BLS LLB 5 Years Sem 1 Logic 1 2024-25 - ATKT Set 2 75/25 Question Paper with Solutions

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

Its characteristics:

1. It asserts something: it affirms or denies, and does not merely name.
2. It is necessarily either true or false, and never both and never neither.
3. It has three parts: subject, predicate and copula.
4. It has a quantity, universal or particular, and a quality, affirmative or negative.
5. It is expressed in a sentence but is not the sentence: one proposition may be carried by many sentences.

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:

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 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".

Primary induction is induction proper: the process by which a general proposition is established directly from the observation of particular instances, by observation and experiment, and on the strength of the law of universal causation and the uniformity of nature.

Secondary induction is the process in which no fresh observation is made and a new general truth is deduced from laws already established by primary induction, as when the law of falling bodies is derived from the law of gravitation.

A propositional function is an expression which contains one or more variables and which becomes a proposition when the variable is given a value or is bound by a quantifier. By itself it is neither true nor false.

Example: "x is a lawyer", written Lx. It becomes a proposition in two ways:

1. By instantiation, substituting a constant: "Ambedkar is a lawyer", La, which is true.
2. By generalisation, prefixing a quantifier: (x)Lx, everything is a lawyer, which is false; or (∃x)Lx, something is a lawyer, which is true.

The idea belongs to modern symbolic logic and is due to Bertrand Russell.

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 depends on the form of the argument, not 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 law of inverse variation

As the connotation of a term increases, its denotation decreases, and the other way round.

man → Indian man → educated Indian man → educated Indian man practising law

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.

The three kinds of connotation

1. Subjective: the attributes a particular individual privately associates with the term. Varies from person to person and is useless for logic.
2. Objective: all the attributes the things denoted actually possess, known and unknown. Unusable, because it includes what nobody has yet discovered.
3. Conventional: the attributes fixed by the usage of the language community. This is the only one a definition can state, and the only one logic works with.

Terms with only one of the two

For full marks, cover: what a definition is and its two parts; the purposes under numbered heads; the modern classification of definitions by purpose; the limits of definition; and a legal application.

What a definition is

A definition is a statement which sets out the connotation of a term, that is, the attributes a thing must possess before the term applies to it. The term is the definiendum, the defining expression the definiens.

The purposes it serves

1. To fix the meaning of a term and remove ambiguity. This is the primary purpose. An argument in which a key word shifts its sense is worthless, and a definition anchors it. Breach of that anchoring is the fallacy of equivocation.

2. To mark the term off from every other. A good definition says not only what a thing is but what it is not, by naming the differentia that separates it from the other species of its genus.

4. Theoretical: embodies a theory of the thing defined.
5. Persuasive: attaches a favourable or unfavourable attitude to a term while wearing the appearance of a report of meaning.

What cannot be defined

1. The summum genus, the highest class, such as being or substance, because there is no wider class to serve as genus.
2. Individuals and proper names, because an individual has no differentia within a species.
3. Simple unanalysable qualities, such as red or pleasure, because there is nothing in them to take apart.

Legal application

A definition clause in a statute is a stipulative or precising definition and governs the whole Act unless the context otherwise requires. It is the legislature fixing the connotation so that courts may work out the denotation, and it is why the interpretation section is the first place a lawyer reads. Section 2(h) of the Indian Contract Act 1872 is the model: agreement is the genus, enforceable by law the differentia.

Rule 2. The dividing members must be mutually exclusive. Fallacy: overlapping division. Breach: "Human beings into men, women and doctors."

Rule 3. The division must be exhaustive. Fallacy: incomplete division. Breach: "Vertebrates into fishes, birds and mammals", omitting 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", leaping over prose.

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."

Division against partition

Division separates a class into kinds; partition separates an individual object into its parts. The test is to predicate the name of the whole of each member: "an epic is a poem" passes, "a handle is an umbrella" fails. Partition is not a fault in itself, only when it is offered as a logical division.

Distributed: neither term.

Reason: "Most" is a sign of particular quantity. Logic recognises only two quantities, so most, many, several and a few all reduce to "some": the proposition speaks of a part of the subject and does not say how large a part. Being affirmative as well as particular, it distributes neither term.

(ii) A few men succeeded.

Some men are persons who succeeded. (I proposition)

Distributed: neither term.

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

(iii) Spiritual persons are necessarily sincere.

All spiritual persons are sincere persons. (A proposition)

Distributed: the subject, "spiritual persons", only.

(i) "If and only if you abide by the law, you will not be punished"

The paper prints "It and only if"; the sense is plainly "If and only if", and the answer takes it so.

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

• Let p = You abide by the law.
• Let q = You will not be punished.

Symbolic form: p ≡ q

The consequent is itself negative, so if r = you will be punished, then q is ~r and the proposition may equally be written p ≡ ~r. Both are correct; the first is simpler and is used in the table.

For every x, if x is a doctor then x is kind. An A proposition.

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

(x)(Sx ⊃ ~Ax)

For every x, if x is a scholar then x is not ambitious. An E proposition. It may equally be written ~(∃x)(Sx · Ax): there is nothing that is both.

3. A few women are soldiers. (Wx, Sx)

(∃x)(Wx · Mx)

There is at least one x which is a woman and is a soldier. An I proposition.

⚠️ The paper's own key uses "Sx" twice, for "scholars" in item 2 and for "soldiers" in item 3. Two different predicates cannot share a letter within one piece of work, so "soldier" has been given Mx here. Say so in the answer: choosing distinct symbols is part of symbolising correctly, and an examiner will credit the candidate who notices rather than the one who copies the clash.

Reasons: "animal" is the genus and nothing more has been said. Every horse, dog and crow is an animal, so the definition covers vastly more than the term defined, and it fails the test of convertibility in one direction: every man is an animal, but not every animal is a man. The differentia has been left out.

Corrected: man is a rational animal, which is the classical definition per genus et differentiam.

(ii) Eyes are the windows of the soul.

Fallacy: the definition is expressed in figurative or metaphorical language. It breaks rule 4.

Reasons: an eye is not a window and there is no literal sense in which anything looks through it into a soul. A definition must be clearer than the term defined, and a metaphor substitutes a picture for an analysis. The sentence also states no genus and no differentia, so it is not a definition at all but an epigram, and it is not co-extensive either.

Corrected: the eye is the organ of sight in animals.

(iii) Light is the absence of darkness.

Fallacy: the definition is negative where an affirmative is possible, and it is circular by correlatives. It breaks rules 5 and 2.

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) Man into Indians, Chinese, and educated.

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

The wider point: the mistake here is a leap upward. The division has slipped from the genus "coin" to the genus "currency" in the middle of the list, which is why an alien member could get in at all.

(iii) Indians into Hindus and not Hindus.

No fallacy. This is a division by dichotomy and it is formally faultless.

Reasons: one fundamentum divisionis is used, and because the two members are contradictories the division is necessarily exhaustive, since every Indian must be one or the other, and necessarily mutually exclusive, since none can be both. Every rule is satisfied.

The criticism that may fairly be made is of usefulness, not validity. The negative member "not Hindus" is wholly indeterminate: it says only what its members are not, and it lumps Muslims, Christians, Sikhs, Buddhists, Jains, Parsis and the unaffiliated into a single remainder. Dichotomy is a first step, and at the next step the negative member must be broken into positive species.

For full marks, cover: the etymology and three definitions with the criticism of the oldest; the nature of logic under four heads, each argued rather than asserted; the scope divided into deductive, inductive and applied, with the contents of each; the relation of logic to the neighbouring subjects, which is where "nature" is really tested; and a conclusion.

Definition

The word comes from the Greek logos, meaning word, thought or reason. The subject was founded by Aristotle, whose logical works are collected as the Organon, the instrument, and he is called the father of logic.

1. The traditional definition: logic is the science of the laws of thought.
2. Whately (Elements of Logic, 1826): logic is the science, and also the art, of reasoning.
3. Copi: logic is the study of the methods and principles used to distinguish correct from incorrect reasoning.

3. It is 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, and why an argument can be conducted on symbols with the subject matter removed entirely.

4. It is a general science. Every other discipline reasons; logic examines reasoning itself. That is the sense in which it is called the science of sciences and the instrument of all enquiry, though it is the science of their method and not their master: it can tell a physicist whether an argument is valid, never whether a premise is true.

The scope of logic

Deductive logic, in which the conclusion follows necessarily. It covers terms (their kinds, connotation and denotation, definition and division), propositions (their classification, quantity, quality and distribution), immediate inference (opposition and eduction), mediate inference (the categorical syllogism and its rules, the hypothetical and disjunctive syllogisms), and modern symbolic logic (truth functions, truth tables, propositional functions and quantification).

Conclusion

Logic is the science and the art of reasoning: a normative science, because it states how we ought to reason; a formal science, because validity is a property of the structure of an argument; and a general science, because it examines the instrument every other discipline uses. Its scope runs from the term through the proposition to the syllogism on the deductive side, and from observation through hypothesis to causal law on the inductive side.

2. By QUALITY, that is, whether the copula joins or separates.

• Affirmative: All contracts are agreements.
• Negative: No minor is competent to contract.

⚠️ The quality is carried by the copula and nothing else. "All men are not-honest" is affirmative with a negative predicate.

3. By RELATION, that is, whether the assertion is made outright or under a condition.

• Categorical: All men are mortal.
• Hypothetical: If a person commits theft, then he is punishable. Its parts are the antecedent and the consequent, and neither is asserted by itself.
• Disjunctive: Either the accused confesses or the prosecution proves the charge. Its parts are the alternatives.

4. By MODALITY, that is, how strongly the predicate is asserted of the subject. This is Kant's division.

• Assertoric or pure: This agreement is void.
• Problematic: This agreement may be void.
• Apodeictic or necessary: This agreement must be void.

The table

The mnemonic: universals distribute the subject; negatives distribute the predicate. Quantity governs the subject; quality governs the predicate.

Why each cell is what it is

A, "All men are mortal". The proposition speaks of every man, so the subject is distributed. It does not speak of every mortal; it says only that the men are somewhere among the mortals, so the predicate is undistributed.

E, "No men are angels". It speaks of every man, so the subject is distributed. It also shuts men out of the whole class of angels, because to exclude a thing from a class is to exclude it from every member of that class, so the predicate is distributed too.

3. Opposition. The square is built on quantity and quality, which are the two things distribution depends on.

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). It reads a universal as a denial, which asserts nothing to exist, so A no longer implies I and E no longer implies O.

Conclusion

The traditional classification examines a proposition from four sides: how much of the subject, whether the copula joins or separates, whether the assertion is outright or conditional, and how strongly it is made. Quantity and quality together give the four forms, and the four forms give the distribution table, which is the single most useful thing in traditional logic: nearly every rule of eduction and of the syllogism is a consequence of it.

The four forms

Taking S as advocates and P as graduates:

The square of opposition

Diagram: draw a square with A at the top left, E at the top right, I at the bottom left and O at the bottom right. The top edge is contraries, the bottom edge sub-contraries, the two sides subalterns, and the two diagonals contradictories. The drawing is reproduced at the end of this answer.

• A true, so I true. I false, so A false. A false, so I doubtful. I true, so A doubtful.

Inference by opposition: the complete table

"Doubtful" is a result, not an evasion: it means the truth value is not settled by the premise.

The practical use: to defeat a rule stated as "all X are Y", establish its contradictory, "some X are not Y", which a single instance proves. Establishing the contrary is harder and unnecessary, and that is exactly what a distinguishing case does.

Conclusion

Opposition is the simplest of the immediate inferences and the one most used in argument, because it settles three propositions the moment a fourth is settled. Its four kinds rest on the laws of thought, and only contradiction binds in both directions, which is why one exception is a complete answer to a rule stated without one.

For full marks, cover: the definition and the form of the argument; its place between deduction and induction; the conditions of soundness as numbered criteria, explained and not merely listed; the marks of a bad analogy and the fallacy of false analogy; the use and the limits of analogy in law; and a conclusion.

Definition

Analogy is that form of inference in which, from the resemblance of two things in certain respects, we conclude that they resemble each other in some further respect.

A and B resemble each other in the properties p, q and r.
A has the further property s.
Therefore B also has the property s.

Example: Mars resembles the Earth in having an atmosphere, water, seasons and a moderate temperature; the Earth is inhabited; therefore Mars is probably inhabited.

4. The number of instances compared should be large. An analogy drawn from many pairs is stronger than one drawn from a single pair, because it begins to approach an induction.

5. The instances should be varied. Instances that differ from each other in every respect except the relevant one are worth more than instances alike in everything, because variety rules out the accidental.

6. The conclusion should be modest. The weaker the property claimed, the more probable the conclusion. "Mars probably supports some form of life" is far better supported than "Mars is inhabited by beings like ourselves".

The marks of a bad analogy

Few resemblances, or superficial ones; resemblances irrelevant to the property inferred; material differences ignored or suppressed; a conclusion far stronger than the premises support; and a comparison between things of different orders, as in the argument that a State should be run like a household.

Where these are present the argument commits the fallacy of false analogy. The fault is never that the two things are unlike, since no two things are alike in everything; it is that the likeness relied on has nothing to do with the conclusion drawn.

Conclusion

An analogical argument is never valid or invalid; it is strong or weak, 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.

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) All men are perfect. Give contrary and contradictory. (2 marks)

An A proposition.

• Contrary (E): No men are perfect.
• Contradictory (O): Some men are not perfect.

Contraries cannot both be true but may both be false. Contradictories can neither both be true nor both be false.

(ii) No men are perfect. Give contrary and contradictory. (2 marks)

An E proposition.

• Contrary (A): All men are perfect.
• Contradictory (I): Some men are perfect.

Step 1, obvert: All men are non-angles. (A) Step 2, convert by limitation: Some non-angles are men. (I), the partial contrapositive. Step 3, obvert again: Some non-angles are not non-men. (O), the full contrapositive.

Answer: partial contrapositive, "Some non-angles are men"; full contrapositive, "Some non-angles are not non-men".

The paper prints "angles"; if "angels" was intended the working is identical, because the item turns on the E form and not on the terms.

(v) No men are angles. (Give inverse) (4 marks)

The same E proposition. Inversion infers a proposition whose subject is the contradictory of the original subject.

Step 1, convert: No angles are men. (E) Step 2, obvert: All angles are non-men. (A) Step 3, convert by limitation: Some non-men are angles. (I)

Answer: "Some non-men are angles."

The results for the four forms are: A gives the inverse "Some non-S is not P"; E gives "Some non-S is P"; I and O have no inverse at all, because neither can begin the chain.