BLS LLB 5 Years Sem 1 Logic 1 February 2026 - 75/25 Question Paper with Solutions

The Law of Identity is the first of the three fundamental laws of thought. It states that whatever is, is: everything is identical with itself.

Symbolically it is written A is A, or in propositional form p ⊃ p, and in class form "everything that is a man is a man".

Its practical force in reasoning is that a term must keep the same meaning throughout an argument. A word that shifts its sense between the premise and the conclusion breaks the law, and the resulting fallacy is the fallacy of equivocation.

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:

1. the subject, that about which something is asserted,
2. the predicate, that which is asserted of the subject, and
3. the copula, the part which joins the two and asserts or denies the relation between them.

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

Let Mx stand for "x is a man" and Tx for "x is mortal".

(x)(Mx ⊃ Tx)

Read: for every x, if x is a man, then x is mortal.

The (x) is the universal quantifier and the ⊃ is the sign of material implication.

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 criminal offence, punishable under Section 290 IPC (Section 292 BNS). Examples are obstruction of a public highway, fouling a public water source, and carrying on a noxious trade in a crowded locality.

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

For full marks, cover: the statement of the theory; its history; its merits; the three standard objections; the two rival theories in contrast; and its place in the law of evidence.

The theory stated

The correspondence theory holds that a proposition is true if, and only if, it corresponds to a fact, and false if it does not. Truth is therefore a relation between what is asserted and what is the case; it is not a property the proposition has by itself.

The oldest statement of it is Aristotle's: to say of what is that it is, and of what is not that it is not, is true. In modern philosophy it was developed by G.E. Moore, Bertrand Russell and, in the Tractatus, by Ludwig Wittgenstein, whose picture theory treats a true proposition as a picture whose structure matches the structure of a fact.

Its merits

1. It matches ordinary usage. When we ask whether a statement is true we do look for the fact it reports.

The pragmatic theory (William James, John Dewey) holds that a proposition is true if it works, that is, if acting on it produces satisfactory results. It captures how beliefs are tested in practice but confuses truth with usefulness, since a false belief can be useful and an inconvenient truth is still true.

Its place in law

Judicial fact-finding is correspondence-based. Section 3 of the Indian Evidence Act 1872, now Section 2(1) of the Bharatiya Sakshya Adhiniyam 2023, defines a fact as "proved" when the court believes it to exist, and the entire law of relevancy is machinery for testing an assertion against what happened.

The third term: sentence

A sentence is the grammatical unit. Every proposition is expressed in a sentence, but not every sentence expresses a proposition: questions, commands, requests and exclamations affirm nothing and are neither true nor false.

The differences

Why logic works on the proposition

An act of the mind 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.

The law of inverse variation

As the connotation of a term increases, its denotation decreases, and as the connotation decreases, the denotation increases.

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

Each step adds an attribute and shrinks the class.

The limits of the law. It holds only along a single line of subordination, genus to species to sub-species. It does not hold where the added attribute belongs to every member already, since "rational man" adds a word and takes away nothing from the class; and it does not hold between two terms that are not related as genus and species.

Terms that have one and not the other

1. Proper names (Rama, the Ganga) denote an individual but, on the traditional view, connote nothing, because they identify without describing.
2. Abstract terms (whiteness, justice, honesty) connote a quality but denote no class of individuals.
3. General terms have both, and they are what logic ordinarily works with.

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 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. Only soundness guarantees a true conclusion. Validity by itself guarantees only that no truth has been lost between the premises and the conclusion.

(a) Reduction to logical form, with the distributed terms

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. The rule of distribution is:

(iii) Only citizens of the United States can vote in the U.S. elections.

All persons who can vote in the United States elections are citizens of the United States. (A proposition)

Distributed: the subject, "persons who can vote in the United States elections", only.

This is an exclusive proposition, and the rule is that "Only S is P" becomes "All P is S": the subject and predicate change places when the sign "only" is removed. Writing it as "All citizens of the United States can vote" would be a different and false proposition, since it would take in minors.

(b) "If a person commits theft, then he is punishable under law"

Identification: a compound proposition; the connective is the conditional or implication, signalled by "if ... then".

• Let p = A person commits theft.
• Let q = He is punishable under law.

Symbolic form: p ⊃ q

Here p is the antecedent and q the consequent.

The classification being applied

Modern logic classifies propositions first as simple (containing no other proposition as a part) or compound (containing at least one), and then, within the simple ones, as singular (a predicate attributed to a named individual) or general (a quantified statement about a class, either universal or existential).

(i) Seems impossible to solve.

Classification: a simple proposition, and an elliptical one, since the subject is not printed.

(iii) A mother loves her child.

Classification: a general proposition, universal in force, and relational.

"A mother" here means every mother; the indefinite article states a rule and not an instance. As a simple one-place symbolisation:

(x)(Mx ⊃ Lx)

where Mx = x is a mother and Lx = x loves her child.

The proposition is really about a relation between two things, and the fuller symbolisation shows it:

(x)(y)[(Mx · Cyx) ⊃ Lxy]

where Mx = x is a mother, Cyx = y is the child of x, and Lxy = x loves y. Read: for any x and any y, if x is a mother and y is x's child, then x loves y.

Reasons: only surgeons perform surgery. A physician, a paediatrician, a psychiatrist and a radiologist are all doctors and none of them performs surgery, so the definition leaves out most of what the term denotes. It fails the test of convertibility in one direction: every person who performs surgery is a doctor, but not every doctor performs surgery. It also states an accident rather than the essence, since performing surgery is something some doctors happen to do and not what makes a person a doctor. A sound definition would be: a doctor is a person qualified and licensed to practise medicine.

(ii) Alternation is a combination of contradictory opposed determinations in the existence of one and the same thing.

Fallacy: the definition is expressed in obscure language. It breaks rule 4.

The rules of logical division

1. There must be only one fundamentum divisionis at each step; breach is cross-division.
2. The dividing members must be mutually exclusive.
3. The division must be exhaustive: the species together must equal the genus.
4. The division must proceed step by step, without a leap (saltus in dividendo).
5. Every dividing member must be a species of the genus divided.

(i) Umbrella into rod, handle, spokes and cloth.

Fallacy: this is not a division at all. It is a partition, and the two have been confused.

The criticism that may fairly be made is one of usefulness, not of validity. The negative member "non-introvert" is wholly indeterminate: it tells us only what its members are not, and it lumps extroverts together with everything else that is not introverted. Dichotomy is therefore formally perfect and practically thin, which is why it is used chiefly as a first step, to be replaced at the next step by positive members.

(iii) Fruits into apple, banana, orange, big, red, ripped.

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

Reasons: four different bases have been used at one step. Apple, banana and orange divide fruit by kind; "big" divides it by size; "red" by colour; and "ripped", which is plainly meant for "ripe", by state of maturity. Rule 1 is broken, and rule 2 falls with it, because the members now overlap: one and the same apple may be big, red and ripe, so it belongs to four members at once. The division is also not exhaustive as a division by kind, since mangoes, grapes and the rest are left out.

For full marks, cover: the definition and the form of an analogical argument; its place between deduction and induction; the tests of a good analogy stated as numbered conditions; the marks of a bad analogy and the fallacy of false analogy; the use of analogy in law under precedent, statutory interpretation and the filling of gaps, with cases; the limits of legal analogy; 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.

Its form is:

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.

5. The instances compared should be varied, so that the resemblance is not an accident of one setting.
6. The two things should be of the same order, so that the comparison is between comparable subjects.

Characteristics of a bad analogy

1. Few points of resemblance, or points that are superficial.
2. Resemblances that are irrelevant to the property inferred. A resemblance in colour tells us nothing about weight.
3. Material differences ignored or suppressed.
4. A conclusion far stronger than the premises support.
5. A comparison between things of different orders, such as an argument that a State should be run like a household, which treats a political community as though it were a family.

Where these faults are present the argument commits the fallacy of false analogy, one of the material fallacies. The fallacy is not that the two things are unlike, since no two things are alike in everything, but that the likeness relied on has nothing to do with the conclusion drawn.

4. Analogy fills gaps in the law. Where no rule covers a case, courts extend the nearest rule to it. English law's treatment of new kinds of negligence after Donoghue v Stevenson [1932] AC 562 is the classic instance: 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, drew on the Convention on the Elimination of All Forms of Discrimination against Women to frame binding guidelines, reasoning from the analogy of the constitutional guarantees in Articles 14, 19(1)(g) and 21.

The limits of analogy in law

1. In criminal law, analogy is forbidden. No act is an offence unless it is made so by law, and a penal statute is construed strictly. To punish by analogy would defeat Article 20(1) of the Constitution.
2. Analogy cannot override an express provision. It operates only where the law is silent or ambiguous.
3. Analogy proves nothing by itself. Its conclusion is probable, and a court that reasons only by analogy has given a reason and not a demonstration.

For full marks, cover: the propositional function and its two ways of becoming a proposition; singular against general propositions; the two quantifiers with their symbols and readings; the universal and existential forms with the connective each requires; the traditional A, E, I and O forms in quantified dress; the table of differences; the four equivalences of quantifier negation; existential import and what it does to the square of opposition; and a legal illustration.

The starting point: the propositional function

A propositional function is an expression containing a variable which becomes a proposition when the variable is dealt with. "x is a lawyer", written Lx, is neither true nor false as it stands.

It becomes a proposition in two ways:

1. Instantiation, putting a constant in place of the variable: La, "Ambedkar is a lawyer".
2. Generalisation, prefixing a quantifier: (x)Lx or (∃x)Lx.

An existential proposition asserts that at least one thing satisfies a condition. Its normal form joins the two predicates by a conjunction:

(∃x)(Sx · Px) there is at least one x which is both S and P.

Why the connectives differ, and why it matters. A universal proposition does not claim that anything exists; it says only that nothing is S without being P, and an implication says exactly that. To write (x)(Sx · Px) would assert that everything in the universe is both S and P. An existential proposition does claim that something exists, and a conjunction says exactly that. To write (∃x)(Sx ⊃ Px) would be satisfied by any object that simply is not S, since a conditional with a false antecedent is true, and the formula would therefore assert almost nothing.

The traditional four forms in quantified dress

Quantifier negation

The two quantifiers are interdefinable, and these four equivalences are what let a formula be rewritten:

1. ~(x)Fx is equivalent to (∃x)~Fx
2. ~(∃x)Fx is equivalent to (x)~Fx
3. (x)Fx is equivalent to ~(∃x)~Fx
4. (∃x)Fx is equivalent to ~(x)~Fx

So "not every subject is difficult" becomes "some subject is not difficult", which is why the contradictory of an A proposition is an O proposition.

Existential import

A proposition has existential import if it asserts that its subject class has at least one member. Traditional logic gave import to all four forms, which is why it could infer I from A and O from E.

Conclusion

The general proposition, formed by quantifying a propositional function, is the unit that modern logic works with. Universals and existentials differ in the quantifier, in the connective each requires, and above all in whether they assert that anything exists. That last difference is not a technicality: it is what separates the modern square of opposition from the traditional one.

The four categorical forms

Taking S as advocates and P as graduates:

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

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.

The types of opposition

1. Contradictory opposition: A with O, and E with I

In the ordinary case of a mixed class, both are true together.

4. Subaltern opposition: A with I, and E with O

Same quality, differing in quantity. The universal is the subalternant, the particular the subalternate. Truth descends and falsity ascends.

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

Strictly this is not opposition at all but subordination, since the two do not conflict; it is included in the square because the inference is immediate.

Inference by opposition: the complete table

3. Traditional logic, needing to fit singulars into the syllogism, treated them as universal, on the ground that the whole of the subject is taken. On that treatment "Socrates is wise" behaves as an A proposition and "Socrates is not wise" as an E proposition, and the two then count as contraries. This is a convention adopted for the syllogism and it is not the natural reading, because two contraries may both be false while these two cannot.
4. Modern logic settles it cleanly. A singular proposition is symbolised Wa, a predicate with an individual constant, and its contradictory is ~Wa. No quantifier is involved, so no question of quantity arises, and the single relation of contradiction is all there is.

A caution: the subject must be the same individual in both. "Socrates is wise" and "Plato is not wise" are not opposed.

The modern square and existential import

The traditional square assumes that the subject class has at least one member. Modern logic reads a universal as a denial, so "All advocates are graduates" asserts only that there is no advocate who is not a graduate, and it stays true even if there are no advocates at all.

Conclusion

Opposition is the simplest kind of inference in logic and the most used in argument, because it tells us what follows about three propositions the moment one of them is settled. Its four relations, and the fact that only contradiction binds both ways, are what a lawyer relies on when a single exception is enough to answer a rule stated without one.

Example: the genus triangle, divided on the basis of the length of the sides, gives equilateral, isosceles and scalene triangles.

Its relation to definition

Definition and division are the two operations that fix the meaning of a general term, and they work on opposite aspects of it. Definition sets out the connotation of a term, by stating genus and differentia. Division sets out the denotation, by breaking the class into the species it covers. Neither can be done well without the other, which is why the two are always examined together.

The rules of division and the fallacies they exclude

Rule 1. There must be only one fundamentum divisionis at each step.

Fallacy: cross-division.

Example of breach: "Books into English, historical and cheap" divides at one step by language, by subject and by price.

Rule 2. The dividing members must be mutually exclusive.

Fallacy: overlapping division.

Example of breach: "Umbrella into rod, handle, spokes and cloth". A spoke is a part of an umbrella, not a kind of umbrella.

Division distinguished from three neighbouring operations

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 an error in itself; it is an error only when offered as a logical division.

Division against enumeration. Division names the species of a class; enumeration names its individuals. Dividing "advocate" gives senior advocates and other advocates; enumerating it gives a list of names.

Division against classification. They are the same operation performed in opposite directions. Division moves downward, from the genus to its species; classification moves upward, gathering individuals into species and species into genera.

Where a taxing or licensing schedule mixes bases, the categories overlap and the assessee falls under two entries at once, which is a standing source of litigation. Article 246 of the Constitution divides legislative power on a single basis, the subject matter, into the Union, State and Concurrent Lists, and the doctrine of pith and substance exists precisely to deal with the entries that nevertheless overlap.

Conclusion

Logical division is the orderly separation of a class into its species on one basis, and its five rules are five ways of insisting on that one basis. Rule 1 is the parent of the rest: once two bases are admitted at a single step, the members overlap, completeness can no longer be tested, and the division ceases to be an operation of logic at all.

Obversion, in which the quality is changed and the predicate replaced by its contradictory. It works for all four forms:

(i) All senators are citizens. (Give contradictory)

An A proposition.

Contradictory (O): Some senators are not citizens.

Obverse (E): No nurses are non-considerate.

The quality is changed from affirmative to negative, and the predicate "considerate" is replaced by its contradictory, "non-considerate". Note that the predicate must be the contradictory and not the contrary: writing "No nurses are inconsiderate" would assert more than the original, because "non-considerate" covers everything that is not considerate while "inconsiderate" names a positive fault.

(iv) All that glitters is not gold. (Give subaltern)

Read as it is printed, this is an E proposition: "No glittering things are gold."

Subaltern (O): Some glittering things are not gold.

The subaltern of a universal is the corresponding particular of the same quality, and truth descends from the universal to it.

Note that O can be contraposed although it cannot be converted, because obversion turns it into an I proposition first, and an I proposition converts simply.

(vi) Elephants never forget. (Give inverse)

Reduced to strict logical form this is an E proposition: "No elephants are creatures that forget."

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

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." (I)

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