BLS LLB 5 Years Sem 1 Logic 1 2022-23 - ATKT 75/25 Question Paper with Solutions

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

Two distinctions:

1. A component must itself be a proposition; a constituent need not be. In "Rama is honest and Shyam is diligent", "Rama is honest" is a component; in the same sentence "Rama" is a constituent and no more.
2. A component must survive substitution; a constituent need not. Replace "Rama is honest" with any other proposition 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, because substituting another proposition for it produces nonsense.

This is a singular proposition, that is, a class-membership proposition: a predicate attributed to a named individual. No quantifier is used.

Gc

where G = "... is a great leader" and c = Chanakya.

The past tense is carried into the predicate, not into the symbolism: G may be read as "... was a great leader", because logic has no tense and the copula in strict logical form is always present.

The membra dividentia are the dividing members: the species into which a class is separated by a logical division.

They are one of the three elements of every division:

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

Example: in the division of triangle into equilateral, isosceles and scalene, the three kinds are the membra dividentia.

The distinction is Kant's.

An analytic proposition is one in which the predicate is contained in the subject, so that it merely unfolds what the subject already means. It is true by definition, adds no new knowledge, and its denial is self-contradictory.

Example: "All bachelors are unmarried." "A triangle has three sides."

A synthetic proposition is one in which the predicate is not contained in the subject, so that it adds something to it. It adds new knowledge, its denial is not self-contradictory, and it is known by experience.

Example: "All bachelors are unhappy." "Water boils at 100 degrees Celsius."

For full marks, cover: both definitions with examples; the point that the test is meaning and not spelling; privative terms; negative against contrary terms; infinite terms; the place of negative terms in obversion; and a legal application.

The two kinds

A positive term connotes the presence of a quality or attribute in the thing it denotes: man, honest, brave, legal, competent.

A negative term connotes the absence of that quality, and is usually formed by prefixing not, non, un, in or dis: not-man, dishonest, illegal, incompetent.

The two are contradictory: between them they exhaust the universe of discourse, so everything is either a man or a not-man.

The test is the meaning, not the spelling

1. Negative in form, positive in meaning: invaluable means of the greatest value; unbounded means immense.

Infinite terms

A negative term such as "not-man" is called an infinite or indefinite term, because its denotation is unlimited: it takes in stones, ideas and numbers. That is why a negative term is of little use as the subject of a proposition and is chiefly useful as a predicate, and why the universe of discourse has to be fixed before a negative term becomes determinate.

Where negative terms are needed: obversion

Obversion changes the quality of a proposition and replaces the predicate by its contradictory, so it cannot be performed without negative terms. "All men are mortal" obverts to "No men are non-mortal". Contraposition and inversion, which are built out of obversion, need them too.

Legal application

Statutes use negative terms constantly: a non-resident, an unregistered document, a disqualified director, an illegal agreement. The drafting rule that follows is that a negative term is only as clear as the positive term it denies, so a definition clause that defines the positive settles the negative with it.

For full marks, cover: what quantification is; the propositional function it operates on; free and bound variables; the two quantifiers with the connective each requires; the four traditional forms quantified; quantifier negation; what quantification gained; and the second meaning of the word, Hamilton's quantification of the predicate.

What quantification is

Quantification is the operation of prefixing a quantifier to a propositional function so as to turn it into a proposition. It is the device on which the whole of modern predicate logic rests.

A propositional function is an expression containing a variable which is neither true nor false as it stands: "x is a lawyer", written Lx. It becomes a proposition either by instantiation, putting a constant for the variable, La; or by quantification.

A variable a quantifier governs is bound; one no quantifier reaches is free. An expression with a free variable is a propositional function; one with none is a proposition.

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 A is O.

What quantification gained

1. Relations can be expressed: "Mohan is wiser than Rohan" is Wmr, and "everybody loves somebody" is (x)(∃y)Lxy.
2. The order of quantifiers matters: (x)(∃y)Lxy is not (∃y)(x)Lxy.
3. Existence is carried by the quantifier, so "unicorns do not exist" is ~(∃x)Ux and existence never has to be treated as a predicate.
4. Existential import becomes explicit, which is why the modern square keeps only the contradictories.

For full marks, cover: the theory stated; its founders and their differences; what "works" means; the merits; the objections; the two rival theories in contrast; and a legal application.

The theory stated

The pragmatic theory holds that a proposition is true if it works: if acting on it leads to satisfactory results, and false if it does not. Truth is not a static relation between a belief and a fact, but something that happens to a belief in the course of being tested by experience.

Its founders

C.S. Peirce framed the pragmatic maxim, that the meaning of a conception lies in its practical consequences, and thought of truth as the opinion on which enquiry would finally converge.

William James made the theory famous and put it most loosely: an idea is true so long as it is profitable to our lives, and truth is "the expedient in the way of our thinking".

1. It confuses truth with utility. A false belief may be useful, and an inconvenient truth is still true. A patient wrongly told he will recover may do better for the belief, which does not make it true.
2. "Works" is vague, and works for whom and over what period? A belief useful today may be disastrous later.
3. It is circular: to say that a belief works is to say that certain predictions come true, so the notion of truth is used in defining truth.
4. Many true propositions have no practical consequences at all, for example about the remote past, and the theory has nothing to say about them.

The rivals

The correspondence theory: a proposition is true if it corresponds to a fact. Objection: the two sides cannot be inspected and compared.

The coherence theory: a proposition is true if it coheres with a system of accepted beliefs. Objection: a consistent fiction would be true.

For full marks, cover: the definition and the form; its place among the forms of inference; the tests of a good analogy; the fallacy of false analogy; the use of analogy in law; and its limits.

The 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 probably has s.

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

The fallacy of false analogy

Where the resemblances are few, superficial or irrelevant, where material differences are suppressed, or where the two things are of different orders, 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.

Analogy in law

Precedent is analogical reasoning. To follow a case is to argue that its material facts resemble the present ones in the respects that produced the earlier result; to distinguish it is to argue that they do not. Analogy also fills gaps where no rule covers a case, as when Donoghue v Stevenson [1932] AC 562 was extended from a manufacturer of ginger beer to manufacturers generally, and ejusdem generis is a rule for reasoning from likeness.

Its limits

In criminal law analogy is forbidden, because no act is an offence unless the law makes it one, and Article 20(1) would be defeated. It cannot override an express provision. And it proves nothing by itself: the conclusion is probable, and a court reasoning only by analogy has given a reason, not a demonstration.

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.

Reason: "rarely" belongs with "seldom" and "few": it means not often, and the negative force is part of its meaning. "Rarely S are not P" therefore carries two negations, which cancel, leaving an affirmative particular.

(iii) Lawyers are wealthy persons.

All lawyers are wealthy persons. (A proposition)

Distributed: the subject, "lawyers", only.

Reason: the sentence prints no quantity sign, and an indefinite proposition stating a general truth of its kind is read as universal. Being affirmative, the predicate is undistributed: the proposition says nothing about every wealthy person.

Reading 2: the negation governs only the first component.

(~p) · q: London is not the coldest city, and it is in Rome

The English "It is false that X and Y" ordinarily takes the whole of "X and Y" as what is denied, so Reading 1 is the natural one and is used for the table.

Truth table for ~(p · q)

Reading of the table: the proposition is false in one case only, where both components are true. That is the denial of a conjunction, and by De Morgan's law it is equivalent to ~p v ~q: at least one of the two is false.

(ii) Identify and symbolise the following simple propositions

1. Run.

The rules a definition must satisfy

1. It must state the essential attributes, per genus et differentiam.
2. It must not be circular.
3. It must be co-extensive: neither too wide nor too narrow.
4. It must not be in obscure or figurative language.
5. It must not be negative where it can be affirmative.

(i) A compulsive gambler is a person who gambles compulsively.

Fallacy: the definition is circular. It breaks rule 2, and this is the purest form of the fault, circulus in definiendo.

(iii) Love is the tendency not to hate others.

Fallacy: the definition is negative where an affirmative is possible, and it is too wide. It breaks rules 5 and 3.

Reasons: love is a positive state of feeling and can be defined affirmatively, so the negative form is not forced on the definer as it is with a privative term such as "orphan". And the definition is too wide, because mere indifference is also a tendency not to hate: a person who feels nothing whatever about another satisfies the definition and does not love him.

Corrected: love is a feeling of deep affection and attachment towards another, accompanied by a concern for that person's good.

Reasons: it is incomplete, because the essay, biography, history and criticism are all literature and none of them appears, so the members fall short of the genus. It is also a cross-division, because two bases are used at one step: poetry and drama are species of literature by literary form, while the novel and the short story are subdivisions of prose fiction by length.

A sound division: literature into poetry, drama and prose, by form; then prose into fiction and non-fiction; then fiction into the novel and the short story, by length. Three steps, one basis each.

(ii) Laptop into display screen, motherboard, RAM, keyboard.

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

Reasons: logical division separates a class into its kinds, so that each member is a species of the genus; a motherboard is not a kind of laptop. What has been done is partition, the physical breaking up of a single object into the parts of which it is made. Rule 5 is broken, and the test settles it at once: the name of the whole can be predicated of every species in a division, so "an ultrabook is a laptop" is true, whereas "a keyboard is a laptop" is false.

It is a NORMATIVE science, not a positive one. Physics and psychology describe what is; a normative science sets up a standard and judges by it, saying what ought to be. Logic belongs with ethics and aesthetics.

It is also an ART. A science states principles, an art gives rules for practice, and logic does both. Whately's definition says so expressly, and the definition under discussion omits it.

2. "Valid"

This is the word that does the repair, and it does it well.

The older definition spoke of the "laws" of thought, which invited two misreadings: a law of nature, which is descriptive and would be falsified by a single person reasoning badly, and a law of the State, which commands and punishes. A law of thought is neither: it is a standard, and the penalty for breaking it is not punishment but unintelligibility.

What the definition gets right, and what it still misses

Right: it is normative, because "valid" carries a standard; it is narrower than "the science of thought", because only valid reasoning is in question; and it locates logic correctly among the sciences.

Still missing, or wrong:

1. It omits the art. Logic lays down rules for practice as well as stating principles.
2. "Thought" is still too wide unless it is read as reasoning, and the definition does not say so.

Conclusion

The definition is a genuine improvement on "the science of the laws of thought", because the word valid puts the normative character of logic into the definition itself instead of leaving it to be explained away. It remains defective on three counts: it omits the art, it leaves "thought" undefined and therefore too wide, and it applies to thoughts a word that properly belongs to arguments. Copi's definition avoids all three, and is to be preferred.

A singular proposition is treated as universal, because the whole of its subject is spoken of, so that it may be used in a syllogism.

2. Quality: whether the copula joins or separates

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

3. The two combined: the fourfold scheme

⚠️ In a conditional proposition neither part is asserted. "If a person commits theft, he is punishable" does not assert that anyone has committed theft.

5. Modality: how strongly the predicate is asserted

The propositions that need reducing, illustrated

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). Relation becomes a matter of truth-functional connectives, and modality is set aside from truth-functional logic, because a modal proposition is not a function of its component's truth value.

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, from which nearly every rule of traditional deductive logic follows.

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.

All the crows I have seen are black.
No crow that is not black has been observed.
Therefore all crows are black.

What "uncontradicted experience" means, and what it does not

It means that in all the instances so far observed the two things have gone together, and no instance has been met in which they came apart.

It does not mean that the connection has been shown to be causal; it does not mean that a counter-instance has been searched for; and it does not mean that the instances were chosen or varied with any care. The ground is the absence of a refutation, and that is a much weaker thing than the presence of a proof.

Hence the two postulates the method leans on: the uniformity of nature, which every induction assumes, but not the law of universal causation, which scientific induction uses and this method never reaches.

Mill's qualified defence

Mill accepted the criticism and kept the method. Where instances are very numerous and very various, and a contrary instance has been actively sought and not found, the probability may become indistinguishable in practice from certainty. Our belief that all men are mortal rests on nothing better, and nobody doubts it.

The conditions of strength

1. A large number of instances.
2. Variety among them.
3. A deliberate search for a counter-instance, unsuccessfully made.
4. No known reason to expect an exception.
5. A modest and revisable conclusion.

Conditions 2 and 3 are the ones that convert bare uncontradicted experience into something worth relying on.

Compared with the other inductions

3. It is the only method available where experiment is impossible, as in history, astronomy and the social sciences.
4. In law it is the argument from an unbroken line of precedent, and it carries the same warning: a rule never challenged is not thereby proved right.

Verdict on the statement

The statement is accurate as a description and incomplete as an endorsement. Simple enumeration does establish generalisations, and it does so on the basis of uncontradicted experience and nothing else. What the sentence leaves out is that "uncontradicted" is a negative ground, so what is established is a probability and not a truth, and that the probability is worth having only where the experience has been wide, varied and actively tested against.

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

3. To make disputes soluble. Many disputes are merely verbal: the parties agree on the facts and use a word differently. Defining the word ends the quarrel or shows that it was real after all.

4. To supply the major premise of an argument. "A contract is an agreement enforceable by law" is both a definition and 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, and every science.

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

What division is

Logical division is the process of separating a class, the genus, into the sub-classes or species contained under it, on the basis of a single attribute. Its three elements are the totum divisum, the membra dividentia and the fundamentum divisionis.

Division sets out the denotation of a term, as definition sets out its connotation.

The five rules

For full marks, cover: each item with the given proposition reduced and its form named, both answers written out where they exist, and the rule that produces each; and the two items where what is asked cannot be given, answered with the reason and with the relation that does exist.

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.

(iii) No umpires are partisans. (Give Sub-contrary and subaltern)

An E proposition.

• Sub-contrary: an E proposition has no sub-contrary. Sub-contrariety holds only between the two particulars, I and O, at the foot of the square. An E proposition is universal; what it has at the top is a contrary, the A proposition "All umpires are partisans".
• Subaltern (O): Some umpires are not partisans.

Give the subaltern, name the contrary, and say why there is no sub-contrary.

(iv) Many great scientists are not college graduates. (Give Obverse)

An O proposition. "Many" is a plain particular sign, exactly like "some".

Obverse (I): Some great scientists are non-college-graduates.

(v) Hardly students are logicians. (Give Converse)

Reduced: "Some students are not logicians." An O proposition, because "hardly" belongs with "few", "seldom" and "rarely": it means not many, and the negative force is part of the word.