BLS LLB 5 Years Sem 1 Logic 1 2023-24 Question Paper with Solutions

A component of a compound proposition is a part of it which is itself a proposition, and which, if it were replaced by any other proposition, would leave the whole a meaningful proposition.

A constituent is any part of a proposition whatever, whether or not it is itself a proposition, and whether or not it can be replaced in that way.

Examples. In "Rama is honest and Shyam is diligent", the part "Rama is honest" is both a constituent and a component: put any other proposition in its place and the whole remains a proposition.

In "The man who is tall is clever", the part "the man is tall" is a constituent but not a component: substituting another proposition for it produces nonsense, not a proposition.

This is a relational proposition: it asserts a relation between two named individuals rather than attributing a predicate to a subject.

Tms

where T = "... is taller than ...", m = Mahesh and s = Suresh.

The order of the constants is part of the proposition. Tms and Tsm say opposite things, which is exactly what a subject-predicate form could not record.

The definiendum is the term to be defined: the word whose meaning a definition sets out to fix. The expression which does the defining is the definiens.

Example: in "A contract is an agreement enforceable by law", the definiendum is "contract" and the definiens is "an agreement enforceable by law".

A singular term denotes a single definite individual and only that individual. Its kinds are proper names (Rama, the Ganga), descriptive phrases that pick out one individual (the present Chief Justice of India), and demonstratives with a common noun (this book).

A general term denotes each of an indefinite number of individuals, and connotes the attributes they share. Examples: man, table, advocate, contract. It applies to its members distributively, one at a time.

A third kind, the collective term, denotes a group taken as a whole and does not apply to the members individually: army, jury, Parliament.

For full marks, cover: all three terms, since this version of the question names soundness explicitly; what each applies to; the six combinations; the one combination that cannot occur; the definition of soundness and why it is the practical goal; and the legal application.

Truth and validity

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 and not on the material truth of what is asserted, which is why logic is called a formal science.

The six combinations

Soundness

An argument is sound when it is valid AND all its premises are true.

Soundness is what anyone actually wants, because only a sound argument guarantees a true conclusion. Validity by itself guarantees only that no truth has been lost between the premises and the conclusion; feed it falsehoods and it will carry them faithfully through, which is what the second and third rows show.

The three ideas therefore stack:

1. Truth is a property of each premise, tested against the facts.

For full marks, cover: the definition with examples; the three kinds of connotation; the law of inverse variation with its limits; the terms that have no connotation; the link to definition; and the legal application.

The definition

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.

It is contrasted with the denotation, or extension, which is the range of individuals or classes the term applies to. Connotation is what the word means; denotation is what it covers.

The three kinds

Its limits. The law holds only along a single line of subordination, genus to species to sub-species. It does not hold where the attribute added belongs to every member already, since "rational man" adds a word and removes nobody, and it does not hold between two terms not related as genus and species.

Terms without connotation

Proper names, on the traditional view, denote an individual and connote nothing, because they identify without describing. J.S. Mill put it that a proper name is a mark set on an individual, not a description of him: "Rama" would go on naming the same person if every one of his attributes changed.

It follows that a proper name cannot be defined per genus et differentiam, since an individual has no differentia within a species, and can only be explained by describing the individual.

For full marks, cover: who Wigmore was and what problem the method solves; the two devices, the Key-List and the Chart; the ladder of probanda; how the method is worked in seven steps; what "fact management" adds in modern practice; the logic behind it; and its merits and criticisms.

The problem it solves

John Henry Wigmore (1863 to 1943), the American authority on the law of evidence, devised a method for the logical analysis of a mass of mixed evidence in a contested case. He set it out in The Principles of Judicial Proof (1913), later revised as The Science of Judicial Proof (1937).

His complaint was that the law of evidence taught admissibility, that is, what may be put before the court, and taught nothing at all about proof, that is, what the mass of admitted evidence actually establishes. The chart method was his answer.

2. The penultimate probanda: the material facts which together establish the ultimate probandum, that is, the elements of the offence or the cause of action.
3. The interim probanda: the intermediate propositions inferred from the evidence and offered in support of the penultimate probanda.
4. The evidentiary facts: what the witnesses and documents actually say.

How the method is worked

1. Clarify the standpoint: whose case is being analysed, at what stage and for what purpose.
2. Formulate the ultimate probandum from the substantive law.
3. Break it into penultimate probanda, the elements that must each be proved.
4. Formulate a provisional key-list of every relevant proposition, numbered.
5. Chart the propositions, drawing an arrow for each inferential step.
6. Test the chart, asking of each arrow what generalisation licenses it and how strong that generalisation is.
7. Revise, since the first chart always exposes gaps and unstated assumptions.

Merits and criticisms

Merits: it makes every inferential step explicit; it shows the whole case as one structure instead of a list of witnesses; it exposes gaps early; and it is neutral, so the same chart can be built for either side.

Criticisms: it is laborious and quickly becomes unreadable on a large case; the symbols have to be learned; it gives no way of measuring the weight of an inference beyond the analyst's judgement; and no advocate has time to chart every case, so in practice it is used on the difficult ones and taught as a discipline rather than a routine.

2. Everything must be either S or not-S, by the Law of Excluded Middle, so nothing can fall outside them.

It also uses a single fundamentum divisionis by construction, namely the presence or absence of the chosen attribute. Dichotomy is therefore the only form of division that is formally guaranteed to be correct.

Its defect

The negative member is wholly indeterminate. "Non-introvert" tells us only what its members are not, and it lumps together everything in the universe of discourse that is not introverted. The division is safe and says almost nothing about half of what it divides.

A second and related defect is that the division may be unequal to the point of uselessness: dividing animals into elephants and non-elephants is faultless and worthless.

Its proper use

Dichotomy is used as a first step, and the negative member is then replaced by positive species at the next step:

Kind: O proposition, particular negative. Distributed: the predicate, "vegetarians", only.

Reason: "frequently" is a word of time doing the work of a particular quantity sign, exactly as "always" does the work of a universal one. It asserts of a part of the subject and no more. The proposition is negative, so its predicate is distributed.

(ii) Not a single artist is a painter.

No artists are painters.

Kind: E proposition, universal negative. Distributed: both terms, "artists" and "painters".

Reason: "not a single" is an emphatic form of "no", and denies the predicate of the whole of the subject. A universal negative distributes its subject because the whole of it is spoken of, and its predicate because to shut the subject out of a class is to shut it out of every member of that class.

(iii) Lawyers are invariably analytical.

All lawyers are analytical persons.

Kind: A proposition, universal affirmative. Distributed: the subject, "lawyers", only.

(i) "He is clever and he is either rich or lucky"

Identification: a compound proposition with two connectives. The main connective is the conjunction; a disjunction is embedded in its second conjunct.

• Let p = He is clever.
• Let q = He is rich.
• Let r = He is lucky.

Symbolic form: p · (q v r)

The brackets are part of the answer. Without them, p · q v r is ambiguous, and the other reading, (p · q) v r, is a different proposition altogether: it would be made true by his merely being lucky, which the English does not allow.

An O proposition. The equivalence is the first law of quantifier negation: ~(x)Fx is the same as (∃x)~Fx.

(b) A few women are emotional. (Wx, Ex)

(∃x)(Wx · Ex)

An I proposition. "A few", with the article, is affirmative.

(c) All animals are rational. (Px, Cx)

(x)(Px ⊃ Cx)

An A proposition, using the paper's own key, in which P = "... is an animal" and C = "... is rational".

Two notes. The key's letters do not match the words, since A and R would be the natural choices; the answer follows the key as printed, because a symbolisation is judged by whether it is consistent, not by whether the letters are mnemonic. And the proposition is false as a matter of fact, which does not affect the symbolisation at all: logic symbolises what is asserted, not what is true.

Its limits: an ostensive definition depends entirely on the situation, so it fails the moment nothing is at hand to point at; and it cannot show which feature of the object the word picks out, so a listener shown a desk might take "desk" to mean anything wooden, anything brown, or that particular piece of furniture.

(ii) Hexagon means a polygon having six sides.

Kind: a definition by genus and difference, that is, per genus et differentiam. A connotative technique, and a lexical definition. It is correct.

Reasons: the proximate genus is "polygon" and the differentia is "having six sides". It is exactly co-extensive: every hexagon is a six-sided polygon and every six-sided polygon is a hexagon, so it converts in both directions. It is not circular, not obscure, not negative and not redundant. This is the only one of the three that is a real definition, because it alone analyses the connotation.

(iii) Planet means Neptune, Mars, Earth, Jupiter and Saturn.

Kind: an extensive definition, that is, a definition by enumeration. A denotative technique. It is defective.

The rules of logical division

1. Only one fundamentum divisionis at each step; breach is cross-division.
2. The members must be mutually exclusive.
3. The division must be exhaustive.
4. It must proceed step by step; breach is the saltus in dividendo.
5. Every member must be a species of the genus divided.

(i) Students into rich, poor, tall and short.

Fallacy: cross-division, and the members are not mutually exclusive.

(iii) Watch into time-piece and guard.

Fallacy: this is not a division at all. The word "watch" is being used in two different senses, so no single class is being divided.

Reasons: a watch in the sense of a timepiece and a watch in the sense of a guard or sentry are two different things sharing one word. A division must divide one class, and here there is no one class: the supposed genus is an ambiguous term, so the two members are not species of anything.

This breaks rule 5, and behind it the Law of Identity, which requires a term to keep the same meaning throughout. The proper name for the fault is equivocation, and what has been divided is a word and not a class.

Compare: "Bank into a financial institution and the side of a river" would be the same mistake.

For full marks, cover: a short definition; then the significance under numbered heads covering the whole working life of a lawyer, not just the courtroom, since the question says "comprehensive use in the field of law"; each head with a section, a case or a worked example; the fallacies met in practice; the limits; and a conclusion.

Definition, briefly

Logic is the science and the art of reasoning, the study of the methods and principles by which correct reasoning is distinguished from incorrect. It is normative, because it states how we ought to reason, and formal, because validity is a property of the structure of an argument.

Law is made of propositions and worked by inference from them, which is why the connection is not decorative.

Schedules and lists of categories are divisions and are judged by the rules of division. A taxing or licensing schedule that mixes bases produces overlapping entries, an assessee who falls under two at once, and litigation. Section 10 of the Contract Act is a conjunction of conditions, so all must be satisfied; a section penalising one who "sells or offers for sale" is a disjunction, so either will do. Whether a clause reads "and" or "or" is a question the propositional calculus answers exactly.

3. Advice and opinion writing

An opinion is an argument, and its worth depends on soundness, which is validity plus true premises. Separating the two is what allows a lawyer to say honestly that a client's case is good in law and weak on the facts, or the reverse.

4. Pleadings and issues

Order XIV of the Code of Civil Procedure requires issues to be framed on the material propositions affirmed by one party and denied by the other, which is an exercise in isolating contradictory pairs. A written statement that admits and denies the same fact offends the Law of Contradiction and is bad for that reason.

Precedent is argued by analogy, and the test of a good analogy, relevance of the resemblance to the point inferred, is the whole technique of following and distinguishing.

7. Judgment writing

A judgment must show that its conclusion follows from its findings. A conclusion that does not follow is a non sequitur, and it is a ground of appeal.

The fallacies met in practice

Naming one is often the whole of answering it: ad hominem, attacking the person; ad misericordiam, appealing to pity; ad populum, appealing to popular feeling; petitio principii, assuming what must be proved; ignoratio elenchi, proving something other than the point in issue; false analogy; hasty generalisation; and the undistributed middle, which is the commonest formal fault in a legal argument that sounds plausible.

For full marks, cover: that this is the classification by relation, and where it sits among the four bases; the categorical proposition with the A, E, I, O scheme; the conditional in both its kinds, hypothetical and disjunctive, each with its parts, its terminology and its examples; the inference built on each, that is, the three syllogisms with their valid moods and their characteristic fallacies; the reduction of conditionals to categoricals; the modern re-expression; and a conclusion.

Where this classification sits

Traditional logic classifies propositions on four bases: quantity, quality, relation and modality. The statement in the question is the classification by relation, that is, by how the predicate is related to the subject, and specifically by whether the assertion is made outright or under a condition.

The categorical proposition

A categorical proposition asserts the predicate of the subject unconditionally, without any if or any either.

If a person commits theft, then he is punishable.

Its parts are the antecedent (the if-clause) and the consequent (the then-clause). ⚠️ Neither part is asserted. The sentence above does not assert that anyone has committed theft, nor that anyone is punishable; it asserts only the connection between the two.

2. The disjunctive proposition, of the form "Either ... or ...".

Either the accused confesses or the prosecution proves the charge.

Its parts are the alternatives or disjuncts, and again neither is asserted by itself. The alternatives may be taken in the weak or inclusive sense, at least one and possibly both, or in the strong or exclusive sense, one but not both.

Some texts add a third kind, the conjunctive proposition, of the form "not both ... and ...": A man cannot be both a judge and an advocate in the same cause.

The inference built on each

This is where the classification earns its keep, because each kind of proposition supports a different syllogism.

This is valid in both senses of "or". But arguing from the affirmation of one alternative to the denial of the other, "Either p or q. p. Therefore not q", is valid only in the exclusive sense, and is a fallacy where the disjunction is inclusive.

Reduction of conditionals to categoricals

Traditional logic treats the categorical as the basic form and shows that a conditional can be reduced to it. "If a person commits theft, then he is punishable" becomes the A proposition "All persons who commit theft are persons who are punishable". A disjunctive can be reduced through a hypothetical: "Either p or q" becomes "If not p, then q".

The reduction is what allows one set of rules, those of the categorical syllogism, to govern the whole of traditional deductive logic.

The modern re-expression

Modern logic keeps the distinction and recasts it as a matter of truth-functional connectives. The hypothetical becomes the conditional p ⊃ q, false only when the antecedent is true and the consequent false; the disjunctive becomes p v q, false only when both disjuncts are false; the conjunctive becomes ~(p · q).

For full marks, cover: induction and the inductive leap; the definition of simple enumeration with examples; its characteristics as numbered points; the grounds on which it rests; Bacon's criticism and Mill's qualified defence; the conditions under which it grows strong; the comparison with perfect induction and with scientific induction in a table; its value and its place in enquiry; the legal application; and a conclusion.

Induction and the inductive leap

Induction is the process of inferring a general proposition from the observation of particular instances. Because the conclusion says more than the premises, there is always a gap between the evidence and what is concluded, and that gap is called the inductive leap. Every kind of induction except perfect induction takes it.

The definition

Induction by simple enumeration is that form of induction in which a general conclusion is drawn merely from the fact that all the observed instances agree, and no contrary instance has been observed.

The grounds it rests on

Like every induction it assumes the uniformity of nature, that the future will resemble the past and the unobserved the observed. Unlike scientific induction it does not rest on the law of universal causation, because it discovers no cause at all: it records that A and B have gone together and infers that they always will.

That is its whole weakness. A run of agreeing instances is consistent with a causal connection and equally consistent with a coincidence, and simple enumeration has no way of telling the two apart.

Bacon's criticism

Francis Bacon, in the Novum Organum, dismissed the method: inductio per enumerationem simplicem, ubi non reperitur instantia contradictoria, res puerilis est, that is, induction by simple enumeration, where no contradictory instance is found, is a childish thing.

His objections were three. It counts instances instead of weighing them. It is passive, waiting for nature to present cases instead of putting questions to nature by experiment. And it does not search for the negative instance, which is the one thing that would test it; it merely notes that none has turned up.

Its value and its place

It is not worthless, and an answer that treats it as merely a mistake has missed the point.

1. Most of the working beliefs of ordinary life rest on it, and they serve well enough.

Conclusion

Induction by simple enumeration is the weakest form of induction and the most used. It generalises on a run of agreeing instances without discovering why they agree, so its conclusion is probable and one exception destroys it. Bacon was right that it is not the method of science; Mill was right that it cannot be dispensed with. Its proper place is as the beginning of enquiry, to be replaced by scientific induction wherever a cause can be found and a hypothesis tested.

~(p · ~p)

Example: "This agreement is void" and "This agreement is not void" cannot both be true of the same agreement at the same time.

3. The Law of Excluded Middle. Everything must either be or not be; there is no third possibility between a proposition and its denial.

p v ~p

Example: an agreement either is void or is not void; there is no middle state.

What they are

They are called laws of thought, but they are not descriptions of how minds work, since people contradict themselves daily. They state the conditions on which thought can be valid, which is why logic is a normative and not a positive science. They are also axioms: they cannot be proved, because any proof offered would already use them.

The three are one requirement seen from three sides. Identity says what a thing is; Contradiction denies that it can also be its opposite; Excluded Middle denies that it can be neither.

2. Change escapes them: Hegel and the dialectical school argued that a thing in the process of becoming both is and is not what it is turning into.
3. Many-valued logics abandon Excluded Middle, admitting a third value for propositions that are neither true nor false.

The usual answer is that they are not empirical claims but the conditions of consistent discourse: an argument that abandons them cannot be contradicted, and so cannot be argued with at all.

(ii) Logical Division (6 marks)

The definition and its elements

Logical division is the process of separating a class, called the genus, into the sub-classes or species contained under it, on the basis of a single attribute.

1. the totum divisum, the whole or genus divided;
2. the membra dividentia, the dividing members, that is, the species;
3. the fundamentum divisionis, the single attribute on which the division rests.

Example: triangle, divided by the length of the sides, gives equilateral, isosceles and scalene.

The test between the first two is to predicate the name of the whole of each member: "an epic is a poem" passes, "a handle is an umbrella" fails.

Dichotomy

Division by a pair of contradictory terms. Because contradictories are exclusive and exhaustive, dichotomy can never break rules 2 or 3, which makes it the only formally guaranteed division; its defect is that the negative member is indeterminate.

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. Only universals have contraries; only particulars have sub-contraries; only universals have subalterns beneath them.

Conversion, subject to the rule that no term may be distributed in the converse unless it was distributed in the convertend: A converts by limitation to I; E and I convert simply; O cannot be converted.

Obversion, which changes the quality and replaces the predicate by its contradictory, works for every form: A gives E, E gives A, I gives O, O gives I.

(i) Some poets are not Spartans. (Give Sub-contrary and Contrary) (2 marks)

An O proposition.

• Sub-contrary (I): Some poets are Spartans. Two sub-contraries cannot both be false, though they may both be true.

Reason: the rule of contraries is that they cannot both be true, but may both be false. So the truth of one settles the other as false, and the falsity of one settles nothing. Here E is false, which tells us only that it is not the case that no spoiled child is attractive; some may be attractive and some not, in which case A is false too, or all may be attractive, in which case A is true. Both remain open.

Note what the item asks for. It does not ask for a proposition; it asks for a truth value, and "doubtful" or "undetermined" is the complete answer. What is settled by E being false is the contradictory, the I proposition "Some spoiled children are attractive", which must be true.

(iv) Some Marxians are Socialists. (Give Converse and Obverse) (2 marks)

An I proposition.

• Converse (I): Some Socialists are Marxians. An I proposition converts simply, because it distributes neither term, so nothing can go wrong.
• Obverse (O): Some Marxians are not non-Socialists.

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

Answer: All non-idealists are non-reformers.