BLS LLB 5 Years Sem 1 Logic 1 2018-19 Question Paper with Solutions

Inductive reasoning is the process of inferring a general proposition from the observation of particular instances, the conclusion going beyond the evidence and being therefore only probable.

Example:

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

Because the conclusion asserts more than the premises, there is always a gap between them, and that gap is the inductive leap. Induction rests on two postulates: the uniformity of nature and the law of universal causation.

Obversion is a form of eduction, that is, of immediate inference, in which the quality of the proposition is changed and the predicate is replaced by its contradictory, the meaning remaining exactly the same.

Example: "All men are mortal" (A) obverts to "No men are non-mortal" (E).

It is valid for all four forms:

An extensive definition, also called a definition by denotation, defines a term by enumerating the individuals or the species which it denotes, instead of stating the attributes which it implies.

Example: "By a metal is meant a substance such as gold, silver, copper, iron and aluminium."

It is a verbal or nominal definition, not a real one: it shows what the word covers without saying what makes those things what they are.

Metaphysical division is the separation of a whole into the attributes or qualities which exist together in it and cannot exist apart from it or from one another.

Example: dividing man into animality and rationality; or an orange into its colour, shape, taste and smell.

⚠️ It is not a logical division, because the members are not species of the thing divided: animality is not a kind of man.

Example: "All men are mortal."

(x)(Mx ⊃ Tx), where M = "... is a man" and T = "... is mortal".

A general proposition is one formed by quantifying a propositional function, that is, one about a class rather than a named individual. It is universal where the quantifier is the universal quantifier (x), and existential where it is (∃x).

For full marks, cover: what each word applies to; the definitions; the six combinations; the one 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.
• 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 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.

The differences

The pattern

The same relation appears three times: conception and term, judgement and proposition, inference and argument. In each pair the first belongs to psychology and the second to logic.

Why logic works on the proposition

A mental act cannot be examined by anyone but the person who performs it. Put into words it becomes something another person can test, contradict and infer from.

The conditions of validity

The inference is generally valid, but only if the added phrase satisfies two conditions.

1. It must be added to both terms, and in the same words.
2. It must mean exactly the same in both places. The Law of Identity governs the addition, and the moment the phrase shifts its sense the inference fails.

Where it fails

⚠️ It fails where the added relation does not hold uniformly of the wider class.

A dog is an animal.
Therefore a picture of a dog is a picture of an animal. (valid)

A cow is an animal.
Therefore the owner of a cow is the owner of an animal. (valid)

A poison is a substance.
Therefore a fear of poison is a fear of a substance. (fails)

The last fails because "fear of" creates an intensional context: a person may fear poison without fearing substances in general, since what is feared is the thing under a description.

For full marks, cover: what a definition does; the two rules chosen, each stated, explained and shown breached with a worked example and a correction; and the remaining rules listed briefly. The question asks for any two, so two are taken in full.

A definition marks off the connotation of a term. The term defined is the definiendum, the defining expression the definiens.

Rule 1 chosen: a definition must be co-extensive with the term defined

Statement. The definition must apply to everything the term applies to and to nothing else: neither too wide nor too narrow. The test is convertibility, that is, reading the definition backwards.

Fallacy, too wide: "A square is a four sided plane figure." Every rectangle and rhombus satisfies it, so it covers far more than the term. The differentia has been left out.

2. It must not be in obscure or figurative language: "necessity is the mother of invention" is a metaphor.
3. It must not be negative where it can be affirmative, except where the term is privative.

Public nuisance

Section 268 of the Indian Penal Code 1860, now Section 270 of the Bharatiya Nyaya Sanhita 2023: an act or illegal omission causing common injury, danger or annoyance to the public, or to people in general in the vicinity. It is a crime, punishable under Section 290 IPC (Section 292 BNS).

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 (Campbell v Paddington Corporation [1911] 1 KB 869).

The differences

5. Its probability rises with the number and the variety of instances.
6. It is passive: it waits for instances instead of contriving them.

Bacon and Mill

Bacon dismissed it as childish, res puerilis: it counts instances instead of weighing them, waits on nature instead of questioning her, and never searches for the negative instance.

Mill kept it: where the instances are very numerous and very various and no exception is found despite search, the probability becomes indistinguishable in practice from certainty. Our belief that all men are mortal rests on nothing better.

Compared with scientific induction

Reason: "a few", with the article, is a particular sign and is affirmative, unlike the bare "few", which carries a negative force and would give an O proposition. The past tense verb is carried into the predicate term so that the copula can be the bare present tense "are".

(ii) Educated persons are not the best.

No educated persons are the best persons. (E proposition)

Distributed: both terms.

Reason: the sentence prints no quantity sign, and an indefinite proposition stating a general truth is read as universal; the "not" attaches to the copula, so it is negative. An E proposition distributes both terms.

The alternative reading: if only some educated persons are meant, the proposition is O, "Some educated persons are not the best persons", and only the predicate is distributed. Where an indefinite proposition is genuinely open, state the reading taken and why.

(iii) Old paths are not the best.

No old paths are the best paths. (E proposition)

Distributed: both terms.

(i) "If and only if voters are well informed, democracy will be successful"

Identification: a compound proposition; the connective is the biconditional, or material equivalence.

• Let p = Voters are well informed.
• Let q = Democracy will be successful.

Symbolic form: p ≡ q, equivalently (p ⊃ q) · (q ⊃ p).

Truth table

The refinement worth a line: on Russell's theory of descriptions, a definite description is not a name at all and the proposition unpacks into three claims, that at least one man wrote to me today, that at most one did, and that he thinks clearly. At this level Ca is the expected answer, with the note that the subject is a description and not a name.

2. Jan Sangh hates the Communist Party.

A relational proposition: it asserts a relation between two named entities.

Hjc, where H = "... hates ...", j = Jan Sangh and c = the Communist Party.

The order matters, since Hjc and Hcj say different things. "Hates" is neither symmetrical nor transitive.

3. The drama we staged is a comedy.

A singular proposition, again with a definite description as subject.

Cd, where C = "... is a comedy" and d = the drama we staged.

(i) Peace is freedom from war.

Fallacy: the definition is negative where an affirmative is possible. It breaks the rule against needless negative definitions.

Reasons: peace is a positive condition, a state of concord and public tranquillity, and can be defined affirmatively; the negative form is therefore not forced on the definer as it is with a privative term such as "orphan". It is also too wide on the negative reading, since a state of armed hostility short of declared war is not peace and yet is freedom from war.

The paper prints "freedom from way", which is plainly a misprint for "war"; the answer takes it so.

Corrected: peace is a condition of public order and concord among States or persons, secured by the absence of hostilities.

(ii) Wedding is a ceremony in which two persons undertake to become one, one undertakes to become nothing, and nothing undertakes to become supportable.

Fallacy: the definition is obscure, and it is in truth a persuasive definition dressed as a paradox. It breaks the rule against obscure and figurative language.

Reasons: one fundamentum divisionis is used, and because the two members are contradictories the division is necessarily exhaustive and mutually exclusive. Every rule is satisfied.

The fair criticism is of usefulness, not validity: the negative member is indeterminate, so dichotomy is used as a first step and refined at the next.

(v) Cobras into poisonous and crawling creatures.

Fallacy: cross-division, and a member which is not a species of the genus.

Reasons: "poisonous" divides cobras by whether they are venomous; "crawling creatures" divides by mode of locomotion, so two bases are used at one step. Worse, the two members do not even exclude one another: every cobra is both poisonous and a crawling creature, so the division sorts nothing at all. And "crawling creatures" is a class wider than cobras, so it is not a species of the genus divided but a genus above it.

(vi) Race horses into insured, non-insured, pedigreed, and non-pedigreed.

Fallacy: cross-division. Two separate dichotomies have been run together at one step.

3. It is not confined to any subject matter. Physics studies matter, biology life; logic studies the form of the reasoning they share.
4. It was so treated historically. Aristotle's logical works are the Organon, the instrument, and the medieval schools taught logic first.

The objection, which should be stated. The phrase suggests that logic is superior to the other sciences or can settle their questions. It cannot. Logic can tell a physicist whether an argument is valid; it cannot tell whether the premises are true, and it discovers no fact about the world. It is best read as the science of the method of the sciences, not their queen.

2. Is logic an art?

Yes, and a science as well. It is both, and the two do not compete.

As a science, logic is a systematic body of general truths about the conditions of valid inference; 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.

1. Deduction is not purely formal in use. A valid argument with a false premise is worthless, and what anyone wants is soundness, which is validity plus true premises. Where those premises come from is induction's business.
2. Induction is not purely material. It has a form of its own: Mill's methods are general rules 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, 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, and 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: what a categorical proposition is; the classification by quantity and by quality with sign words; the fourfold scheme with examples and the origin of the letters; the distribution table with the reasoning behind each cell; the propositions that do not fit and how they are forced in; the uses of the scheme; and the modern re-expression.

The categorical proposition

A categorical proposition asserts the predicate of the subject unconditionally, without any if and without any either. It has three parts: subject, predicate and copula.

It is contrasted with the conditional proposition, hypothetical or disjunctive, which asserts under a condition; that is the classification by relation, and it is not what this question is about.

Classification by quantity

Settled by how much of the subject is spoken of.

The letters come from the Latin AffIrmo, I affirm, and nEgO, I deny.

Distribution

Universals distribute the subject; negatives distribute the predicate.

The reasoning. "All men are mortal" speaks of every man, so the subject is distributed; it does not speak of every mortal, only of enough of them to include the men, so the predicate is not. "No men are angels" shuts men out of the whole class of angels, because to exclude a thing from a class is to exclude it from every member, so both terms are distributed.

1. Opposition: the square is built on quantity and quality; contraries differ in quality, subalterns in quantity, contradictories in both.
2. Eduction: the rules of conversion follow from distribution, so A converts by limitation, E and I simply, and O not at all.
3. The syllogism: its rules are rules about distribution, and the undistributed middle and illicit process are breaches of them.

The modern re-expression

Modern logic keeps the four forms and writes them with quantifiers: (x)(Sx ⊃ Px), (x)(Sx ⊃ ~Px), (∃x)(Sx · Px), (∃x)(Sx · ~Px). A universal is read as a denial, asserting nothing to exist, so A no longer implies I and E no longer implies O.

Conclusion

The traditional classification of categorical propositions is quantity and quality taken together, and it is the smallest scheme that will carry the traditional machinery of inference. Two questions, each with two answers, give four forms; the four forms give the distribution table; and the distribution table gives most of the rules of traditional logic.

A compound proposition contains at least one other proposition as a component, joined by a connective.

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.

The 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.

3. Disjunction (p v q), "p or q". The components are disjuncts. Inclusive: false only when both are false. The exclusive sense is (p v q) · ~(p · q).

4. Implication (p ⊃ q), "if p then q". Antecedent and consequent. False only when the antecedent is true and the consequent false. This is material implication.

5. Equivalence (p ≡ q), "p if and only if q". True when both components have the same truth value.

2. Relational, asserting a relation between individuals: "Jan Sangh hates the Communist Party", Hjc. The order matters.
3. General, formed by quantifying a propositional function: universal, (x)(Sx ⊃ Px), or existential, (∃x)(Sx · Px). A universal affirmative is a class-inclusion proposition.

The four traditional forms reappear as (x)(Sx ⊃ Px), (x)(Sx ⊃ ~Px), (∃x)(Sx · Px) and (∃x)(Sx · ~Px).

Statement forms by their truth tables

Tautologous, true on every row, as p v ~p; contradictory, false on every row, as p · ~p; contingent, true on some and false on others, as p ⊃ q.

Legal illustration

Statutes are written in these connectives. 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. Implication is the form of every proviso. Equivalence is what a definition clause asserts.

For full marks, cover: the definition with its strict conditions; the four categorical forms; the drawn square; each of the four forms of opposition with its rule and an illustration; the complete table of inferences; opposition of singular propositions; the modern square; and a legal illustration.

Definition

Opposition is the relation between two propositions which have the same subject and the same predicate, but which differ in quantity, or in quality, or in both.

Three conditions are strict: the subject term must be the same, the predicate term must be the same, and both must be taken in the same sense and at the same time.

Opposition is a form of immediate inference, because the conclusion comes from one premise with no middle term.

The four categorical forms

Taking S as advocates and P as graduates: A, all advocates are graduates; E, no advocates are graduates; I, some advocates are graduates; O, some advocates are not graduates.

Illustration: if it is false that some advocates are graduates, then none is, so it must be true that some are not.

4. Subaltern: A with I, E with O. Same quality, differing in quantity. Truth descends and falsity ascends.

Illustration: if all advocates are graduates, certainly some are; but if it is false that all are, the particular remains doubtful.

The complete table of inferences

The practical use: to defeat a rule stated as "all X are Y", establish its contradictory, which one instance proves. That is what a distinguishing case does.

Its place among the forms of inference

Analogy is neither deduction nor complete induction. Not deduction, because the conclusion can be false while the premises are true; not a full induction, because it does not generalise to a class but moves from particular to particular. Its conclusion is always probable, so an analogy is strong or weak, never valid or invalid.

The conditions of a good analogy

1. The number of resembling points should be large.
2. The resemblances must be relevant to the property inferred. The decisive condition, to which the others are subordinate. Two cars of the same colour tell us nothing about their engines; two of the same make and model tell us a great deal.
3. The differences should be few, and none should bear on the property inferred.
4. The number of instances compared should be large.
5. The instances should be varied, so the resemblance is not an accident of one setting.
6. The conclusion should be modest. The weaker the property claimed, the more probable the conclusion.

3. Statutory interpretation uses analogy in fixed forms. Ejusdem generis, by which general words following an enumeration are confined to the same kind, and noscitur a sociis, by which a word takes colour from its neighbours.

4. Analogy fills gaps. 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.

5. Indian courts use it expressly. In Vishaka v State of Rajasthan AIR 1997 SC 3011 the Supreme Court, finding no statute on sexual harassment at the workplace, framed binding guidelines by reasoning from the constitutional guarantees in Articles 14, 19(1)(g) and 21.

The limits

1. In criminal law analogy is forbidden: no act is an offence unless the law makes it one, penal statutes are construed strictly, and punishing by analogy would defeat Article 20(1).
2. It cannot override an express provision.
3. It proves nothing by itself: the conclusion is probable, and a court reasoning only by analogy has given a reason, not a demonstration.

For full marks, cover: each item with the form of the given proposition named, both answers written out where they exist, and the rule that produces each; 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.

• Contradictory (I): Some men are fallible.
• Subaltern (O): Some men are not fallible.

(iv) Some ideals are not attainable. (Converse and obverse) (2 marks)

An O proposition.

• Converse: an O proposition cannot be converted. It has no converse.

Reason: the attempted converse would be "Some attainable things are not ideals". In the original, "ideals" is the subject of a particular proposition and is undistributed; in the attempted converse it has become the predicate of a negative proposition and is therefore distributed. A term distributed in the converse but undistributed in the convertend breaks the rule of conversion.

• Obverse (I): Some ideals are non-attainable.

(v) No matter is destructible. (Obverse and converse) (2 marks)

An E proposition. The paper prints "distractible", plainly for "destructible"; the form is E either way and the working is identical.

• Obverse (A): All matter is non-destructible, that is, all matter is indestructible.