Antecedents		Consequents
1. Length of lesson,		Work not properly prepared.
2. Definiteness of lesson,
3. Amount of interest shown,
4. Physical condition the same.

Second trial.

1. Length of lesson the same,		Work properly prepared.
2. Lesson made more definite,
3. Interest the same,
4. Physical condition the same.

The foregoing symbolized:

L D I C   W

L – I C   —

It is seen that indefiniteness of lesson assignment is the cause of the unprepared home work.

(4) Advantages and disadvantages of the Method of Difference.

The main difficulty attending the use of the method of difference is the complexity of phenomenon. The very nature of the method insists as an essential requirement that only one material antecedent shall be varied at a time. In life the variations are more or less confused, and it is often not only impossible to observe cases of a single variation, but frequently error comes through overlooking antecedents which are material to the case under investigation. For these reasons the Method of Difference is more a method of experiment than it is a method of observation. By controlling the circumstances it becomes possible to vary but one antecedent at a time, and also to bring into prominence all of the material antecedents.

Bacon claims that all “crucial instances” are merely applications of the Method of Difference. By crucial instance he means any fact which will enable us to determine at once which supposition is the correct one. For example, the physician may not know whether it is malaria or typhoid fever till he takes a blood test; such a test typifies “crucial instances.” The various tests in chemistry are likewise cases of crucial instances, and, in consequence, this science makes use of “Difference” more than any other method.

(5) Characteristic features of Method of Difference.

There are three distinguishing marks of the Method of Difference: these are, (1) The phenomenon does not always happen; (2) One antecedent is variable; (3) The other antecedents are more or less invariable.

The following symbolizations will make these three characteristics evident:

	Antecedents	Consequents
(1)	A B C	P
	A – C	–
(2)	– B C	–
	X B C	P
(3)	L M T K	P
	L M – K	–

AGREEMENT AND DIFFERENCE COMPARED.

(a) The methods of Agreement and Difference are complementary as may be discerned by comparing their characteristic features: In Agreement the phenomenon always occurs; in Difference the phenomenon does not always occur: In Agreement there is one invariable antecedent; whereas in Difference there is one variable antecedent: In Agreement the other antecedents are more or less variable; but in Difference the other antecedents are more or less invariable.

(b) According to Mill the Method of Agreement insists that what can be eliminated is not connected; whereas the Method of Difference implies that what cannot be eliminated is connected.

(c) The Method of Agreement is more a method of observation, since it is chiefly concerned with the discovery of causes. The Method of Difference is distinctly a method of experiment, because its usual aim is to discover effects.

(d) The Method of Agreement is so called because the object is to compare several instances to determine in what respect they agree; but in the case of Difference instances are compared to determine in what respects they differ.

(e) The conclusions of the Method of Difference involve greater certainty than those of Agreement and, therefore, the former method should be adopted when there is a choice.

4. THE JOINT METHOD OF AGREEMENT AND DIFFERENCE.

(1) Principle stated.

The uncertainty of the conclusions of Agreement and the impossibility at times of employing directly the Method of Difference, give rise to the use of the combination of Agreement and Difference known as the Joint Method. As stated by Mill, the principle conditioning the Joint Method is this: “If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance, the circumstance in which alone the two sets of instances differ is the effect or the cause or an indispensable part of the cause, of the phenomenon.” More briefly the notion may be stated in this wise: Among many instances, if one circumstance is invariably present when the phenomenon occurs, and invariably absent when the phenomenon does not occur this circumstance is probably the cause or the effect of the phenomenon.

This principle differs from the one underlying the Method of Difference in that the instances considered are more varied and more numerous. The principle of Difference requires but two sets of instances, while the Joint Method demands at least three; two when the phenomenon occurs and one when it does not occur. A study of the symbolizations and illustrations will clarify this distinction.

(2) Joint Method symbolized.

If we use circumstances and phenomenon in place of antecedents and consequent, then one symbolization may be made to stand for ascertaining either the invariable antecedent, or the invariable consequent.

	Circumstances	Phenomenon
1.	A B C D	P₁
2.	A D E F	P₂
3.	A L M N	P₃
4.	A O P Q	P₄
5.	O P Q	—
6.	L M N	—
7.	D E F	—
8.	B C D	—

It is obvious that the first, second, third and fourth groups of instances illustrate the principle of Agreement; whereas the first and eighth, the second and seventh, the third and sixth, and the fourth and fifth illustrate in each case, the principle of Difference.

(3) Concrete Examples illustrating Joint Method.

The problem: Too much whispering.

	Antecedents		Consequent
1.	Insufficient work,		Much whispering.
	Lack of interest,
	Seated near a friend.
2.	More work,		Much whispering.
	Lack of interest,
	Seated near a friend.
3.	More work,		Much whispering.
	More interest,
	Seated near a friend.
4.	More work,		Not much whispering.
	More interest,
	Not seated near friend.
5.	More work,		Not much whispering.
	Lack of interest,
	Not seated near friend.
6.	Insufficient work,		Not much whispering.
	Lack of interest,
	Not seated near friend.

From this it may be concluded that the undue amount of whispering is caused by seating particular friends near each other.

The problem: Poor recitations.

	Antecedents		Consequent
1.	Long lesson,		Poor recitation.
	Faulty assignment of lesson,
	Fear of teacher.
2.	Lesson made shorter,		Poor recitation.
	Faulty assignment,
	Fear of teacher.
3.	Lesson made shorter,		Poor recitation.
	A more careful assignment,
	Fear of teacher.
4.	Lesson made shorter,		Good recitation.
	A more careful assignment,
	Removal of fear of teacher.
5.	Lesson made shorter,		Good recitation.
	Faulty assignment,
	No fear of teacher.
6.	Lesson long,		Good recitation.
	Faulty assignment,
	No fear of teacher.

Fear of teacher is the cause of the poor recitation.

(4) Distinguishing features.

Being a combination of Agreement and Difference the Joint Method possesses the characteristics of each, though more or less modified. The distinguishing marks may be summarized as follows:

Briefly, the one principle concerned is this: There must be an invariable conjunction between the phenomenon involved and the antecedent suspected of being the cause.

(5) Advantages and Disadvantages of the Joint Method.

Since the Joint Method permits a consideration of the negative aspect of the question as well as the affirmative, the opportunities for testing the many instances concerned are doubled. In consequence, the conclusions of the Joint Method are more positive than those of the other methods. It follows that this same opportunity to multiply the instances would tend to lessen the other objections raised against the Method of Agreement; viz., plurality of causes, immaterial antecedents, complexity of phenomenon.

The student must regard the given illustrative symbolizations and concrete examples as being of the simplest form; in life such are the exceptions rather than the rule. When investigating questions, like the cause of the high cost of living, the effect of high tariff, the reason for the typhoid epidemic, etc., there is often a confusion of circumstances which makes the Joint Method unsatisfactory, even though it furnishes a larger opportunity for the multiplication of instances.

The strongest case which the Joint Method is able to present is when the negative instances repeat the positive in every detail, with the one exception of the variable antecedent. To wit:

Strong Argument:
	Circumstances	Phenomenon
	A B C	P₁
	A L M	P₂
	– L M	—
	– B C	—
Weak Argument:
	A B C	P₁
	A L M	P₂
	– R S	—
	– T K	—

Despite the disadvantages, the conditions of the Joint Method are more or less ideal; since the positive branch of the argument suggests the hypothesis, while the negative branch proves the accuracy or inaccuracy of such.

5. METHOD OF CONCOMITANT VARIATIONS.

(1) Principle stated.

Mill’s statement is this: “Whatever phenomenon varies in any manner whenever another phenomenon varies in a particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.”

To put it differently: If when one phenomenon varies alone, another also varies alone, the one is either the cause or the effect of the other.

(2) Concomitant Variations symbolized.

Circumstances	Phenomenon
A	P
A + a	P + p
(A + a) − a	(P + p) − p

It is evident from this that little “a” is the cause or the effect of little “p.” To put it in concrete form:

Let A = X number of calories of heat,

And P = 68° F., the original temperature of room,

And a = candle burning in room for ½ hour,

And p = 2° F.

Then

Antecedents	Consequent
X no. of cal. of heat in room	= 68° F. temp. of room
X no. of cal. of heat + burning candle	= 68° + 2° = 70°
X (no. of cal. of heat + burning candle) − burning candle	= (68° + 2°) − 2° = 68°

As large “A” is increased and decreased by little “a” so large “P” appears to be increased and decreased by little “p.” This strongly suggests a causal connection between little “a” and little “p.”

(3) Other concrete illustrations.

Problem: To ascertain nature of sound.

Antecedent	Consequent
Bell rung when within a glass jar filled with air,	Loud sound.
Some of the air pumped out of the jar,	Sound not so loud.
More air pumped into jar again,	Sound louder again.

The conclusion must be that air has something to do with the production of sound.

Problem: To find best feed for egg production.

100 lbs. beef scraps,		30 doz. eggs.
100 lbs. wheat,
100 lbs. oats,
100 lbs. corn,
50 lbs. beef scraps,		27 doz. eggs.
100 lbs. wheat,
100 lbs. oats,
100 lbs. corn,
90 lbs. beef scraps,		28 doz. eggs.
100 lbs. wheat,
100 lbs. oats,
100 lbs. corn,

Since the variation in the amount of beef scraps is accompanied by a like variation in the number of eggs produced, it may be assumed that beef scraps are essential to large egg production.

(4) Distinguishing features.

(5) Advantages and disadvantages.

Concomitant Variations is applicable in cases when it is impossible to use Difference. Recourse is made to the latter when the phenomenon can be made to appear or disappear at will, but there are times when it is impossible to cause the phenomenon to disappear altogether. For example, in the case of the varying degrees of heat in the room it would be scientifically impossible to take all of the heat out of the room; or in experimenting with gravitation, to do away with its influence entirely, is beyond the power of man. It is thus evident that Concomitant Variations may be used in cases where the conditions forbid doing away entirely with the phenomenon.

The special function of Concomitant Variations seems to be to establish the exact quantitative relation between the varying cause and the varying effect. To illustrate: As a general law it is known that bodies attract each other in varying degrees according to their distances apart and according to their relative sizes; by Concomitant Variations this law has been given definite quantitative value and reads like this: “Bodies attract each other directly as the product of their masses, and inversely as the square of the distance between them.” This illustration suggests that the variation between antecedent and consequent may be direct or inverse.

The error most common in this method is the assumption that the quantitative relation between two varying phenomena will always be according to a constant ratio. For example, when being reduced from a high temperature to 39¹⁄₅° F., water steadily contracts; but at 39¹⁄₅° F. it commences to expand until it becomes ice. Thus the ratio of contraction of water is constant only within certain limits. In any event the established ratio of variation can with absolute safety be applied only to the instances investigated. Another disadvantage incident to this method, is the situation of two elements varying together constantly, and yet having no causal connection whatever.