CHSE Odisha Class 12 Logic Solutions Chapter 6 Experimental Methods of Mill

Odisha State Board CHSE Odisha Class 12 Logic Solutions Chapter 6 Experimental Methods of Mill Questions and Answers.

CHSE Odisha 12th Class Logic Chapter 6 Question Answer Experimental Methods of Mill

Group – A

Short type Questions with Answers
I. Answer with in Two/Three sentence.

Question 1.
What do you mean by experimental methods?
Answer:
(i) The word experiment relates to experience, so experience includes both observation and experiment.
(ii) So the methods are called experimental methods as they are based on experience.

Question 2.
Write the names of different methods of mill.
Answer:
(i) Mill has given 5 experimental methods.
(ii) These are, agreement, difference, joint method of agreement and difference, method of concomitant variation and the method of residues.

Question 3.
State the mills cannon of method of Agreement.
Answer:
Mill’s Canon of method of Agreement:
“If two or more instances of the phenomenon undo- investigation have only one circumstance in common, the circumstance in which alone all the instances agree is the cause of the given phenomenon.”

Question 4.
State the advantages of the method of Agreement.
Answer:
(i) Being a method of observation, it has a wide range and it can applied where experiment is not possible.
(ii) By this method we can discover cause from effect and effect from cause, since it is a method of observation.

Question 5.
Give a symbolic example of the method of residues.
Answer:
ABC are followed by abc.
BC are known to the cause of be
∴ A is the cause of a.

Question 6.
Why are the experimental methods called inductive method?
Answer:

1. The experimental methods are called inductive methods because they are inductive in character.
2. Here the whole process is empirical.
3. Here the major premise is a result of inductive generalisation, the minor premise is obtained by observation and the conclusion is probable.

Question 7.
What is the principle of elimination applied by the method of agreement?
Answer:
The method of agreement is based on the first principle of elimination which states that
“whatever antecedent can be left out without prejudice to the effect can be no part of the cause”.

Question 8.
What methods of mill are not fundamental and why?
Answer:

1. The joint method of agreement, the method of concomitant variations and the method of residues are not fundamental.
2. They are not fundamental because they are the modifications of either the method of agreement or the method of difference.

Question 9.
How can the method of concomitant variations be applied in permanent causes?
Answer:

1. The method of concomitant variations can be applied in the causes of permanent cases of permanent causes.
2. The permanent causes cannot be dropped but they can be increased or decreased.
3. We can apply this method in the cases of heat, friction, gravitation and study their effects on the human body and other bodies.

Question 10.
Is the method of residues essentially a method of deduction?
Answer:
(i) The method of residues is essentially a method of deduction because we follow it to reach at the conclusion.
(ii) Here we adopt the method of substraction.

Question 11.
How can the method of concomitant variations be a graphic method ?
Answer:
(i) The method of concomitant variation is called as graphic method or statistical method because it is based on the quantiative aspect of causation.

(ii) Here the quantitative charges in both the causes and the effect can be represented graphically.

Question 12.
State the characteristics of the method of difference.
Answer:
(i) The method of difference is said to be a method of single difference.
(ii) It is a method of experiement
(iii) It is a method of proof and the result of this method is certain.

Question 13.
State any three advantages of the method of difference.
Answer:
(i) This method requires only two instances in order to find out the cause.
(ii) It conclusively establish a causal relation between cause and effect.
(iii) The result of this method is reliable and dependable.

Question 14.
What are the merits of joint method.
Answer:
(i) It’s scope is very wide.
(ii) It’s conclusion is more certain.
(iii) This method is free from the defects arising out of the plurality of causes.

II. Answer within Five/Six sentence :

Question 1.
State the advantages of the method of difference.
Answer:
(i) As this method requires only two instances, so the trouble of collecting a large number of instances in avoided.
(ii) Under ideal conditions it can establish a casual relation conclusively.
(iii) The result of this method is reliable when based on experiment.

Question 2.
State three disadvantages of the joint method.
Answer:
(i) The joint method cannot distinguish Cause from condition.
(ii) It cannot distinguish causation from coexistence.
(iii) It is inapplicable in cases of permanent causes.

Question 3.
State the advantages of the method of concomitant variation.
Answer:
(i) This method is applicable in case of permanent causes.
(ii) As this method is a modification of the method of difference, so it called a method of proof.
(iii) It is useful in cases of intermixture of effect.

Question 4.
Why the method-of Residues is called as a method of discovery?
Answer:
The method of Residues is said to be method of discovery rather than method of proof, because many important disooveries have been made by this method. The planet Neptune and the gas Argon were discovered by this method. As, the result of this method in hypothetical and does not end with the deductive work of substructure, so mellone called if as finger post to fee unexplained.

Question 5.
Which experimental method of , mill is called method of double Agreement?
Answer:
Joint method of mill is called the method of double Agreement, because it has an agreement in the presence in the positive set and also a agreement in the absence in the negative set. So fowler calls it, the double method of Agreement and Bain calls it the method of double Agreement.

Question 6.
Why the method of Agreement is said to be the method of single agreement? Explain.
Answer:
The method of Agreement is said to be the method of single agreement, because in third method different instances agree on a single point. But if the instances agree on a number of circumstances it will be difficulty by this method to ascertain which one of them in the cause. The singleness of the agreement generates our belief in the causal connection. This method in called “the method of single agreement”.

Group-B

Long Type Questions With Answers

Question 1.
State and Explain Mill’s method Difference.
Answer:
Mill’s Canon : “If an instance in which the Phenomenon under investigation occurs and an instance in which it does not occur have every circumstance in which alone the two instance differ in the effect, or the cause or an indispensable part of the cause of the phenomenon”.

Concrete examples :
(i) In a lighted rooom the furniture are visible at night. When the light is switched off, visibility is lost. So we conclude that light the cause of human visibility.

(ii) Some one sips coffee and finds that it does not taste well. Then he adds a bit of sugar and it becomes tasteful. So sugar is an indispensable part of the cause of the taste of coffee. Principle of elimination : The method of difference is based on the principle. “Which an antecedent can not be left out without the consequent disappearing, such an antecedent must be the cause or a part of the cause”.

Characteristics :
(1) The method of difference is said to be the method of single difference : In this method on comparison of two instance, we find that they differ only in one respect. It is the singleness of the difference that constitutes the ground of proof. Hence Mellone and coffey call this method the method of single difference.

(2) The method of difference is mainly a method of experiment : The essential requirement of this method is the two instances are special type. They must be exactly alike except that in one, the phenomenon under investigation is present and in other it is absent. It can be found only when the situation is under our control. Unless we examine, how can we know that our thirst can be quenced after drinking of water. So this method is truly called a method of experiment.

(3) The method of difference is said to be a method of proof in cases of experimental observation: The method of agreement does not prove the causal connection, but merely suggests something to be the cause. But the method of difference can prove the causal connection by experiment. Since this method faces many difficulties, it can not completely establish the causal connection.

Advantages :

1. The method of difference requires only two instance, though of a special kind. So the trouble of collecting a large number of instance is a avoided.
2. It can concresively establish a causal relation, under ideal conditions.
3. When applied experimentally the result obtained by this method is reliable.

Disadvantages :
(1) The method of difference can not eradicate the difficult arising out of the plurality of causes : According to Mellone, it proves a cause but not the cause. In a particular case, this method can conclusively prove that x is the cause of y, but it does not prove that x is the only cause of y.

(2) In the method of difference, we are liable to commit the fallacy of post hoc er go proper hoc : This fallacy literally means, after this therefore due to this. When this method is wrongly used in out daily lige. We commit this fallacy. Anew bridge is the cause of my father’s death then my argument is certainly fallacious.

(3) This method falls to distinguish between cause and condition : Sometimes this method takes one condition to be cause. But this view is wrong. For example : Addition of bit of sugar brings about the right taste of coffee, but sugar alone is not the cause of the taste.

(4) The method of difference can not proceed from effect to cause : By experiment we can proceed from cause to effect, but not from effect to cause. So as a method of experiment, this method cannot proceed from effect to cause.

(5) The method of difference is inapplicable in the case of permanent causes : Because the permanent causes like friction, gravitation on the surface of the Earth etc. cannot totally eliminated. So this method is not applicable in such cashes.

(6) The scope of the method of difference is limited : As the method of difference is based on the method of experiment and experiment is not possible in all cases, that is why it’s scope is limited.

(7) The method of difference can not be applied where the phenomenon is complex and beyond our control.

Question 2.
State and explain the joint method of Agreement and difference.
Answer:
Mill’s Canon : “If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instance 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.

Symbolic example :

• Positive Set :
  1. ABC are followed by abc.
  2. ACD are followed by acd.
  3. ADC are followed by ade.
• Negative Set :
  1. BCD are followed by bed.
  2. DEC are followed by dec.
  3. EFD are followed by efd.

From these two sects of instances we inter that ‘A’ is the cause of a. Concrete examples :
(i) We collect the instances and find that where there is malaria fever, there are mosquitoes and where there is no malaria fever, there is no mosquitoes. So mosquito is the cause if malaria fever.

(ii) The students who are sincere in their study succeed in their examination and who are insincere in their study do not succeed in the examination. So sincere study is cause of success in the examination.

Principle of Elimination : The joint method is a modification of the method of a agreement and the method of difference. So the principles of elimination of both these methods are employed here.

1. Whatever antecedent can be left out without prejudice to the effect can be no part of the cause.
2. When an antecedent can not be left out without consequent disappearing, such antecedent must be the cause or part of the cause.

Characteristics :
(1) The joint method is a method of double agreement: This method is not a primary method, but it is considered to be a double application of the method of agreement and method of difference. It has an agreement in the presence in the positive set and also an agreement in the absence in the negative set. So fowler calls it, the double method of  agreement. Mill calls this method, “The indirect method of difference, “Because the negative instances are obtained noi by experiment but indirectly by showing what would be the result it experiment could be made.

(2) According to venn, this method is called as the method of exclusion because in this method all the circumstances are eliminated except one.

(3) This method is said to be a method of proof, because the causal connection suggested by the positive instances is confirmed by the negative instances.

Advantages :

1. The joint method being a method of observation has a wide range of application. The scope of this method is wider because observation is universally applicable. Besides it helps discover by suggesting a hypothesis about cause.
2. This method is free from the difficulties arising out of the plurality of causes. This method takes that the common antecedent and common consequent are causally related. If the negative instances contain all the circumstances other that what is uniformaly present in the positive set, there is no change of plurality of causes.
3. The joint method is applicable in the field of experiment.
4. This method can confirm the casual connection.

Disadvantages :

1. The joint method fails to distinguish between cause and co-existence. Lightning and thunder are co-effects but they may be supposed to be cause and effect by this method.
2. It is inapplicable in cases of permanent causes, such as heat, gravitation etc. Because in the these cases we can not find a negative set.
3. It cannot distinguish between cause and condition. If may be shown that salt is uniformly present in several dishes of palatable curries but the same dishes become unpalatable when no salt is given. By this method we are liable to conclude that salt alone is responsible for palatableness of the curries.
4. This method requires that different antecedents and consequent must be separately distinguishable. So it can not deal with the intermixture of effects where the separate effects are indistinguishable.

Question 3.
State and explain the method of residues.
Answer:
Mill’s Cannon : “Subduct from any given Phenomenon such part as is known by previous inductions to the effect of certain antecedents and the residue of the phenomenon is the effect of the remaining antecedents.”

Symbolic example :
ABC ………… abc
BC …………… be
∴ A is the cause of a.

Concrete example :
(1) The shopkeeper weights the empty tin and then filled up the tin with oil, then weights it. Now by subtracting the weight of the empty tin from the total weight he determines the weight of the oil.

(2) You purchased a shirt, a banyon and a pant, you paid Rs 70. You know the price of the shirt is Rs 25 and banyan is Rs 5. So the price of the.pant is Rs 40.

The Principle of Elimination :
This method is based on the principle that what is the cause of one thing cannot be the cause of a different thing. When we are dealing with a complex set a phenomena and we already know the cause of same of them, we conclude that the cause of the reminder or residual phenomenon is to be found among the remaining antecedents.

• Characteristics :
  1. It is a method of discover rather than proof.
  2. It is a special modification of method of difference.
  3. It is essentially method of deduction. We calculated the effect of known causes and substract this calculated effect from the total effect.
• Advantages :
  1. This method is applicable of complex phenomena.
  2. It is capable of proving casual relation where the data are collected by experiment.
• Disadvantages :
  1. The method of residues is inapplicable in cases where previous knowledge of causal relation is a lacking.
  2. The method of residues is in applicable when the complex effect is unanalysed.

Question 4.
Di$cuss critically mill’s method of Agreement.
Answer:
Mill’s common : If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree is the cause (or effect) of the given phenomenon.

Symbolic Example :
ABC are followed by abc
ADE are followed by ade
AFG are followed by afg.
∴ A is the cause of a or a is the effect of A.

Concrete example : Female anopheles mosquito biting is the common antecedent of malaria fever. So female anopheles mosquito biting is the cause of malaria.

Principle of Elimination : The method of Agreement is based on the principle ‘Whatever antecedent can be left out without prejudice to the effect can be no part of the cause”.

Characteristics :
(1) The method of Agreement is said to be the method of single agreement:
Agreement on a single point or factor is important in this method. If the instance agree on a number of points, the real cause among them will be difficult to be found. Because we can not be sure which are among these points of agreement is the cause of the event.

(2) It is a method of observation : By observation we find instances for this method. Because it is based on observation, it has a wider range of application, there are certain areas where we can not experiment. For example, the planetory movement. In such cases, observation is the only way. This method of agreement has some advantages and some disadvantages. Let us discuss them.

Advantages:

1. The scope of the method is wide. This method is based on observation and observation has a wider range of application than experiment. So this method is applicable to cases where experiment is not possible.
2. In observation, we can go from cause to effect and also from effect to cause. So, as this method is based on observation, we get this advantage in the method. We can discover the cause and also the effect of an event.

Disadvantages :
1. This method fails to eradicate the difficulties arising out of pluraling of causes. Salt is the common element in all tasty curries. The method of agreement may declare salt as the cause of taste which is not the case.

2. There is the possibility of committing the fallacy of non-observation in this method. As this method collects its instances from observation, there always remains a possibility that we have overlooked certain circumstances. Those hidden factors may be responsible for the creation of effect of course, this difficulty can be removed to some extent by careful observation and manipulation of instances.

3. The method of Agreement fails to distinguish between cause and condition. We may mistake a condition to be the cause. The common antecedent may be a part of the cause but not the total cause!

4. It can not deal with intermixture of effects. There may be a case where the effect is a joint one from several causes. In atherwords, several causes together have produced an intermixture of effects. In such case, it is almost impossible to point out which effect is of which cause.

5. The result of this method is probable : This method depends only on the observation. There is always a possibility that we have not observed all the relevant facts. The probability only increases with more and more number of cases. Inspite of some disadvantages the method of agreement of mill is regarded as very important method.

Question 5.
What is the method of concomitant variation. Discuss critically.
Answer:
Mill’s Common : “Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner is either a cause or an effect of that phenomenon, or is connected with it through same fact of causation”.

Symbolic Example :
A₁BC followed by a,be.
A₂BC followed by a₂bc.
A₃BC followed by a₃bc.
∴ A is the cause of a.

Concrete Example:
When the temperature rises, the mercury in the thermometer expands. So temperature is the cause of the expansion of mercury.

Principle of Elimination:
The method is based on the principle – “An antecedent and a consequent rising and falling together in numerical concomitance are to be held as cause and effect”.

Characteristics: The characteristics of the method given by J.S. Mill are

1. This method of comitant variation is the modification of either the method of agreement or the method of difference. It is not a primary method.
2. It is based on the quantitative aspect of causation. It can be represented graphically. So, this method is also called graphic method or statistical method.
3. It is a method of discovery. When two phenomenon change simultaneously in a particular pattern, a casual connection is suspected.
4. This method of concomitant variation has some advantages and also some disadvantages. All of them are discussed below.

Advantages :

1. It is a method of proof. Because this method is a modification of the method of difference.
2. This method is useful in case of permanent cause. Permanent causes like that, friction, etc. can not be totally eliminated. This method does not need total elimination of any factor.
3. It is useful in case of intermixture of effects. Even when effects are blended together, variation in antecedent indicate a variation consequent and thereby a casual connection is established.

Disadvantages :

1. This method is applicable to a certain limit. Exercise increase physical strength. But excess of it deterioates the strength.
2. The method is not applicable in case of qualitative change. Only in case of variation in quantity this method is applicable.
3. It fails to distinguish between cause and condition. Two things may changer together But that does not mean that they are cause and effect. One may be a condition.
4. It is not free from the difficulties arising out of plurality of causes when it is a method of observation.
5. This is the method of concomitant variation advanced by J.S. Mill to find the cause of an event.

Question 6.
Elaborate the advantages and the dis-advantages of the joint method of agreement and difference.
Answer:
The joint method of agreement and difference is a logical approach employed in the process of reasoning and argumentation. This method, developed by the philosopher John Stuart Mill, is a part of Mill’s broader system of inductive reasoning known as the Methods of Experimental Inquiry. The joint method involves identifying the common factors (agreements) and differences in a set of instances to arrive at more reliable conclusions. While the joint method has advantages in certain contexts, it also has its limitations and potential drawbacks.

Advantages of the Joint Method of Agreement and Difference :

1. Increased Reliability : The joint method combines the strengths of both the method of agreement and the method of difference. By considering both agreements and differences among instances, the reasoning process becomes more comprehensive and less susceptible to errors.

2. Robust Inductive Reasoning : The joint method enhances the robustness of inductive reasoning by addressing multiple aspects of the observed phenomena. It allows for a more thorough analysis of patterns, making the conclusions drawn from the data more reliable and well-founded.

3. Effective in Identifying Causal Factors : When used to analyze causal relationships, the joint method is effective in identifying factors that are consistently present (agreements) or consistently absent (differences) when a phenomenon occurs or does not occur. This can contribute to a better understanding of causation.

4. Versatility in Application : The joint method is versatile and applicable across various disciplines. Whether in scientific research, social sciences, or other fields, it provides a structured approach to analyze patterns and draw meaningful conclusions from observed data.

5. Guided Hypothesis Formation : The joint method guides the process of hypothesis formation by focusing on factors that consistently accompany or differ in instances under investigation. This can streamline the formulation of hypotheses that can be tested and refined through further empirical research.

6. Facilitates Comparative Analysis : Comparative analysis is inherent in the joint method, as it involves comparing instances to identify both commonalities and differences. This facilitates a nuanced understanding of the subject matter, contributing to more accurate generalizations.

Disadvantages of the Joint Method of Agreement and Difference :
1. Complexity and Resource Intensiveness : The joint method may be complex and resource-intensive, particularly when dealing with a large number of instances. Analyzing agreements and differences in a comprehensive manner requires careful consideration and detailed examination, which can be time-consuming.

2. Dependency on Quality of Data : The effectiveness of the joint method depends on the quality and reliability of the data used for analysis. If the data are flawed, incomplete, or biased, the conclusions drawn from the joint method may also be compromised.

3. Assumption of Relevance : The joint method assumes that the factors being considered (agreements and differences) are relevant to the observed phenomena. If important factors are overlooked or if irrelevant factors are included, the conclusions may be inaccurate or incomplete.

4. Limited Applicability in Some Contexts : While versatile, the joint method may not be equally applicable to all types of research questions or phenomena. In some cases, alternative methods or approaches may be more suitable for drawing meaningful conclusions.

5. Difficulty in Identifying Unobservable Factors : The joint method may face challenges in situations where certain factors are unobservable or difficult to measure. If critical factors are not considered due to their invisibility or complexity, the conclusions may be incomplete.

6. Potential for Overfitting : Overfitting occurs when an analysis is tailored too closely to the specific data set at hand, leading to conclusions that do not generalize well to new data. The joint method, if not carefully applied, may be susceptible to overfitting.

7. Limited Insight into Dynamic-Processes : The joint method may not be as effective in capturing dynamic or evolving processes where causal factors change over time. In such cases, alternative methods that account for temporal dynamics might be more appropriate.

Conclusion:
The joint method of agreement and difference offers a valuable approach to inductive reasoning by considering both commonalities and variations among instances. Its advantages lie in the increased reliability of conclusions, the robustness of inductive reasoning, and its effectiveness in identifying causal factors.

However, the method has its limitations, including potential complexity, dependency on data quality, and challenges in identifying unobservable factors. Researchers and analysts should carefully consider the context and nature of the phenomenon under investigation when deciding whether to apply the joint method or explore alternative methods that may better suit their specific research questions.

Question 7.
What is the special feature of the method of residues?
Answer:
The method of residues, also known as the method of differences, is a mathematical technique used for solving problems in various fifelds, including calculus, number theory, and physics. This method, attributed to the French mathematician Francis Viete, is characterized by its special feature of isolating and manipulating the “residue” or remaining part of a given quantity after subtracting known components. Understanding the special features of the method of residues requires exploring its principles, applications, and advantages.

Principles of the Method of Residues :
1. Isolation of the Residue : The primary objective is to isolate and identify the residue in the given quantity. This is achieved by subtracting known components from the total quantity, leaving behind the residue. The residue represents the part of the quantity that is not accounted for by the known components.

2. Manipulation of Residue : Once the residue is isolated, mathematical operations are performed on it to obtain the desired result or solution. This manipulation often involves simplifying the residue or transforming it into a more manageable form.

Special Features of the Method of Residues :
1. Focused on the Remaining Quantity : The distinctive feature of the method of residues is its focus on the remaining or residual part of a quantity after subtracting known components. This allows mathematicians to direct their attention specifically to the part that requires further analysis or computation.

2. Utilization in Various Mathematical Disciplines : The method of residues finds applications in diverse mathematical disciplines, including calculus, complex analysis, number theory, and combinatorics. Its adaptability and versatility contribute to its widespread use in solving different types of problems.

3. Integration with Complex Analysis : In complex analysis, the method of residues is closely linked to the theory , of residues, which involves evaluating complex integrals using the residues of functions. The residues are residues of meromorphic functions, and their properties contribute to the evaluation of integrals.

4. Application in Combinatorics : The method of residues is employed in combinatorics to count certain types of objects or arrangements. The residue is often associated with a counting function, and its manipulation leads to the determination of combinatorial quantities.

5. Solution of Linear Recurrences : In number theory, the method of residues is utilized to solve linear recurrence relations, providing a systematic approach to finding closed-form expressions for sequences defined by recurrence relations.

6. Inverse Process of Synthesis : The method of residues can be viewed as an inverse process of synthesis, where a complex quantity is broken down into simpler components. This inverse approach is particularly valuable in solving problems where direct synthesis is challenging

7. Advantageous for Simplification : The method of residues is advantageous for simplifying complex mathematical expressions. By isolating the residue, mathematicians can focus on specific terms or components, making it easier to analyze and manipulate the expression.

8. Application in Indeterminate Analysis : The method of residues is applied in indeterminate analysis, where it aids in finding limits and evaluating indeterminate forms. The residue plays a crucial role in expressing the limit of a function as the residue of a corresponding function.

Applications and Examples:
1. Calculus and Integration: In complex analysis, the method of residues is applied to calculate integrals by identifying residues at singular points. This is particularly useful for integrals involving rational functions or functions with poles.

2. Combinatorial Enumeration: In combinatorics, the method of residues is used to count the number of objects satisfying certain conditions. The residue is often associated with a generating function, and its manipulation yields information about the combinatorial structure.

3. Linear Recurrence Relations: In number theory, the method of residues is employed to solve linear recurrence relations. By transforming the recurrence relation into a generating function, the residue can be used to find a closed-form expression for the sequence.

4. Evaluation of Indeterminate Forms: The method of residues is applied to evaluate indeterminate forms, such as limits involving 0/0 or ∞/∞. By isolating the residue, mathematicians can simplify the expression and determine the limit.

Advantages of the Method of Residues :
1. Versatility and Applicability : The method of residues is versatile and applicable to various mathematical disciplines, providing a unified approach to solving problems in calculus, complex analysis, number theory, and combinatorics.

2. Focus on Essential Components : By isolating the residue, the method of residues allows mathematicians to focus on the essential, remaining part of a quantity. This simplifies the analysis and facilitates the solution of complex problems.

3. Inverse Synthesis Approach : The method of residues offers an inverse approach to synthesis, breaking down complex quantities into simpler components. This can be advantageous in situations where the direct synthesis is challenging or impractical.

4. Integration with Complex Analysis : In complex analysis, the method of residues is a powerful tool for evaluating integrals involving meromorphic functions. Its integration with complex analysis enhances its effectiveness in solving problems in this domain.

Limitations of the Method of Residues :
1. Dependency on Quality of Data : The effectiveness of the method of residues depends on the quality, and accuracy of the data or functions involved. If the initial expressions are poorly defined or contain errors, the conclusions drawn from the method may be
compromised.

2. Complexity and Resource Intensiveness : The method of residues can be complex and resource-intensive, especially when dealing with intricate mathematical expressions. The manipulation of residues may involve intricate algebraic steps, requiring careful analysis

3. Applicability to Specific Problem Types : While versatile, the method of residues may not be equally applicable to all types of mathematical problems. Certain problems may have more efficient or suitable methods for solution, and reliance solely on the method of residues may not be optimal.

Conclusion :
The method of residues, characterized by its focus on isolating and manipulating the remaining part of a quantity, is a valuable mathematical tool with wide-ranging applications. Its principles, versatility, and integration with complex analysis contribute to its effectiveness in solving problems in calculus, number theory, and combinatorics.

However, like any method, the approach has its limitations, including dependence on data quality and potential complexity. Understanding the special features, advantages, and limitations of the method of residues is essential for mathematicians and researchers seeking to apply this method judiciously in various mathematical contexts.

Group -C

Objective type Questions with Answers
I. Multiple Choice Questions with Answers :

Question 1.
How many type principles of elimination are there in Mill’s experimental method?
(i) One
(ii) Two
(iii) Three
(iv) Four
Answer:
(iii) Three

Question 2.
Who has given five experimental methods in inductive logic?
(i) Mill
(ii) Jevons
(iii) Bain
(iv) Carveth Read
Answer:
(i) Mill

Question 3.
Which method is called as graphic or statistical method?
(i) The method of concomitant variations
(ii) The joint method of Agreement and Difference
(iii) The method of Difference
(iv) The method of Agreement
Answer:
(i) The method of concomitant variations

Question 4.
Which method is said to be a method of Deduction?
(i) The method of Agreement
(ii) The method of Difference
(iii) The method of concommitant variation
(iv) The method of Residues
Answer:
(iv) The method of Residues

Question 5.
Which method of is said to be method of proof?
(i) The method of Agreement.
(ii) The method of Difference, Joint method and the method of concomitant variations.
(iii) The method of Residues
(iv) The method of concomitant variations.
Answer:
(ii) The method of Difference, Joint method and the method of concomitant variations.

Question 6.
Which method is called the method of Discovery?
(i) The method of Difference
(ii) The method of Residues
(iii) The method of Agreement and the method of concommitant variations.
(iv) The joint method of Agreement and Difference.
Answer:
(iii) The method of Agreement and the method of concommitant variations.

Question 7.
Which method is called as the method of experiment?
(i) The method of Difference
(ii) The method of Agreement.
(iii) The joint method of Agreement and Difference.
(iv) The method of concommitant variations
Answer:
(i) The method of Difference

Question 8.
In which method we are liable to commit the fallacy of post hoc ergo propter hoc?
(i) The method of Agreement
(ii) The joint method of Agreement and difference
(iii) The method of Difference
(iv) The method of Residues
Answer:
(iii) The method of Difference

Question 9.
Which method is called as a method of exclusion?
(i) Method of Agreement
(ii) Method of difference
(iii) Joint Method of Agreement and difference.
(iv) Method of Residues.
Answer:
(iii) Joint Method of Agreement and difference.

Question 10.
Mill has puforth how many kinds of explanation?
(i) Three types
(ii) Four types
(iii) Five types
(iv) Six types
Answer:
(i) Three types

II. Fill in the blanks :

Question 1.
The method of Agreement in a method of ______.
Answer:
Discovery

Question 2.
The method of ______ is applicable in case of permanent causes.
Answer:
Concomittant variation

Question 3.
The method of ______ cannot distinguish between cause and condition.
Answer:
Agreement

Question 4.
In joint method we are unable to distinguish causation from ______.
Answer:
Co-existence.

Question 5.
The method of concern ittant variation is called ______ method.
Answer:
Graphic or statistical

Question 6.
There are ______ experimental methods in Logic.
Answer:
Five

Question 7.
There are ______ principles of elimination.
Answer:
Three

Question 8.
The experimental methods are called the methods of elimination because ______.
Answer:
By applying these methods the accidental circumstances are discarded and the essential circumstances are determined.

Question 9.
The first principle of elimination says ______.
Answer:
An absent circumstances cannot be the cause.

Question 10.
The method of Agreement is based on ______ principle.
Answer:
The first

Question 11.
The method of difference is based on the ______ principle.
Answer:
Second

Question 12.
The method of concomitant variation is based on the ______ principle.
Answer:
Third

Question 13.
The joint method is modification of ______.
Answer:
The method of Agreement and the method of difference.

Question 14.
The method of Residues is modification of ______.
Answer:
The method of difference

Question 15.
The method of agreement is essentially a method of ______.
Answer:
observation

Question 16.
The method of Agreement is used in. those cases where ______.
Answer:
The phenomenon is beyond our control

Question 17.
The method of Agreement suffers from the defects of ______.
Answer:
Plurality of causes and non-observation

Question 18.
The method of difference is pre-eminently a method ______.
Answer:
Experiment

Question 19.
When the method of difference is applied in the sphere of simple observation if leads to the fallacy of ______.
Answer:
Post hoc propter hoc

Question 20.
______ method is called a graphic or statistical method.
Answer:
Method of concomitant variation

Question 21.
Method of ______ is said to be a method of deduction.
Answer:
Residues

Question 22.
______ method is called a method of double agreement.
Answer:
Joint method

Question 23.
______ calls the method of difference as the method of single difference.
Answer:
Mellone

Question 24.
Venn calls the Joint method as the ______.
Answer:
method of exclusion

Question 25.
The method of ______ is based on the quantitative aspect of causation.
Answer:
Agreement

Question 26.
The method of concomitant variation is useful’in cases of ______.
Answer:
Intermixture of effects

Question 27.
The method of concomitant variation is not applicable in cases of ______ .
Answer:
qualitative variation

Question 28.
The method of concomitant variation does not enable us to distinguish causation from ______
Answer:
Co-existence

Question 29.
According to Mellone, Method of Residues is called ______.
Answer:
A finger post to the unexplained

Question 30.
The method of Residues is useful in cases of ______.
Answer:
Complex effect.

III. Correct the Sentences :

Question 1.
The method of Agreement is said to be the method of single difference.
Answer:
The method of Agreement is said to be the method of single agreement.

Question 2.
The method of Agreement is said to be mainly method of Experiment.
Answer:
The method of Agreement is said to be^nainly a method of observation.

Question 3.
The method of Agreement is said to be a method of proof rather than a method of Discovery.
Answer:
The method of Agreement is said to be a fttetfrod of discovery rather than a method of proof.

Question 4.
In the method of Agreement we are liable to commit the fallacy of Petitio Principii.
Answer:
In the method of Agreement we are liable to commit the fallacy of Non-observation.

Question 5.
The result of the method of agreement is certain.
Answer:
The result of the method of agreement is probable.

Question 6.
The method of difference is said to be the method of single Agreement.
Answer:
The method of difference is said to be the method of single difference.

Question 7.
The method of difference is mainly a method of observation.
Answer:
The method of difference is mainly a method of experiment.

IV. Answer the following questions in one word :

Question 1.
Which method is said to be mainly a method of observation?
Answer:
Agreement

Question 2.
The method of Agreement is said to be a method of discovery rather than a method of what?
Answer:
Proof

Question 3.
The result of tae method of Agreement is only what?
Answer:
Probable

Question 4.
Which method is said to be the method of single difference?
Answer:
Difference

Question 5.
The method of difference is mainly a method of what?
Answer:
Experiment

Question 6.
In which method we are liable to commit the fallacy of “post hocergo propter hoc”?
Answer:
Difference

Question 7.
Which method is not useful in cases of intermixture of effects?
Answer:
Joint method

Question 8.
Which method is called Graphic or statistical method?
Answer:
Concomitant – Variation

Question 9.
Which method is said to be essentially a method of deduction?
Answer:
Residues

Question 10.
Which method is called “a finger post to the unexplained”?
Answer:
Residues

Question 11.
Which method is called the method of Inclusion?
Answer:
Joint method