BLS LLB 5 Years Sem 1 Logic 1 2025-26 - ATKT Set 2 75/25 Question Paper with Solutions

Division by dichotomy is the division of a class into two members by a pair of contradictory terms, one positive and one negative, so that the class is split into what has a given attribute and what does not.

Example: people into introverts and non-introverts; things into material and non-material; flowers into fragrant and non-fragrant.

Its merit is that it can never go wrong formally: because the two members are contradictories, the division is necessarily exhaustive, since everything must be one or the other, and necessarily mutually exclusive, since nothing can be both.

Its defect is that the negative member is wholly indeterminate: "non-introvert" says only what its members are not.

A proposition is a statement in which something is affirmed or denied of something else, and which is therefore necessarily either true or false.

It has three parts: the subject, that about which something is asserted; the predicate, that which is asserted of it; and the copula, the part which joins the two and asserts or denies the relation.

Example: in "All men are mortal", "men" is the subject, "mortal" the predicate, and "are" the copula.

Section 268 of the Indian Penal Code 1860, now Section 270 of the Bharatiya Nyaya Sanhita 2023, defines public nuisance as any act or illegal omission which causes any common injury, danger or annoyance to the public, or to the people in general who dwell or occupy property in the vicinity, or which must necessarily cause injury, obstruction, danger or annoyance to persons who may have occasion to use any public right.

It is a crime, punishable under Section 290 IPC (Section 292 BNS). Examples: obstruction of a public highway, polluting a public water source, carrying on a noxious trade in a crowded locality.

An individual may sue in tort only on proof of special damage, that is, damage over and above that suffered by the public at large.

Eduction is a form of immediate inference in which, from a given proposition, another proposition is inferred whose subject or predicate, or both, is either a term of the original or its contradictory, the meaning being kept unchanged.

Its kinds are:

1. Conversion: the subject and predicate change places. "No horses are bipeds" gives "No bipeds are horses".
2. Obversion: the quality is changed and the predicate is replaced by its contradictory. "All men are mortal" gives "No men are non-mortal".
3. Contraposition: obvert, then convert. "All men are mortal" gives "No non-mortals are men".
4. Inversion: the subject of the inferred proposition is the contradictory of the original subject. "All men are mortal" gives "Some non-men are not mortal".

The five rules

Rule 1. There must be only one fundamentum divisionis at each step. Fallacy: cross-division. Breach: "Books into English, historical and cheap" divides at once by language, by subject and by price.

Rule 2. The dividing members must be mutually exclusive. Fallacy: overlapping division. Breach: "Human beings into men, women and doctors", since a doctor is already a man or a woman.

Rule 3. The division must be exhaustive: the members together must equal the genus. Fallacy: incomplete division. Breach: "Vertebrates into fishes, birds and mammals", which omits amphibians and reptiles.

Rule 4. The division must proceed step by step, to the proximate species. Fallacy: saltus in dividendo, the leap in division. Breach: "Literature into poetry, drama and the novel", which leaps over prose, the species that should have come next.

Rule 5. Every dividing member must be a species of the genus divided. Fallacy: division confused with partition. Breach: "Umbrella into rod, handle, spokes and cloth", since a spoke is a part of an umbrella and not a kind of umbrella.

For full marks, cover: what each word applies to; the definitions; the six combinations with an example of each; the one combination that cannot occur; soundness; and the legal application.

The distinction

Truth and falsity are properties of propositions. Validity and invalidity are properties of arguments. To call a proposition valid, or an argument true, is a category mistake.

• A proposition is true when it corresponds to fact, and false when it does not.
• An argument is valid when the conclusion follows necessarily from the premises, that is, when it is impossible for the premises to be true and the conclusion false.

Validity therefore depends on the form of the argument, not on the material truth of what is asserted.

The six combinations

Soundness

An argument is sound when it is valid and all its premises are true. Only soundness guarantees a true conclusion; validity alone guarantees only that no truth has been lost on the way.

The fallacy of a definition too narrow. The definition covers less than the term.

Example: "A doctor is a person who performs surgery." Physicians and paediatricians are doctors and perform no surgery. Corrected: a doctor is a person qualified and licensed to practise medicine.

A legal instance of both: "A contract is an agreement for the sale of goods" is too narrow, and "a contract is an agreement" is too wide. Section 2(h) of the Indian Contract Act 1872 gets it exactly co-extensive: an agreement enforceable by law.

Rule 2 chosen: a definition must not be circular

Statement. The definition must not contain the term defined, whether directly, or through a synonym, or through its correlative. The fallacy is circulus in definiendo, the circle in defining.

Breach by repeating the term. "Law is a rule laid down by the law-making authority." The word to be explained is used to explain itself.

Breach by synonym. "Freedom is liberty." The two words are equivalent, so the reader who does not understand one is no better off.

For full marks, cover: both definitions with a worked table; the law of inverse variation and its limits; the terms that have one and not the other; and the legal application.

The two aspects

The connotation, or intension, of a term is the sum of the essential attributes which the term implies, that is, the qualities a thing must possess before the term can be applied to it.

The denotation, or extension, of a term is the range of individuals or classes to which the term applies.

Legal application

A definition clause fixes the connotation of a word and the court then decides what falls within its denotation. When a bench asks whether an e-rickshaw is a "motor vehicle", it is testing an object against a connotation the legislature has fixed. The rule of ejusdem generis, by which general words following an enumeration are limited to things of the same kind, is a rule about genus and species and therefore about connotation.

Strict logical form and the distribution rule

A proposition is in strict logical form when it reads

quantity sign + subject term + copula (is or are, present tense) + predicate term

and can be identified at once as A, E, I or O.

Universals distribute the subject; negatives distribute the predicate.

Kind: E proposition, universal negative. Distributed: both terms, "old paths" and "the best paths".

Reason: the sentence carries no quantity sign, and an unquantified proposition that states a general truth is read as universal. The "not" attaches to the copula, not to the predicate, so the proposition is negative and the reduction is E.

The alternative reading, worth one line: if the speaker means only that some old paths are not the best, the proposition is O and only the predicate is distributed. Where an indefinite proposition is genuinely ambiguous, state the reading you have taken and why, which is what an examiner is looking for.

Reading of the table: the conditional is false in one case only, where the antecedent is true and the consequent false. The statement is broken only by an eclipse during which the food is not cooked. It is contingent, neither a tautology nor a contradiction.

(2) Symbolise the following general propositions

(i) No athletes are bookworms. (Ax, Bx)

(x)(Ax ⊃ ~Bx)

For every x, if x is an athlete then x is not a bookworm. This is an E proposition. An equivalent form is ~(∃x)(Ax · Bx): there is nothing that is both.

(ii) Not every applicant was hired. (Ax, Hx)

~(x)(Ax ⊃ Hx), which is equivalent to (∃x)(Ax · ~Hx)

It is not the case that every applicant was hired, that is, there is at least one applicant who was not. This is an O proposition.

The modern classification of definitions

Modern logic classifies definitions in two ways at once. By purpose: stipulative, lexical, precising, theoretical and persuasive. By technique, which is what this question asks about:

1. Denotative or extensional techniques, which point at what the term covers: definition by example, by enumeration of the species, and ostensive or demonstrative definition, which points at an instance.
2. Connotative or intensional techniques, which state what the term means: synonymous definition, operational definition, and definition by genus and difference.

(i) Pen is what you are holding in your hand.

Kind: an ostensive, or demonstrative, definition. It is a denotative technique.

(iii) Flower means rose, lily, sunflower and lotus.

Kind: a definition by enumeration, that is, an extensive or denotative definition.

Reason: the term is defined by listing the species it denotes instead of stating the attributes it connotes.

Its limits: it can never be complete where the class is open, since jasmine, marigold and a thousand others are equally flowers, and it says nothing about what makes them all flowers. Where the class is genuinely closed the technique works well, which is why statutory schedules use it.

Reasons: logical division separates a class into its kinds, so that each member is a species of the genus. A seed is not a kind of mango. What has been done is partition, the breaking up of a single object into the parts of which it is made. The test settles it at once: the name of the whole can be predicated of every species in a division, so "an Alphonso is a mango" is true, whereas "a skin is a mango" is false. Rule 5 is broken.

A sound division of the same genus: mangoes into Alphonso, Kesar, Langda, Dasheri and other varieties, on the basis of variety.

(ii) Grains into Basmati, Kolam and Parimal.

Fallacy: the division is incomplete, and it leaps a step.

Reasons: it is incomplete, because wheat, jowar, bajra, maize and ragi are all grains and none of them appears, so the members fall far short of the genus. It also commits the saltus in dividendo, the leap in division: Basmati, Kolam and Parimal are not species of grain at all but varieties of rice, which is itself only one species of grain. The proper order is grains into rice, wheat, millets and pulses at the first step, and rice into Basmati, Kolam and Parimal at the next.

First division: simple and compound

A simple proposition contains no other proposition as a component part. Example: "Rama is honest."

A compound proposition contains at least one other proposition as a component, joined by a connective. Example: "Rama is honest and Shyam is diligent."

A compound is truth-functional when its truth value is completely determined by the truth values of its components together with the connective. That property is what makes the truth table possible, and it is the heart of the modern classification.

The five connectives

1. Negation (~p), "not p". Reverses the truth value.

2. Conjunction (p · q), "p and q". The components are conjuncts. True only when both are true.

5. Equivalence (p ≡ q), "p if and only if q". True when both components have the same truth value. It is the conjunction of the two conditionals.

Compounds that are not truth-functional

A compound whose truth value is not settled by its components falls outside the scheme: "Rama believes that the earth is flat", "It is necessary that two and two are four", "He died because he was poisoned". Knowing that "the earth is flat" is false does not tell us whether Rama believes it. Modal and belief contexts are handled by separate branches of logic for exactly this reason.

Singular and general propositions

Among the simple propositions, modern logic separates:

Legal illustration

Statutes are written in these connectives and the choice decides cases. Conjunction makes conditions cumulative: Section 10 of the Contract Act requires free consent and competence and lawful consideration and lawful object. Disjunction makes them alternative: a section penalising one who "sells or offers for sale" is satisfied by either. Implication is the form of every proviso and deeming clause. Equivalence is what a definition clause asserts, since the term and its definition must be interchangeable.

Conclusion

The modern classification groups propositions by what settles their truth value: simple or compound, and within the simple, singular or general. It gains a mechanical test of validity, a symbolism free of the ambiguities of English, and the power to express relations and multiple generality, none of which the traditional scheme could offer. It does not discard the old four forms; it re-expresses them in the predicate calculus and shows what they always meant.

Its place among the forms of inference

Analogy is neither deduction nor complete induction. It is not deductive, because the conclusion can be false while the premises are true. It is not a full induction either, because it does not generalise to a class: it moves from particular to particular, which is why it is also called reasoning from example. Its conclusion is therefore always probable, and the whole question about any analogy is how probable.

The conditions of a good analogy

1. The number of resemblances should be large. The more respects in which the two agree, the stronger the inference.
2. The resemblances must be relevant, that is, causally connected with the property inferred. This is the decisive test. Ten irrelevant resemblances are worth less than one relevant one.
3. The number of differences should be small, and no known difference should bear on the property inferred.
4. The conclusion should be modest. The weaker the property claimed, the more probable the conclusion.

The use of analogy in law

1. Precedent is analogical reasoning. To follow a case is to argue that the material facts of the present case resemble it in the respects that mattered; to distinguish it is to argue that they do not. Counsel who says a decision is distinguishable is applying the relevance test, and a judge who accepts it is holding that the resemblance relied on is not the one that produced the earlier result.

2. Finding the ratio decidendi is an exercise in relevance. The binding part of a decision is the principle applied to its material facts, and deciding which facts were material is deciding which resemblances would carry the result across.

3. Statutory interpretation uses analogy in fixed forms. Ejusdem generis, by which general words following an enumeration are read as limited to the same kind, and noscitur a sociis, by which a word takes its colour from its neighbours, are both rules for reasoning from likeness.

4. Analogy fills gaps. Where no rule covers a case, courts extend the nearest one. After Donoghue v Stevenson [1932] AC 562 the duty owed by a manufacturer of ginger beer was extended by analogy to manufacturers of every kind of product.

Conclusion

Analogy is the weakest form of inference in logic and among the most used in law, and there is no contradiction in that. A system that must decide new cases with old rules has no other way of moving from what has been decided to what has not. Its discipline is the test of relevance: the resemblance relied on must be the one that produced the earlier result, and every judgment that follows or distinguishes a precedent says so in as many words.

This crow is black, and that one, and that one. Therefore all crows are black.

The distinction

4. Mill's experimental methods are deductive in form. The Method of Agreement, the Method of Difference and the rest are stated as rules and applied to instances exactly as a major premise is applied to a minor. Induction, at the point where it becomes rigorous, borrows the form of deduction.

5. They differ in emphasis, not in kind. Deduction is concerned with validity, induction with truth. An argument needs both to be sound, so the two are two requirements on one process rather than two rival processes.

Both at work in legal reasoning

Legal reasoning shows the partnership plainly.

The judgment is deductive. The rule of law is the major premise, the facts found are the minor, and the order is the conclusion.

Finding the facts is inductive. A finding on circumstantial evidence is an induction from particulars, and the conditions laid down in Sharad Birdhichand Sarda v State of Maharashtra AIR 1984 SC 1622, that the chain be complete and exclude every hypothesis but guilt, are Mill's method of elimination in judicial dress.

For full marks, cover: the three questions separately and in the order asked; the sense in which logic is the science of sciences, with the objection to the phrase; logic as science and as art, with the relation between them; and the formal-material question answered with a qualified "not quite", showing what is right and what is wrong in it. Close with a conclusion that ties the three together.

1. "Logic is the science of sciences"

The claim. Every science reasons: it observes, frames hypotheses, draws consequences and tests them. Logic is the science that examines reasoning itself, and so it studies the instrument that every other science uses. In that sense it stands to the sciences as grammar stands to the languages, and it is called the science of sciences and the scientia scientiarum.

Four supports for the claim:

2. Is logic an art?

Yes, and it is a science as well. It is both, and the two are not in competition.

As a science, logic is a systematic body of general truths about the conditions of valid inference. It states principles, and its statements are true or false.

As an art, logic is a body of rules for the practice of reasoning: how to define, how to divide, how to test a syllogism, how to detect a fallacy. Its rules are useful or useless, not true or false.

The relation is that of theory to practice, and the standard comparisons carry it. Anatomy is a science, surgery the art founded on it. Physiology is a science, medicine the art. The science states what valid reasoning is; the art tells you how to reason validly and how to catch someone who has not.

Because both are present, logic is called a normative science: it sets a standard rather than describing behaviour, and a standard is only worth having if it can be applied. That is exactly why an art must go with the science.

1. Deduction is not purely formal in use. A valid argument with a false premise is worthless in practice, and soundness, which is validity plus true premises, is what anyone actually wants. The material truth of the premises therefore matters to deduction too, and where those premises come from is induction's business.
2. Induction is not purely material. It has a form of its own. Mill's experimental methods are stated as general rules and applied to instances exactly as a major premise is applied to a minor, and an induction that misapplies the Method of Difference fails for a formal reason.
3. Induction rests on premises of its own, the uniformity of nature and the law of universal causation, and it uses them deductively.
4. The distinction is one of degree. Deduction gives more attention to form, induction more to matter; neither can dispense with the other.

Conclusion on the third question: it is correct to say that deductive logic is primarily formal and inductive logic primarily material. It is not correct to say that either is purely so.

For full marks, cover: each item with the given proposition reduced and its form named, the answer written out in full, and the rule that produces it; and the one item where what is asked cannot be given, answered with the reason.

The tables the answers depend on

Opposition: contradictories are A with O and E with I; contraries are A with E; sub-contraries are I with O; subalterns are A with I and E with O.

Contrary (A): All rainy days are bright days.

Contraries are the two universals differing in quality. They cannot both be true, but they may both be false, which is in fact the case here.

(ii) Many businessmen do not pay proper taxes. (Give Sub-contrary) (2 marks)

Reduced: "Some businessmen are not persons who pay proper taxes." An O proposition, because "many" is a sign of particular quantity and the proposition is negative.

Sub-contrary (I): Some businessmen are persons who pay proper taxes.

Sub-contraries are the two particulars differing in quality. They cannot both be false, though they may both be true, and here they are.

(iii) Not every goods manufacturer in the country is exported. (Give Converse) (2 marks)

Reduced: "Some goods manufactured in the country are not exported goods." An O proposition, because "not every" is the sign of a particular negative.

An O proposition cannot be converted. It has no converse.

(v) All reformers are idealists. (Give Full Contrapositive) (2 marks)

An A proposition. Contraposition is obversion followed by conversion; the full contrapositive adds a second obversion.

Step 1, obvert: No reformers are non-idealists. (E) Step 2, convert: No non-idealists are reformers. (E). This is the partial contrapositive. Step 3, obvert again: All non-idealists are non-reformers. (A). This is the full contrapositive.

Answer: All non-idealists are non-reformers.

(vi) Elephants never forget. (Give Inverse) (3 marks)

Reduced: "No elephants are creatures that forget." An E proposition.

Inversion infers a proposition whose subject is the contradictory of the original subject. For an E proposition the chain is:

Step 1, convert: No creatures that forget are elephants. (E) Step 2, obvert: All creatures that forget are non-elephants. (A) Step 3, convert by limitation: Some non-elephants are creatures that forget. (I)

Answer: Some non-elephants are creatures that forget.