BLS LLB 5 Years Sem 1 Logic 1 2022-23 - ATKT 60/40 Question Paper with Solutions

Two terms are contradictory when one is the simple negative of the other, so that between them they exhaust the universe of discourse: nothing falls under both and nothing falls outside both. Examples: man and not-man; white and not-white; legal and illegal.

Two terms are contrary when they are the two extremes of the same series, so that they exclude each other but do not exhaust the field. Examples: white and black; rich and poor; hot and cold. A thing may be neither white nor black, but everything must be either white or not-white.

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 connotation of "man" is animality together with rationality.

The denotation, or extension, of a term is the range of individuals or classes to which the term applies. The denotation of "man" is Rama, Shyam, Fatima and every other human being.

Law of inverse variation: as connotation increases, denotation decreases. Man, then Indian man, then educated Indian man.

The Law of Excluded Middle is the third of the three laws of thought. It states that everything must either be or not be: between a proposition and its denial there is no third possibility.

p v ~p, or "everything is either A or not-A"

Example: an agreement either is void or is not void. There is no middle state and no third thing it could be.

Induction 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.

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 the two, and that gap is the inductive leap. Induction rests on two postulates: the uniformity of nature and the law of universal causation.

A variable is BOUND when it falls within the scope of a quantifier that governs it, and FREE when no quantifier reaches it.

Examples:

• In (x)(Fx ⊃ Gx), every occurrence of x is bound, because the quantifier (x) governs the whole formula.
• In Fx ⊃ (x)Gx, the first x is free and the second is bound, because the quantifier governs only Gx.

The test that matters: an expression containing a free variable is a propositional function and is neither true nor false; an expression with no free variable is a proposition and is true or false.

For full marks, cover: both definitions; the act against its expression; the third term, sentence; the table of differences; the parts of a proposition; the pattern that runs through logic; and why logic works on the proposition.

Judgement

A judgement is the mental act by which the mind affirms or denies something of something else. It is an act; it happens at a moment, in a particular mind; and it belongs to psychology.

Proposition

A proposition is a judgement expressed in words: a statement in which something is affirmed or denied of something else, and which is therefore either true or false. It belongs to logic.

It has three parts: the subject, the predicate and the copula. In "All men are mortal", "men" is the subject, "mortal" the predicate and "are" the copula.

In every 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. The moment the judgement is put into words it becomes something another person can test, contradict and infer from. Logic therefore leaves the act to psychology and works on the expression.

General term

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, that is, one at a time: "man" applies to Rama, and separately to Shyam.

Collective term

A collective term denotes a group taken as a whole and does not apply to the members individually. Examples: army, jury, Parliament, library. A soldier is not an army, whereas a man is a man.

⚠️ The same word may be general or collective according to use. "Jury" in "the jury returned a verdict" is collective; in "juries are drawn from the electoral roll" it is general, standing for each jury.

The differences

The modern treatment

Modern logic writes a singular term as an individual constant, a, b, c, and a general term as a predicate, Sx, Px. A singular proposition is therefore Ma, a predicate with a constant, and needs no quantifier; a general proposition needs one.

The fallacies of composition and division

The fallacy of composition argues from what is true of each member to what is true of the whole: every juror is fallible, so the jury is fallible. The fallacy of division argues the other way: the committee decided unanimously, so each member decided unanimously. Both come of treating a collective term as though it were general.

4. To supply the major premise of an argument. "A contract is an agreement enforceable by law" is a definition and also the premise from which it follows that an unenforceable agreement is no contract.

5. To introduce or fix a technical vocabulary, which is what every statutory definition clause does.

6. To increase knowledge of the thing itself, since framing a real definition forces an analysis that merely naming it does not.

The modern classification, by purpose

1. Stipulative: assigns a meaning by declaration; cannot be true or false.
2. Lexical: reports the meaning a word already has; can be true or false.
3. Precising: reduces vagueness at the borderline, as when a statute fixes an age of majority.
4. Theoretical: embodies a theory of the thing defined.
5. Persuasive: attaches an attitude to a term while wearing the appearance of a report of meaning.

For full marks, cover: the statute; both definitions with their sections; the fourfold classification and the two bases it uses; the deeming provisions and the Schedule; the compensation that follows; and, because this is a logic paper, what the classification illustrates about definition and division.

The statute

Disablement is defined by the Employee's Compensation Act 1923, formerly the Workmen's Compensation Act, which makes an employer liable to pay compensation for personal injury caused to an employee by accident arising out of and in the course of employment.

Disablement in this sense is the loss or reduction of earning capacity resulting from such an injury. The Act measures it by earning capacity, not by the physical injury as such: two people with the same wound may suffer different disablement.

Why the definitions are drawn as they are

The two limbs of Section 2(1)(g) differ deliberately and that is where the cases are fought. Temporary partial disablement is measured against the employment he was in; permanent partial disablement against every employment he could then have undertaken. A clerk who loses a finger may be unable to do his own job for a month and yet remain able to do it permanently, so the two limbs give different answers on the same facts.

Compensation

Section 4 fixes the amount and follows the classification: death and permanent total disablement by a percentage of monthly wages multiplied by a relevant factor; permanent partial disablement by the percentage of loss of earning capacity in Schedule I; and temporary disablement by a half-monthly payment.

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

Universals distribute the subject; negatives distribute the predicate.

(iii) A few doctors are social workers.

Some doctors are social workers. (I proposition)

Distributed: neither term.

Reason: "a few", with the article, is a sign of particular quantity and is affirmative, unlike the bare "few", which carries a negative force and would give an O proposition.

Reading of the table: the biconditional is true when both components have the same truth value and false when they differ. It is contingent, neither a tautology nor a contradiction.

What the proposition claims. Because it is a biconditional, merit is asserted to be both a sufficient and a necessary condition of admission: not only will the meritorious be admitted, but nobody else will. The third row is what makes that a strong claim, since it is falsified by a single admission granted without merit.

(ii) Symbolise with quantifiers

(a) All doctors are kind. (Dx, Kx)

(x)(Dx ⊃ Kx), an A proposition.

(b) No scholars are ambitious. (Sx, Ax)

(x)(Sx ⊃ ~Ax), an E proposition. Equivalently ~(∃x)(Sx · Ax).

The kinds of definition to choose from

By technique. Denotative: by example, by enumeration, ostensive. Connotative: synonymous, operational, genus and difference. By purpose: stipulative, lexical, precising, theoretical, persuasive.

(i) The child is father of the man.

Kind: not a definition at all, but a figurative statement. Judged as a definition it breaks the rule against metaphorical language.

Reasons: the line is Wordsworth's, and it means that the character formed in childhood shapes the adult. Taken as a definition it is absurd, since a child is not literally anybody's father, and it states no genus and no differentia. A definition must be clearer than the term defined, and this offers a paradox in place of an analysis.

Reasons: the term is defined by listing members instead of stating attributes, and the word "etc." is the definition's own admission that the list is incomplete. It is also badly chosen: the four named are all mammals, so a reader would have no reason to include a fish, a bird or an insect, and the definition therefore misleads about the extent of the class as well as failing to exhaust it.

Above all it says nothing about why those creatures belong together, so a reader meeting a new creature could not tell whether it was an animal.

Corrected: an animal is a living organism which feeds on organic matter, has sense organs and a nervous system, and can move of its own accord.

Reasons: two bases at one step. "Vocal" and "instrumental" divide music by the medium, that is, whether it is sung or played; "classical" and "film music" divide it by genre or origin. Rule 1 is broken, and rule 2 falls with it, because classical music may be vocal and film music is usually both, so a single piece belongs to three members at once.

Sound divisions: music into vocal and instrumental, by medium; or into classical, folk, film and popular, by genre.

(ii) Cloth into cotton, silk, rayon, nylon and costly.

Fallacy: cross-division, and the members overlap.

Reasons: cotton, silk, rayon and nylon divide cloth by the material of which it is made; "costly" divides it by price. Two bases at one step, so rule 1 is broken, and the members are no longer exclusive, because a costly cloth may perfectly well be silk. Remove "costly" and the remaining four are a sound division by material, though still not exhaustive while wool and linen are missing.

(iii) Educational institutions into schools and colleges.

Fallacy: the division is incomplete.

The first is too wide. Thought includes memory, imagination and daydreaming, and logic examines none of them. The later definitions narrow the subject to reasoning, and Copi's adds that logic discriminates: it separates the correct from the incorrect rather than describing either.

The nature of logic

1. Both a science and an art. A science, because it is a systematic body of general truths about the conditions of valid inference; an art, because it lays down rules for the practice of reasoning and the detection of fallacies. The relation is that of anatomy to surgery. The question is never "which", but "in what sense each".

2. A normative science. It studies how we ought to reason, not how we do. Psychology describes actual mental processes, mistakes included; logic supplies the standard by which they are judged, which is why it belongs with ethics and aesthetics.

3. A formal science. It is concerned with the form of an argument and not with the material truth of its premises. That is why one test serves an argument about crows and an argument about contracts.

For full marks, cover: what the fourfold scheme is and the two bases it rests on; quantity with its sign words; quality with the rule about the copula; the four forms in a table with examples; the letters and their origin; the distribution table with the reasoning behind each cell; the propositions that do not fit; the uses of the scheme; and the modern re-expression.

What the fourfold scheme is

The fourfold scheme divides categorical propositions into A, E, I and O, by combining the two divisions of quantity with the two divisions of quality. It is the foundation of the whole of traditional deductive logic, because opposition, eduction and the syllogism are all stated in terms of these four forms.

Quantity

Settled by how much of the subject is spoken of.

1. Universal: the predicate is affirmed or denied of the whole subject. Signs: all, every, any, no, none, whoever, always, never, not a single.

The letters come from the Latin AffIrmo, I affirm, and nEgO, I deny: the first two vowels of each give the two affirmative and the two negative forms.

Distribution, which follows from the pair

A term is distributed when the proposition speaks of every member of the class it names.

Universals distribute the subject; negatives distribute the predicate. Quantity governs the subject, quality governs the predicate.

2. Eduction. Conversion and obversion are stated form by form, and their rules follow from distribution: A cannot be converted simply, O cannot be converted 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 and rewrites them with quantifiers: (x)(Sx ⊃ Px), (x)(Sx ⊃ ~Px), (∃x)(Sx · Px), (∃x)(Sx · ~Px). The change is not cosmetic: a universal is read as a denial that asserts nothing to exist, so A no longer implies I and E no longer implies O, and subalternation, contrariety and sub-contrariety all fail.

Conclusion

The fourfold scheme 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. That economy is why the scheme lasted two thousand years.

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 never valid or invalid, only strong or weak.

The conditions of a good analogy

1. The number of resembling points should be large. Every further agreement makes coincidence less likely.

2. The resemblances must be relevant to the property inferred. This is the decisive condition and every other is subordinate to it. A resemblance strengthens the argument only if it is causally connected with the property concluded to. Two cars of the same colour tell us nothing about their engines; two of the same make and model tell us a great deal. Ten irrelevant resemblances are worth less than one relevant one.

3. The differences should be few, and none should bear on the property inferred.

4. The number of instances compared should be large. An analogy drawn from many pairs begins to approach an induction.

Analogy also fills gaps where no rule covers a case, as when the duty of care in Donoghue v Stevenson [1932] AC 562 was extended from a manufacturer of ginger beer to manufacturers generally; and ejusdem generis and noscitur a sociis are rules for reasoning from likeness.

Its limits are firm. In criminal law analogy is forbidden, because no act is an offence unless the law makes it one and Article 20(1) of the Constitution would be defeated. It cannot override an express provision. And it proves nothing by itself.

Conclusion

An analogical argument is strong or weak, never valid or invalid, and its strength is measured by the six conditions above, of which relevance is the master condition. Analogy is the weakest form of inference in logic and among the most used in law, because 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.

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.

Conversion: 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) A few writers are teachers. (State Contradictory)

An I proposition. Contradictory (E): No writers are teachers.

(ii) Girls obey their parents. (State Contrary)

Reduced: "All girls are persons who obey their parents." An A proposition, since the indefinite sentence states a general truth.

Contrary (E): No girls are persons who obey their parents.

Reason: a singular proposition has no quantity in the ordinary sense, because its subject is one individual and the whole of it is spoken of. Contrariety, sub-contrariety and subalternation all depend on a difference of quantity, so none of them is available. Changing the quality alone gives the contradictory, and the two cannot both be true and cannot both be false.

⚠️ Traditional logic treats a singular as universal so that it may be used in a syllogism, which would make these two contraries and allow both to be false. That convention gives the wrong answer here, and modern logic avoids it by writing the pair as Ta and ~Ta, plain contradictories.

(vii) No virtuous man is addicted to drugs. (State Converse)

An E proposition. Converse (E): No person addicted to drugs is a virtuous man.

E converts simply, because both its terms are distributed in the original.

(viii) Ideal teachers never involve in politics. (State Converse)

Reduced: "No ideal teachers are persons who involve in politics." An E proposition, since "never" is a universal sign.

Obverse (I): Some men are non-poets.

Which is meant: ordinary English uses "all... not" for the second, and it is also the true proposition, so the idiomatic reading is the natural one. Say which you have taken and why, because a reduction that hides an ambiguity is worse than one that names it.