Welcome to HSLC Guru. In this lesson we present complete question and answer notes for Class 11 Logic and Philosophy Chapter 7 — Inference, prepared strictly for the ASSEB (Assam State School Education Board) English-medium syllabus. Inference is the heart of logic. Every argument we construct, every proof in geometry, every scientific generalisation and every courtroom verdict relies on the act of inferring a new judgement from one or more judgements already accepted. This chapter teaches you what inference is, how it differs from related mental processes such as implication, the major divisions of inference (deductive and inductive, immediate and mediate), and the rules that govern the four kinds of immediate inference — conversion, obversion, contraposition and inversion. Worked examples are given for each operation so that you can confidently solve any objective or descriptive question that appears in the final examination.
Chapter Summary
Inference is that mental process by which the mind passes from one or more known propositions, called premises, to another proposition, called the conclusion, which is logically connected with them. Symbolically, “P, therefore Q” represents the form of every inference. Inference is a verbal expression of reasoning; reasoning is the mental process, while inference is its linguistic form.
Logicians distinguish inference from implication. Implication is an objective relation that holds between propositions independently of any thinker — if proposition P is true, then Q must also be true, whether or not anyone notices the connection. Inference, on the other hand, is a subjective mental act in which a thinker actually moves from premises to a conclusion. Implication is the ground of inference; without implication, inference would not be valid.
Inference is divided primarily into two types: deductive inference and inductive inference. In a deductive inference the conclusion is never wider in meaning than the premises; the movement is from the general to the particular or from the equally general to the equally general. In an inductive inference the conclusion goes beyond the premises and is more general than them; the movement is from the particular to the general, from the observed to the unobserved.
Deductive inference is further classified as immediate and mediate. Immediate inference draws a conclusion from a single premise without the help of any middle term. Mediate inference, also called syllogism, draws a conclusion from two premises through the help of a middle term. Mediate inference is again divided into categorical, hypothetical and disjunctive syllogism.
Immediate inference is of four important kinds: conversion, obversion, contraposition and inversion. In conversion the subject and predicate of the original proposition (the convertend) are interchanged to obtain a new proposition (the converse), keeping the quality unchanged. The rules of conversion give the following pattern: A converts to I (per accidens or by limitation), E converts to E (simple), I converts to I (simple), and O cannot be converted at all.
In obversion the quality of the proposition is changed and the predicate is replaced by its contradictory, while the subject and the truth-value remain the same. The original is the obvertend and the new proposition is the obverse. Every categorical proposition admits of obversion: A becomes E, E becomes A, I becomes O, and O becomes I.
Contraposition is a combined operation: first obvert the original, then convert the obverse. The original is the contraponend and the new proposition is the contrapositive. A gives a valid contrapositive of the form E, E gives only a limited contrapositive of the form O, I has no valid contrapositive, and O gives a contrapositive of the form I.
Inversion is a more complex immediate inference in which both the subject and the predicate of the original are replaced by their contradictories. Only A and E propositions admit of inversion; I and O do not. By inversion, “All S is P” yields “Some not-S is not P”, and “No S is P” yields “Some not-S is P”.
A clear understanding of these operations is essential because they form the logical basis of mediate inference and of all later work in syllogistic and symbolic logic.
Textbook Questions and Answers
Very Short Answer Type Questions
1. What is inference?
Answer: Inference is the mental process by which we pass from one or more known propositions, called premises, to another proposition, called the conclusion, that is logically connected with the premises.
2. Define deductive inference.
Answer: Deductive inference is that inference in which the conclusion is never wider than the premises; the movement is from the general to the particular or from a known judgement to another judgement of equal generality.
3. Define inductive inference.
Answer: Inductive inference is that inference in which the conclusion is wider than the premises; the movement is from the particular to the general, from observed cases to unobserved cases.
4. What is immediate inference?
Answer: Immediate inference is a kind of deductive inference in which the conclusion is drawn directly from a single premise, without the help of any middle term.
5. What is mediate inference?
Answer: Mediate inference, also called syllogism, is a kind of deductive inference in which the conclusion is drawn from two premises through the help of a middle term.
6. What is conversion?
Answer: Conversion is a kind of immediate inference in which a new proposition is formed by interchanging the subject and the predicate of the original proposition while keeping the quality unchanged.
7. What is obversion?
Answer: Obversion is a kind of immediate inference in which the quality of the original proposition is changed and the predicate is replaced by its contradictory, while the subject and the truth-value remain unchanged.
8. What is contraposition?
Answer: Contraposition is a kind of immediate inference obtained by first obverting the original proposition and then converting the obverse so produced.
9. What is inversion?
Answer: Inversion is a kind of immediate inference in which both the subject and the predicate of the original proposition are replaced by their contradictories.
10. What is a syllogism?
Answer: A syllogism is a mediate deductive inference consisting of three propositions — two premises and one conclusion — in which the conclusion is drawn through a middle term that occurs in both premises.
11. Name the three kinds of mediate inference.
Answer: The three kinds of mediate inference are categorical syllogism, hypothetical syllogism and disjunctive syllogism.
12. What is the convertend?
Answer: The original proposition that is being converted is called the convertend.
13. What is the converse?
Answer: The new proposition that results from conversion is called the converse.
14. What is the obvertend?
Answer: The original proposition that is being obverted is called the obvertend.
15. What is the obverse?
Answer: The new proposition that results from obversion is called the obverse.
16. What is the contraponend?
Answer: The original proposition that is being contraposed is called the contraponend.
17. What is the contrapositive?
Answer: The new proposition that results from contraposition is called the contrapositive.
18. Convert the proposition “No man is perfect.”
Answer: The converse is “No perfect being is man.” (E converts to E by simple conversion.)
19. Convert “Some students are intelligent.”
Answer: The converse is “Some intelligent persons are students.” (I converts to I simply.)
20. Why can the O proposition not be converted?
Answer: Because conversion of an O proposition would require distributing the predicate term in the converse although it was undistributed in the convertend, which violates the rule that no term can be distributed in the converse unless it was distributed in the convertend.
21. Obvert “All metals are conductors.”
Answer: The obverse is “No metals are non-conductors.” (A becomes E.)
22. Obvert “Some boys are not honest.”
Answer: The obverse is “Some boys are dishonest.” (O becomes I.)
23. Give one example of an inductive inference.
Answer: “Crow1 is black, crow2 is black, crow3 is black, … therefore all crows are black” is an inductive inference, because the conclusion is wider than the observed premises.
24. State whether the following is true or false: “Every inference is a syllogism.”
Answer: False. Only mediate inferences are syllogisms; immediate inferences are not.
25. Is implication the same as inference?
Answer: No. Implication is an objective logical relation between propositions, while inference is the subjective mental act of drawing a conclusion based on that relation.
Short Answer Type Questions
26. Distinguish between inference and implication.
Answer: Inference is a mental act in which a thinker passes from premises to a conclusion. Implication is the objective relation between propositions such that the truth of one entails the truth of another. Inference is psychological and personal; implication is logical and impersonal. Inference takes place in time and may or may not occur in any given mind, but implication holds eternally and timelessly. Inference depends on implication for its validity, but implication does not depend on any actual inference. In short, implication is the ground and inference is the act.
27. Distinguish between deductive and inductive inference.
Answer: Deductive inference moves from the general to the particular or remains at the same level of generality. Inductive inference moves from the particular to the general. In deduction the conclusion is never wider than the premises, while in induction the conclusion is wider. Deductive inference yields formal certainty if the premises are true and the form is valid; inductive inference yields only probable conclusions, however high the probability may be. Deduction is concerned chiefly with form; induction is concerned with material truth and depends on observation and experiment.
28. Distinguish between immediate and mediate inference.
Answer: Immediate inference is drawn from a single premise without any middle term, while mediate inference is drawn from two premises with the help of a middle term. In immediate inference the conclusion only restates or partly modifies the meaning already contained in the premise; in mediate inference the conclusion combines the information of two premises into a new judgement. Conversion, obversion, contraposition and inversion are immediate inferences. Categorical, hypothetical and disjunctive syllogisms are mediate inferences.
29. State the rules of conversion.
Answer: The rules of conversion are: (i) the subject of the convertend becomes the predicate of the converse; (ii) the predicate of the convertend becomes the subject of the converse; (iii) the quality of the converse remains the same as that of the convertend; (iv) no term that was undistributed in the convertend can be distributed in the converse. Following these rules, A converts to I (per accidens), E converts to E, I converts to I, and O cannot be converted at all.
30. State the rules of obversion.
Answer: The rules of obversion are: (i) the subject of the obvertend remains the subject of the obverse; (ii) the predicate of the obverse is the contradictory of the predicate of the obvertend; (iii) the quality is changed — affirmative becomes negative and vice versa; (iv) the quantity remains unchanged. By these rules every categorical proposition can be obverted: A becomes E, E becomes A, I becomes O, and O becomes I.
31. State the rules of contraposition.
Answer: Contraposition is performed by first obverting the original proposition and then converting the obverse. The contrapositive therefore has, as its subject, the contradictory of the original predicate and, as its predicate, the original subject (or its contradictory, depending on the form). A gives the contrapositive of the form E, E gives a limited contrapositive of the form O, I has no valid contrapositive, and O gives the contrapositive of the form I.
32. What is conversion by limitation (per accidens)?
Answer: Conversion by limitation, also called conversion per accidens, is the conversion of a universal affirmative proposition (A) into a particular affirmative proposition (I). For example, “All cows are mammals” cannot become “All mammals are cows”, because the predicate “mammals” was undistributed in the convertend. So we limit the quantity and obtain “Some mammals are cows”. The quantity is reduced from universal to particular, hence the name “by limitation”.
33. Distinguish between simple conversion and conversion by limitation.
Answer: In simple conversion the quantity of the converse is the same as the quantity of the convertend; the only change is the interchange of subject and predicate. E and I propositions are converted simply. In conversion by limitation the quantity is reduced from universal to particular, because retaining the universal quantity would distribute a term in the converse that was not distributed in the convertend. Only the A proposition is converted by limitation.
34. Why is obversion called the most reliable immediate inference?
Answer: Obversion is called the most reliable or universally valid immediate inference because it can be performed on every one of the four categorical propositions — A, E, I and O — without exception. Conversion fails for O, contraposition fails for I, and inversion fails for I and O, but obversion is always possible. This is why textbooks usually treat obversion as the safest and most general operation.
35. State and explain the three kinds of syllogism.
Answer: Mediate inference is divided into three kinds. (i) Categorical syllogism — both premises are categorical propositions, for example, “All men are mortal; Socrates is a man; therefore, Socrates is mortal.” (ii) Hypothetical syllogism — at least one premise is a hypothetical proposition, for example, “If it rains, the ground will be wet; it is raining; therefore the ground is wet.” (iii) Disjunctive syllogism — at least one premise is a disjunctive proposition, for example, “Either he is at home or in the office; he is not at home; therefore he is in the office.”
36. Convert and obvert “All saints are virtuous.”
Answer: The proposition is an A proposition. Conversion (per accidens) gives “Some virtuous persons are saints” (I). Obversion gives “No saints are non-virtuous” (E). Both operations preserve the truth-value of the original.
37. Find the contrapositive of “All birds are animals.”
Answer: Step 1 — Obvert the original: “No birds are non-animals” (E). Step 2 — Convert the obverse simply: “No non-animals are birds” (E). Therefore the contrapositive is “No non-animals are birds.” (A gives a contrapositive of the form E.)
38. Find the contrapositive of “Some students are not lazy.”
Answer: Step 1 — Obvert: “Some students are non-lazy” (I). Step 2 — Convert: “Some non-lazy persons are students” (I). Therefore the contrapositive of the O proposition “Some students are not lazy” is “Some non-lazy persons are students.” (O gives a contrapositive of the form I.)
39. Why has the O proposition no converse but a contrapositive?
Answer: In the O proposition “Some S is not P”, the predicate P is distributed but the subject S is undistributed. Direct conversion would distribute S in the converse, which violates the rule of distribution, so the O proposition has no converse. But by first obverting it to the I proposition “Some S is non-P” and then converting that I simply, we obtain “Some non-P is S”, which is a valid contrapositive. Hence O has a contrapositive although it has no converse.
40. Why is the I proposition without a contrapositive?
Answer: The I proposition “Some S is P”, when obverted, becomes the O proposition “Some S is not non-P”. The O proposition has no converse, because direct conversion would distribute its undistributed subject. Since contraposition requires successful conversion of the obverse, and the obverse of I is an O which cannot be converted, the I proposition has no valid contrapositive.
41. Give one example each of conversion of E and I propositions.
Answer: E proposition: “No men are angels”; converse — “No angels are men” (E to E, simple conversion). I proposition: “Some flowers are red”; converse — “Some red things are flowers” (I to I, simple conversion). In both cases the truth-value is preserved.
42. Explain the importance of inference in logic.
Answer: Inference is the central subject-matter of logic. Logic is sometimes defined as the science of inference, since its main task is to study the rules by which valid conclusions may be drawn from given premises. Without inference there would be no demonstration, no proof, no science and no advance of knowledge. The classification of inferences and the formulation of rules for each kind enables us to detect fallacies, evaluate arguments and reason consistently in everyday life, in mathematics, in law and in scientific investigation.
Long Answer Type Questions
43. Define inference and discuss its main divisions.
Answer: Inference is the mental process by which the mind, starting from one or more propositions known or assumed to be true, passes to another proposition that is logically related to them. The propositions from which we start are called premises and the proposition we arrive at is called the conclusion. The form “P, therefore Q” expresses every inference.
The first major division of inference is into deductive and inductive. In deductive inference the conclusion never goes beyond the premises in extent or content. The movement is from the general to the particular, as in “All men are mortal; Rama is a man; therefore Rama is mortal”, or it remains at the same level of generality. Inductive inference, by contrast, moves from particular observed cases to a general unobserved law. From “this swan is white, that swan is white, …” we conclude “all swans are white”. Deduction, if its form is valid and its premises true, gives a conclusion of certainty; induction gives a conclusion of high probability based on observation, generalisation and causal analysis.
Deductive inference is further divided into immediate and mediate. Immediate inference draws a conclusion from a single premise, without the help of any third term. Mediate inference, also called syllogism, draws a conclusion from two premises through a middle term that appears in both premises but not in the conclusion. Immediate inference includes conversion, obversion, contraposition and inversion. Mediate inference includes categorical syllogism, hypothetical syllogism and disjunctive syllogism. Together these divisions cover the entire field of formal deductive logic.
44. Distinguish between deductive and inductive inference with examples.
Answer: Deductive and inductive inference differ in several important respects. (i) Direction — Deduction moves from the general to the particular or from one general truth to another general truth of equal extent; induction moves from particular cases to a general conclusion. (ii) Extent of conclusion — In deduction the conclusion never asserts more than the premises; in induction the conclusion asserts more than the premises and is therefore wider. (iii) Certainty — A valid deduction with true premises is certain; an induction is only probable. (iv) Basis — Deduction depends on the form of the argument; induction depends on observation, experiment and the principle of the uniformity of nature. (v) Function — Deduction explicates and applies general truths; induction discovers new general truths.
Example of deduction: “All metals expand on heating; iron is a metal; therefore iron expands on heating.” Here the conclusion is contained in the premises and adds nothing new.
Example of induction: “This piece of iron expanded on heating, that piece of iron expanded on heating, every observed piece of every metal has expanded on heating; therefore all metals expand on heating.” The conclusion goes beyond the observed cases to a universal law and is therefore inductive.
45. Distinguish between immediate and mediate inference and discuss their kinds.
Answer: Immediate inference is that deductive inference in which a conclusion is drawn from a single premise without any middle term. The conclusion only re-arranges or modifies the meaning already present in the premise. Its kinds are: (i) Conversion — interchange of subject and predicate, keeping quality the same; (ii) Obversion — change of quality and replacement of predicate by its contradictory; (iii) Contraposition — obversion followed by conversion; (iv) Inversion — replacement of both subject and predicate by their contradictories.
Mediate inference, on the other hand, draws a conclusion from two premises by means of a middle term. It is also called syllogism. Its kinds are: (i) Categorical syllogism, in which both premises are categorical, e.g., “All men are mortal; all kings are men; therefore all kings are mortal.” (ii) Hypothetical syllogism, in which at least one premise is hypothetical, e.g., “If a triangle is equilateral, then it is equiangular; this triangle is equilateral; therefore it is equiangular.” (iii) Disjunctive syllogism, in which at least one premise is disjunctive, e.g., “Either Hari is in school or at home; Hari is not in school; therefore Hari is at home.”
The basic difference is that immediate inference unfolds the meaning of one proposition, while mediate inference combines two propositions to yield a new judgement.
46. Explain conversion with all its rules and examples for A, E, I, O.
Answer: Conversion is the form of immediate inference in which a new proposition called the converse is obtained by interchanging the subject and predicate of the original proposition called the convertend, while the quality of the original is preserved.
The rules are: (1) The subject of the convertend becomes the predicate of the converse. (2) The predicate of the convertend becomes the subject of the converse. (3) The quality of the converse is the same as the quality of the convertend. (4) No term may be distributed in the converse unless it was distributed in the convertend.
Application to the four categorical propositions:
(i) A — “All S is P”. Subject S is distributed, predicate P is not. If we say “All P is S”, we would distribute P illicitly. So the converse is limited to a particular: “Some P is S”. A converts to I per accidens. Example: “All teachers are educated persons” — converse — “Some educated persons are teachers”.
(ii) E — “No S is P”. Both S and P are distributed. We may say “No P is S” without violating any rule. E converts simply to E. Example: “No misers are happy” — converse — “No happy persons are misers”.
(iii) I — “Some S is P”. Neither term is distributed. “Some P is S” is therefore valid. I converts simply to I. Example: “Some flowers are fragrant” — converse — “Some fragrant things are flowers”.
(iv) O — “Some S is not P”. Here P is distributed but S is not. Conversion would yield “Some P is not S”, which would distribute S illicitly. Therefore O has no converse.
47. Explain obversion with rules and examples for A, E, I, O.
Answer: Obversion is the form of immediate inference in which the quality of the original proposition (the obvertend) is changed, the predicate is replaced by its contradictory, and the result is the obverse, with the same subject, the same quantity and the same truth-value as the original.
The rules of obversion are: (1) The subject of the obverse is the same as the subject of the obvertend. (2) The predicate of the obverse is the contradictory of the predicate of the obvertend. (3) The quality is changed — affirmative becomes negative, negative becomes affirmative. (4) The quantity remains unchanged.
Application:
(i) A — “All S is P” obverts to “No S is non-P” (E). Example: “All judges are wise” — obverse — “No judges are non-wise”.
(ii) E — “No S is P” obverts to “All S is non-P” (A). Example: “No men are perfect” — obverse — “All men are non-perfect”.
(iii) I — “Some S is P” obverts to “Some S is not non-P” (O). Example: “Some workers are skilled” — obverse — “Some workers are not non-skilled”.
(iv) O — “Some S is not P” obverts to “Some S is non-P” (I). Example: “Some men are not honest” — obverse — “Some men are non-honest” (i.e., dishonest).
Obversion is always valid for every categorical proposition, and the truth-value of the obverse is identical with that of the obvertend.
48. Explain contraposition with rules and examples.
Answer: Contraposition is the immediate inference obtained by first obverting the original proposition and then converting the obverse. The original is the contraponend and the result is the contrapositive. Symbolically, contraposition combines the rules of obversion and conversion. The subject of the contrapositive is, in general, the contradictory of the predicate of the contraponend, and the predicate of the contrapositive is the original subject (or its contradictory in the case of “full” contraposition).
(i) A — “All S is P”. Step 1 (obvert) — “No S is non-P”. Step 2 (convert) — “No non-P is S”. Contrapositive of A is E. Example: “All men are mortal” — contrapositive — “No non-mortal beings are men”.
(ii) E — “No S is P”. Step 1 (obvert) — “All S is non-P” (A). Step 2 (convert per accidens) — “Some non-P is S” (I). The contrapositive of E is therefore an I, but logicians traditionally regard this as a limited contrapositive of the form O (“Some non-P is not S”) because direct contraposition of E gives a weaker conclusion.
(iii) I — “Some S is P”. Step 1 (obvert) — “Some S is not non-P” (O). Step 2 — but O cannot be converted. Therefore I has no contrapositive.
(iv) O — “Some S is not P”. Step 1 (obvert) — “Some S is non-P” (I). Step 2 (convert simply) — “Some non-P is S” (I). Contrapositive of O is I. Example: “Some men are not learned” — contrapositive — “Some non-learned persons are men”.
Contraposition is therefore valid for A and O, limited for E, and impossible for I.
49. Explain inversion with examples.
Answer: Inversion is the form of immediate inference in which both the subject and the predicate of the original proposition (the invertend) are replaced by their contradictories. The result is the inverse. Inversion is performed by a series of obversions and conversions and is valid only for the universal propositions A and E.
(i) A — “All S is P” inverts to “Some non-S is not P”. Example: “All men are mortal” — inverse — “Some non-men are not mortal” (e.g., some stones, which are not men, are not mortal in the sense in which men are mortal).
(ii) E — “No S is P” inverts to “Some non-S is P”. Example: “No tigers are vegetarians” — inverse — “Some non-tigers are vegetarians” (e.g., some cows, which are not tigers, are vegetarians).
(iii) I and O — Particular propositions cannot be inverted, because the rules of distribution do not permit the contradictory of an undistributed term to appear as the distributed subject of a new universal proposition.
Inversion is the most complex of the four immediate inferences and is sometimes treated as derived rather than primitive.
50. Distinguish between conversion and obversion.
Answer: Conversion and obversion are both immediate inferences but they differ in several ways. (i) In conversion the subject and predicate are interchanged, while in obversion they remain in their original positions. (ii) In conversion the quality of the proposition is preserved, while in obversion the quality is changed. (iii) In conversion the predicate is unchanged in form (only its position is changed), while in obversion the predicate is replaced by its contradictory. (iv) The quantity may change in conversion (as in A to I per accidens) but never changes in obversion. (v) Obversion is valid for all four categorical propositions A, E, I and O, while conversion fails for O. (vi) The truth-value of the obverse is always identical with that of the obvertend, but in conversion per accidens the converse is weaker than the convertend.
51. Distinguish between obversion and contraposition.
Answer: Obversion is a single-step immediate inference in which the quality is changed and the predicate is replaced by its contradictory, the subject remaining the same. Contraposition is a two-step immediate inference, the steps being obversion followed by conversion. In obversion the subject of the original remains the subject of the conclusion, while in contraposition the subject of the conclusion is the contradictory of the predicate of the original. Obversion is valid for A, E, I and O, while contraposition is valid for A and O, only limited for E, and not valid for I. The truth-value of the obverse is the same as that of the original; the truth-value of the contrapositive is also preserved when contraposition is valid, but the operation cannot always be performed.
52. Explain with examples how the truth-value is preserved in immediate inferences.
Answer: An immediate inference is logically correct only if the truth of the premise guarantees the truth of the conclusion. In each of the four kinds of immediate inference the rules are designed so that the truth-value is preserved.
In conversion, when carried out according to the four rules, the converse is true whenever the convertend is true. Thus from the true premise “No men are angels” we derive the true conclusion “No angels are men”.
In obversion, since changing the quality and replacing the predicate by its contradictory only restates the same content in a different form, the truth-value is necessarily the same. From “All metals are conductors” we derive the equally true “No metals are non-conductors”.
In contraposition, because it is built up from valid obversion and valid conversion, the truth-value is preserved wherever the operation is admissible. From “All men are mortal” we derive “No non-mortal beings are men”, which is true whenever the original is true.
In inversion, the rules permit the operation only for A and E because only there can the contradictories of the original terms be made to appear consistently as subject and predicate of a new universal or particular proposition without violating distribution rules. Hence wherever inversion is permitted, the inverse is true whenever the invertend is true.
53. Why is logic called the science of inference?
Answer: Logic is called the science of inference because its central concern is to lay down the rules and conditions under which one proposition may be validly inferred from another. Every other topic in logic — terms, propositions, distribution, opposition, conversion, syllogism, induction, fallacy — is studied chiefly because it bears on the validity of inference. The classification of inferences into deductive and inductive, immediate and mediate; the rules of conversion, obversion, contraposition and inversion; the figures and moods of the syllogism; the canons of induction — all these belong to the theory of inference. By studying these we acquire the power to test arguments, detect fallacies and reason rigorously. This is why traditional logicians defined logic as “the science that deals with the methods and principles used to distinguish correct reasoning from incorrect reasoning”, and modern logicians equally regard the theory of valid inference as the heart of the subject.
Additional Important Questions
54. Convert: “All philosophers are thinkers.”
Answer: A proposition. Converse (per accidens): “Some thinkers are philosophers” (I).
55. Convert: “No criminals are happy.”
Answer: E proposition. Converse: “No happy persons are criminals” (E).
56. Convert: “Some books are useful.”
Answer: I proposition. Converse: “Some useful things are books” (I).
57. Convert: “Some men are not honest.”
Answer: O proposition. The O proposition has no converse, because conversion would distribute the originally undistributed subject term.
58. Obvert: “All swans are birds.”
Answer: A. Obverse: “No swans are non-birds” (E).
59. Obvert: “No animals are immortal.”
Answer: E. Obverse: “All animals are non-immortal” i.e., “All animals are mortal” (A).
60. Obvert: “Some leaders are honest.”
Answer: I. Obverse: “Some leaders are not non-honest” (O).
61. Obvert: “Some students are not regular.”
Answer: O. Obverse: “Some students are non-regular” i.e., “Some students are irregular” (I).
62. Find the contrapositive of “All wise men are humble.”
Answer: Step 1 — Obvert: “No wise men are non-humble” (E). Step 2 — Convert: “No non-humble persons are wise men” (E). Contrapositive: “No non-humble persons are wise men”.
63. Find the contrapositive of “Some men are not soldiers.”
Answer: Step 1 — Obvert: “Some men are non-soldiers” (I). Step 2 — Convert: “Some non-soldiers are men” (I). Contrapositive: “Some non-soldiers are men”.
64. Find the inverse of “All cats are animals.”
Answer: Inverse: “Some non-cats are not animals”. (For example, some chairs, which are not cats, are not animals.)
65. Find the inverse of “No metals are non-conductors.”
Answer: Inverse: “Some non-metals are non-conductors” (e.g., wood is a non-metal and is a non-conductor).
66. Identify the kind of inference: “All graduates are educated; Hari is a graduate; therefore Hari is educated.”
Answer: This is a mediate deductive inference, more specifically a categorical syllogism, since the conclusion is drawn from two premises by means of the middle term “graduates”.
67. Identify the kind of inference: “If it rains, the match will be cancelled; it is raining; therefore the match will be cancelled.”
Answer: This is a hypothetical syllogism, a kind of mediate deductive inference.
68. Identify the kind of inference: “Either John is at home or in the office; John is not at home; therefore John is in the office.”
Answer: This is a disjunctive syllogism, a kind of mediate deductive inference.
69. Is the following an immediate inference? “All men are mortal; therefore some mortal beings are men.”
Answer: Yes, it is an immediate inference; specifically it is a conversion (per accidens) of an A proposition.
70. Is the following an immediate inference? “All metals are conductors; therefore no metals are non-conductors.”
Answer: Yes. It is an immediate inference and specifically an obversion of an A proposition.
71. State whether the following is true or false and justify: “Inversion is valid for all four categorical propositions.”
Answer: False. Inversion is valid only for the universal propositions A and E. The particular propositions I and O cannot be inverted, because the rules of distribution would be violated.
72. State whether the following is true or false and justify: “Obversion changes the quantity of the proposition.”
Answer: False. Obversion changes only the quality of the proposition. The quantity remains the same.
73. State whether the following is true or false: “The conclusion of an inductive inference is always certain.”
Answer: False. The conclusion of an inductive inference is only probable, since it asserts more than the premises.
74. State whether the following is true or false: “Implication is a mental act.”
Answer: False. Implication is an objective logical relation between propositions; it is not a mental act. Inference is the mental act.
75. State the relation between inference and reasoning.
Answer: Reasoning is the mental process by which we move from premises to a conclusion; inference is the verbal or symbolic expression of that process. Reasoning is psychological, inference is logical. Every inference is the linguistic counterpart of some act of reasoning, and the validity of the inference is the test of the correctness of the reasoning.
76. Why is induction sometimes called material logic?
Answer: Induction is sometimes called material logic because it is concerned not merely with the form of the argument but with the material truth of the premises, with observation, experiment and the actual constitution of nature. Deduction can be carried out without verifying the matter of the premises, but induction depends essentially on real observation and on the principle of the uniformity of nature.
77. Give one example each of valid and invalid conversion.
Answer: Valid conversion: “No men are stones” — converse — “No stones are men” (E to E). Invalid conversion: “All men are mortal” — proposed converse — “All mortal beings are men”. This is invalid because it distributes the predicate “mortal” which was undistributed in the convertend. The correct converse is “Some mortal beings are men”.
78. Give a complete table showing the obverse, converse and contrapositive of “All S is P”.
Answer: Original (A): “All S is P”. Obverse (E): “No S is non-P”. Converse (I): “Some P is S”. Contrapositive (E): “No non-P is S”. Inverse: “Some non-S is not P”.
79. Why does the validity of mediate inference depend on the validity of immediate inference?
Answer: Mediate inference uses immediate inferences in many of its steps. For example, in proving the validity of certain syllogistic moods we reduce them to other moods by conversion, obversion or contraposition. Therefore the rules of immediate inference are presupposed by the rules of mediate inference, and any error at the immediate stage transmits itself to the mediate stage. Hence a clear mastery of conversion, obversion, contraposition and inversion is the foundation of all syllogistic reasoning.
80. Give the form “A, therefore B” of two everyday inferences.
Answer: (i) “He has fever, therefore he is ill” — an inductive or causal inference. (ii) “All Indians are Asians, therefore some Asians are Indians” — a deductive immediate inference (conversion per accidens).
Glossary
| Term | Meaning |
|---|---|
| Inference | The mental process of passing from one or more premises to a conclusion. |
| Premise | A proposition from which a conclusion is drawn. |
| Conclusion | The proposition that is drawn from the premise(s) in an inference. |
| Implication | The objective logical relation between propositions that grounds inference. |
| Reasoning | The mental process of which inference is the verbal expression. |
| Deductive Inference | Inference whose conclusion is not wider than the premises. |
| Inductive Inference | Inference whose conclusion is wider than the premises. |
| Immediate Inference | Deductive inference from one premise. |
| Mediate Inference | Deductive inference from two premises through a middle term; also called syllogism. |
| Categorical Syllogism | Mediate inference whose premises are categorical propositions. |
| Hypothetical Syllogism | Mediate inference at least one of whose premises is a hypothetical proposition. |
| Disjunctive Syllogism | Mediate inference at least one of whose premises is a disjunctive proposition. |
| Conversion | Immediate inference by interchange of subject and predicate, quality unchanged. |
| Convertend | The original proposition in conversion. |
| Converse | The new proposition obtained by conversion. |
| Conversion per accidens | Conversion by limitation, applicable to A propositions; A converts to I. |
| Obversion | Immediate inference by changing quality and replacing predicate with its contradictory. |
| Obvertend | The original proposition in obversion. |
| Obverse | The new proposition obtained by obversion. |
| Contraposition | Immediate inference obtained by obverting and then converting. |
| Contraponend | The original proposition in contraposition. |
| Contrapositive | The new proposition obtained by contraposition. |
| Inversion | Immediate inference replacing both terms by their contradictories. |
| Invertend | The original proposition in inversion. |
| Inverse | The new proposition obtained by inversion. |
| Distribution | The taking of a term in its full extension; relevant to all rules of immediate inference. |
| Middle Term | The term that occurs in both premises of a syllogism but not in the conclusion. |
| Uniformity of Nature | The principle that the same causes produce the same effects under the same conditions; ground of inductive inference. |
Conversion Reference Table
| Convertend | Form | Converse | Form | Type |
|---|---|---|---|---|
| All S is P | A | Some P is S | I | Per accidens (by limitation) |
| No S is P | E | No P is S | E | Simple |
| Some S is P | I | Some P is S | I | Simple |
| Some S is not P | O | No converse | — | Not convertible |
Obversion Reference Table
| Obvertend | Form | Obverse | Form |
|---|---|---|---|
| All S is P | A | No S is non-P | E |
| No S is P | E | All S is non-P | A |
| Some S is P | I | Some S is not non-P | O |
| Some S is not P | O | Some S is non-P | I |
Contraposition Reference Table
| Contraponend | Form | Contrapositive | Form | Validity |
|---|---|---|---|---|
| All S is P | A | No non-P is S | E | Valid |
| No S is P | E | Some non-P is not S | O | Limited (per accidens) |
| Some S is P | I | No contrapositive | — | Invalid |
| Some S is not P | O | Some non-P is S | I | Valid |
Inversion Reference Table
| Invertend | Form | Inverse | Form | Validity |
|---|---|---|---|---|
| All S is P | A | Some non-S is not P | O | Valid |
| No S is P | E | Some non-S is P | I | Valid |
| Some S is P | I | No inverse | — | Invalid |
| Some S is not P | O | No inverse | — | Invalid |
Worked Examples — Conversion
| No. | Convertend | Form | Converse | Form | Notes |
|---|---|---|---|---|---|
| 1 | All teachers are graduates. | A | Some graduates are teachers. | I | A converts to I per accidens; “graduates” was undistributed in the convertend, so we must reduce the quantity. |
| 2 | All Indians are Asians. | A | Some Asians are Indians. | I | Same pattern; “All Asians are Indians” would distribute “Asians” illicitly. |
| 3 | No men are angels. | E | No angels are men. | E | E converts simply; both terms are distributed in E, so simple conversion is safe. |
| 4 | No criminals are happy persons. | E | No happy persons are criminals. | E | Simple conversion; truth-value preserved. |
| 5 | Some students are intelligent persons. | I | Some intelligent persons are students. | I | I converts simply; neither term is distributed in I. |
| 6 | Some metals are precious. | I | Some precious things are metals. | I | Simple conversion; valid for every I proposition. |
| 7 | Some animals are not carnivores. | O | No converse. | — | O proposition is not convertible because the subject “animals” is undistributed in the convertend but would be distributed in any direct converse. |
| 8 | Some politicians are not honest persons. | O | No converse. | — | Same reason; the predicate “honest persons” is distributed in O, while “politicians” is not. |
Worked Examples — Obversion
| No. | Obvertend | Form | Obverse | Form | Notes |
|---|---|---|---|---|---|
| 1 | All metals are conductors. | A | No metals are non-conductors. | E | Quality changed; predicate replaced by its contradictory; quantity unchanged. |
| 2 | All saints are virtuous. | A | No saints are non-virtuous. | E | Same operation; “non-virtuous” is the contradictory of “virtuous”. |
| 3 | No animals are immortal. | E | All animals are non-immortal (mortal). | A | E becomes A; “non-immortal” is more naturally written as “mortal”. |
| 4 | No misers are happy. | E | All misers are non-happy (unhappy). | A | Same pattern; truth-value preserved. |
| 5 | Some workers are skilled. | I | Some workers are not non-skilled. | O | I becomes O; quantity remains “some”. |
| 6 | Some flowers are fragrant. | I | Some flowers are not non-fragrant. | O | Same operation; quality changes from affirmative to negative. |
| 7 | Some men are not honest. | O | Some men are non-honest (dishonest). | I | O becomes I; the contradictory of “honest” is “non-honest” or “dishonest”. |
| 8 | Some students are not regular. | O | Some students are non-regular (irregular). | I | Same pattern. |
Worked Examples — Contraposition
| No. | Contraponend | Form | Step 1 — Obverse | Step 2 — Contrapositive | Form |
|---|---|---|---|---|---|
| 1 | All men are mortal. | A | No men are non-mortal. | No non-mortal beings are men. | E |
| 2 | All birds are animals. | A | No birds are non-animals. | No non-animals are birds. | E |
| 3 | No men are perfect. | E | All men are non-perfect. | Some non-perfect beings are not men. | O (limited) |
| 4 | No metals are non-conductors. | E | All metals are conductors. | Some conductors are not non-metals. | O (limited) |
| 5 | Some flowers are red. | I | Some flowers are not non-red. | No contrapositive. | — |
| 6 | Some students are intelligent. | I | Some students are not non-intelligent. | No contrapositive (O cannot be converted). | — |
| 7 | Some men are not learned. | O | Some men are non-learned. | Some non-learned persons are men. | I |
| 8 | Some animals are not herbivores. | O | Some animals are non-herbivores. | Some non-herbivores are animals. | I |
Worked Examples — Inversion
| No. | Invertend | Form | Inverse | Form |
|---|---|---|---|---|
| 1 | All cats are animals. | A | Some non-cats are not animals. | O |
| 2 | All men are mortal. | A | Some non-men are not mortal. | O |
| 3 | No tigers are vegetarians. | E | Some non-tigers are vegetarians. | I |
| 4 | No criminals are happy. | E | Some non-criminals are happy. | I |
| 5 | Some men are honest. | I | No inverse. | — |
| 6 | Some men are not honest. | O | No inverse. | — |
Solved Problems on Mediate Inference
Problem 1. Test the validity of: “All Indians are Asians; all Asians are humans; therefore all Indians are humans.”
Answer: This is a categorical syllogism. The middle term “Asians” is distributed in the second premise (it is the subject of an A proposition). The major term “humans” and the minor term “Indians” are not distributed in the conclusion in any way that is not justified by the premises. The conclusion “All Indians are humans” follows validly. Hence the inference is valid.
Problem 2. Test: “If it rains, the ground will be wet; the ground is wet; therefore it has rained.”
Answer: This is a hypothetical syllogism. It commits the fallacy of “affirming the consequent”. The wet ground might be due to other causes (a burst pipe, a sprinkler). Therefore the inference is invalid.
Problem 3. Test: “Either the train is late or my watch is wrong; my watch is not wrong; therefore the train is late.”
Answer: This is a disjunctive syllogism. It is valid in form: “Either p or q; not q; therefore p.” Provided the disjunctive premise is true and exhaustive, the inference is valid.
Problem 4. Test: “All philosophers are thinkers; some thinkers are scientists; therefore some philosophers are scientists.”
Answer: This is a categorical syllogism. The middle term “thinkers” is undistributed in both premises (predicate of an A and subject of an I). Hence the inference commits the fallacy of “undistributed middle” and is invalid.
Problem 5. Test: “If a triangle is equilateral, then it is equiangular; this triangle is equilateral; therefore it is equiangular.”
Answer: This is a hypothetical syllogism in the valid form modus ponens: “If p then q; p; therefore q.” The inference is valid.
Problem 6. Convert the contrapositive of “All sincere students are successful.” Verify that the result is consistent with the original.
Answer: Step 1 — Contrapositive of “All sincere students are successful” (A): obvert to “No sincere students are non-successful” (E); convert to “No non-successful persons are sincere students” (E). Step 2 — Convert this E proposition simply: “No sincere students are non-successful persons” (E). Step 3 — Compare with the obverse of the original “No sincere students are non-successful”. The two are identical, confirming consistency.
Common Errors and How to Avoid Them
| Error | Why it is wrong | Correct procedure |
|---|---|---|
| Converting “All S is P” simply to “All P is S”. | The predicate term “P” was undistributed in the original A proposition. | Use conversion per accidens: “Some P is S”. |
| Converting “Some S is not P” to “Some P is not S”. | The subject “S” was undistributed in the O proposition but would be distributed in the proposed converse. | O proposition is not convertible. Use contraposition instead: “Some non-P is S”. |
| Obverting by changing only the quality without altering the predicate. | An obverse must replace the predicate with its contradictory; merely flipping the quality gives the contradictory or contrary, not the obverse. | Both change the quality and replace the predicate by its contradictory. |
| Treating obversion as changing the quantity. | Obversion never changes quantity; only the quality and the predicate change. | Keep the quantity exactly the same. |
| Trying to invert an I or O proposition. | Particular propositions cannot yield the contradictory subject as a fresh subject of a new universal or particular without violating distribution rules. | Inversion is permitted only for A and E. |
| Confusing implication with inference. | Implication is a logical relation; inference is the mental act based on it. | Use “implies” for the relation between propositions, “infers” for the act of the thinker. |
Multiple Choice Practice
| No. | Question | Answer |
|---|---|---|
| 1 | Inference from one premise is called … | Immediate inference. |
| 2 | Inference from two premises is called … | Mediate inference (syllogism). |
| 3 | The kind of inference in which the conclusion is wider than the premises is … | Inductive inference. |
| 4 | The proposition that is converted is called the … | Convertend. |
| 5 | The new proposition produced by obversion is the … | Obverse. |
| 6 | A proposition converts to an … proposition. | I (per accidens). |
| 7 | E proposition converts to an … proposition. | E (simple). |
| 8 | I proposition converts to an … proposition. | I (simple). |
| 9 | O proposition has … | No converse. |
| 10 | The obverse of “All S is P” is … | “No S is non-P”. |
| 11 | The obverse of “No S is P” is … | “All S is non-P”. |
| 12 | The obverse of “Some S is P” is … | “Some S is not non-P”. |
| 13 | The obverse of “Some S is not P” is … | “Some S is non-P”. |
| 14 | The contrapositive of “All S is P” is … | “No non-P is S”. |
| 15 | The contrapositive of “Some S is not P” is … | “Some non-P is S”. |
| 16 | Inversion is valid only for … | A and E propositions. |
| 17 | The middle term occurs in … | Both premises but not the conclusion of a syllogism. |
| 18 | The kind of mediate inference whose premises are all categorical is … | Categorical syllogism. |
| 19 | “If P, then Q; P; therefore Q” is the form of … | Modus ponens (a valid hypothetical syllogism). |
| 20 | “Either P or Q; not P; therefore Q” is the form of … | Disjunctive syllogism. |
Quick Recap
| Operation | Changes | Valid for |
|---|---|---|
| Conversion | Subject and predicate interchanged; quality same; quantity may change for A | A (per accidens), E, I |
| Obversion | Quality changed; predicate replaced by contradictory; subject and quantity unchanged | A, E, I, O |
| Contraposition | Obvert then convert; subject becomes contradictory of original predicate | A, O fully; E only with limitation; I never |
| Inversion | Both terms replaced by their contradictories | A, E only |
Extended Discussion — Why Each Rule of Conversion Exists
The four rules of conversion may at first appear arbitrary, but every one of them flows from a single deeper principle: no term may be used in a wider extension in the conclusion than it had in the premise. This is sometimes stated as the rule that “no term can be distributed in the converse unless it was distributed in the convertend”.
Consider why the A proposition cannot convert simply. In “All men are mortal”, the subject “men” is distributed (it refers to every individual man), but the predicate “mortal” is not distributed (it does not refer to every mortal being; only to those mortals which are men). If we were to convert it as “All mortal beings are men”, we would be saying something about every mortal being, including stones, plants and dogs, which the original premise never asserted. To avoid this illicit extension we limit the quantity to “Some mortal beings are men”, so that “mortal beings” remains undistributed.
The E proposition behaves differently because both its terms are distributed. “No men are angels” denies any overlap between the entire class of men and the entire class of angels. The same denial flows symmetrically in the converse “No angels are men”, and no term gains in distribution. Hence E converts simply to E.
The I proposition is the easiest case. “Some men are wise” merely asserts that the classes “men” and “wise” overlap. The same overlap is reported by “Some wise persons are men”. Neither term is distributed, so no rule of distribution can be violated.
The O proposition is the trickiest. “Some men are not honest” tells us that there exist men who fall outside the class of honest persons. In O, the predicate “honest” is distributed (we are denying these men of every honest person whatsoever), but the subject “men” is not. A direct conversion would yield “Some honest persons are not men”, which would make a denial about some specifically picked-out honest persons. But in the original we never picked out any honest persons; we only spoke of men. So the proposed converse asserts something not contained in the original, and is invalid. This is why the O proposition has no converse and must instead be approached through contraposition.
Extended Discussion — Why Inversion Fails for I and O
Inversion replaces both the subject and the predicate of the original by their contradictories. For inversion to be valid we must produce a new universal or particular proposition without violating any distribution rule. In the case of A and E this is possible through a chain of obversions and conversions, because at each step a term that was distributed remains distributed and a term that was undistributed remains undistributed (or is reduced in quantity if necessary).
For the I proposition, “Some S is P”, neither S nor P is distributed. If we try to invert it to a proposition about “non-S” and “non-P”, we have no information at all about the extension of the contradictories, because the original premise locates only a small part of “S” inside “P” and tells us nothing about what falls outside “S” or “P”. Hence no valid conclusion can be drawn about non-S and non-P.
For the O proposition, “Some S is not P”, we know that there are some “S” outside the class of “P”, but we do not know whether anything outside “S” is or is not in “P”. Therefore no valid statement about non-S in relation to “P” or non-P can be derived.
In short, particular propositions are too weak to support the inference of inversion, which requires us to make claims about the contradictories of both terms.
Extended Discussion — Inference and the Growth of Knowledge
The reason logicians spend so much effort classifying inferences is that nearly every advance of knowledge can be analysed as a chain of inferences. In mathematics, every theorem is derived deductively from axioms and definitions through a long chain of immediate and mediate inferences. In the natural sciences, generalisations such as “All metals expand on heating” are inductive inferences from observed cases, and individual predictions such as “this iron rod will expand if heated” are deductive inferences from those generalisations. In law, the verdict of a judge is a mediate inference from the legal rule (major premise) and the established facts of the case (minor premise). In daily life, every reasoned decision — to carry an umbrella because clouds are gathering, to call the doctor because of fever — is a small inference. By learning the rules of valid inference we equip ourselves to think clearly, to argue convincingly and to detect the fallacies that cloud public discussion.
A thorough mastery of the definitions, rules and reference tables given above is enough to handle every question on Chapter 7 — Inference in the ASSEB Class 11 examination, whether very short, short or long-answer type. Practise each conversion, obversion, contraposition and inversion on at least five fresh propositions of your own, and the chapter will become second nature.