Unit 6: Logical Reasoning mind map

The building blocks: denotation and connotation, the four standard propositions A, E, I and O, distribution, and kinds of definition.

Denotation and connotation

Denotation and extension

Denotation, also called extension, is the set of objects a term refers to.

• Wider class means larger denotation.
• Order by increasing denotation: cow, bovine, mammal, vertebrate, animal.
• Each step down the list adds more members.

Think of a class as a box of things. Animal is the biggest box, and cow sits inside bovine, mammal and vertebrate. Arrange the terms from the smallest box to the biggest, or the reverse, as the stem asks.

Connotation and intension

Connotation, also called intension, is the set of attributes a term carries. It runs opposite to denotation.

• Narrower term means more attributes, so greater connotation.
• Order by increasing connotation: animal, vertebrate, mammal, bovine, cow.
• Fielder carries more attributes than baseball player, ball player and athlete.

Each step towards the narrower term adds a required attribute. A horse is an animal, a domestic animal, a beast of burden and a horse. So connotation rises as denotation falls.

Ordering questions: a fast method

Every ordering question gives a chain of terms inside one another. Rank by breadth first, then flip for the other direction.

• Find the widest term and the narrowest term first.
• Place the rest between them by asking which contains which.
• Decreasing denotation and increasing connotation give the same order.

Answer one direction and reverse the list for the other. For example, the order animal, mammal, feline, tiger is decreasing extension and also increasing intension.

Intension fixes extension

Two linked facts: intension determines extension, and a term can have intension with an empty extension.

• Fix the attributes and you have fixed which objects fall under the term.
• The word 'God' has intension, so it is meaningful.
• A proper name has denotation but no connotation.

Knowing which objects fall under a term does not tell you the attributes. So the claim that extension determines intension is the one to mark incorrect. The word 'God' is meaningful even if debated, because it conveys a set of attributes.

Denotative and connotative meaning

Denotative meaning is the direct, dictionary meaning. Connotative meaning is the suggestion or feeling attached to the word.

• Dictionary meaning is denotative.
• Attributes shared by all objects a term names: intension, or connotation.
• Extension is the set of those objects, also called denotation.

The dictionary records the denotation. The associations a word picks up in use belong to its connotation.

Categorical propositions

The four standard forms A, E, I, O

Every categorical proposition is one of four forms. Learn the letters and the wording.

• A: All S are P. Universal affirmative.
• E: No S is P. Universal negative.
• I: Some S are P. Particular affirmative.
• O: Some S are not P. Particular negative.

Quantity is universal or particular. Quality is affirmative or negative. 'All children are not greedy' is read in traditional logic as 'Some children are not greedy', which is O. 'Few students did not clear' means 'some did not clear', also O.

Parts of a proposition

A standard-form proposition has a quantifier, a subject term, a copula and a predicate term. The quality is affirmative or negative.

• The copula links subject and predicate: is, are, is not, are not.
• 'All Indians are Citizens': All is quantifier, Indians subject, are copula, Citizens predicate.
• Quality tells whether class inclusion is affirmed or denied.

Every categorical proposition has three attributes: quality, quantity and distribution. Quality is about affirming or denying. Quantity is about all or some. Distribution is about whether a term is covered in full.

Distribution of terms

A term is distributed when the proposition speaks about every member of its class.

• A distributes only its subject.
• E distributes both subject and predicate.
• I distributes neither term.
• O distributes only its predicate.

Two rules do the work. Quantity governs the subject: a universal distributes it. Quality governs the predicate: a negative distributes it. So 'Some girls are not students' distributes the predicate 'students' but not the subject.

Definition

Kinds of definition

Match the kind of definition to its job.

• Stipulative: assigns a new meaning by decision, so it is neither true nor false.
• Lexical: reports how the word is actually used.
• Precising: removes vagueness by drawing a sharper line.
• Theoretical: packages a whole theory, like the definition of a planet.

A stipulative definition is a proposal. Coining 'googol' for ten to the hundredth power is the textbook case. A precising definition fixes a borderline term, such as who counts as poor for a welfare scheme.

Operational definitions and good definitions

An operational definition ties a term to a test and its result. A good definition by genus and difference avoids certain faults.

• Operational: acid is whatever turns blue litmus red.
• Avoid circular definitions that use the defined word.
• Avoid a definiens wider than the definiendum.

The genus and difference rules were asked as what must be AVOIDED, so read the stem for the direction. The right answers named the circular definition and the one that denotes more things than the term defined.

Informative language

Logic works with language that asserts something that can be true or false.

• Informative language conveys information.
• Directive language instructs: 'Step on scale, please'.
• Adding 'please' softens a directive but does not change its function.

Only informative sentences give propositions that can serve as premises and conclusions. That is why logicians are primarily interested in informative language.

Read the four corners of the square, infer truth values from one given proposition, and find equivalent statements by conversion, obversion and contraposition.

Relations on the square

The four relations

Four relations join A, E, I and O. Learn which pair each one joins and what it forbids.

• Contradictories: A with O, E with I. Never both true, never both false.
• Contraries: A with E. Cannot both be true, can both be false.
• Subcontraries: I with O. Cannot both be false, can both be true.
• Subalternation: A over I, E over O. Truth passes down.

Same subject and predicate throughout. Contradictories differ in both quantity and quality. Contraries differ in quality only. Subalterns differ in quantity only. Label each statement A, E, I or O first, and then the pair shows its relation.

Spotting the pair from a stem

The stem describes the relation in words. Translate the words to the relation, then label the four statements.

• Cannot be true together, can be false together: contraries, the A-E pair.
• Cannot be false together, can be true together: subcontraries, the I-O pair.
• Truth of one implies the other, not conversely: subalternation.
• Exactly one must hold: contradictories.

Label first. For example, 'All poets are dreamers' is A and 'No poets are dreamers' is E, so that pair is the answer to the contraries test. The same template returns with fresh nouns in every paper.

If one is true, what follows?

Starting from a TRUE corner, the square settles the other corners.

• True A: E false, O false, I true.
• True E: A false, I false, O true.
• True I: E false, A and O undetermined.
• True O: A false, E and I undetermined.

A true A determines three of the four corners. For 'All apes are mammals' being true, 'No apes are mammals' is false and 'Some apes are not mammals' is false. A true I only rules out its contradictory E. The other two stay open.

If one is false, what follows?

Starting from a FALSE corner, run the logic the other way.

• False A: O true, E and I undetermined.
• False E: I true, O undetermined.
• False I: E true, A false, O true.
• False O: A true, I true, E false.

Take 'Some birds are apes' as false. It is I, so its contradictory E, 'No birds are apes', is true and A is false. A false E, 'No birds are animals', makes I true. Converting 'Some birds are animals' gives 'Some animals are birds'.

Properties of subcontraries and the square as a whole

Test each property against the definition. Subcontraries and the A-E pair are asked as checklists.

• Subcontraries cannot both be false but can both be true.
• If one subcontrary is false, the other must be true.
• If A is true, E is false.
• If I is false, A is false, not true.

The checklist items repeat the four relations in different words. Run each against the square and mark those that hold. Watch for wrong versions such as 'if one subcontrary is true, the other is false', which is not valid.

Equivalents of a proposition

Conversion

Conversion swaps subject and predicate. It is valid for E and I, and for A only by limitation.

• E: No S is P gives No P is S.
• I: Some S are P gives Some P are S.
• A: All S are P gives Some P are S, by limitation.
• O does not convert.

Simple conversion keeps quantity and quality. So the converse of 'All cats are mammals' is 'Some mammals are cats'. 'All mammals are cats' does not follow.

Obversion

Obversion changes the quality and negates the predicate. It is valid for all four forms.

• A: All S are P gives No S is non-P.
• E: No S is P gives All S are non-P.
• I: Some S are P gives Some S are not non-P.
• O: Some S are not P gives Some S are non-P.

When the predicate is already negative, the two negations cancel. 'No cats are non-feline' obverts to 'All cats are feline'. 'All philosophers are non-conservatives' obverts to 'No philosophers are conservatives'.

Contraposition

Contraposition swaps the two terms and negates both. It is valid for A and O, not for E or I.

• A: All S are P gives All non-P are non-S.
• O: Some S are not P gives Some non-P are not non-S.
• E and I do not contrapose.

'All tulips are flowers' contraposes to 'All non-flowers are non-tulips'. 'Some squares are not circles' contraposes to 'Some non-circles are not non-squares'. Obversion plus conversion plus obversion gives the same result.

Picking out equivalent statements

Start from one statement, generate its equivalents, and tick the options that match.

• For an A statement, its obverse and contrapositive are equivalent.
• For an E statement, its converse and obverse are equivalent.
• For an O statement, its obverse and contrapositive are equivalent.
• The subaltern is not equivalent; it only follows.

In 'All poems are artworks', the equivalents are 'All non-artworks are non-poems' and 'No poems are non-artworks'. 'Some poems are artworks' follows by subalternation but is weaker, so it is not equivalent. For E, 'No red is black', 'No black is red' and 'Red is not black' are equivalent, but 'Some red are not black' only follows.

Two premises, one conclusion. Identify terms, mood and figure, apply the rules, and test with Venn diagrams.

Parts and shape of a syllogism

Major, minor and middle term

A categorical syllogism has three terms. The conclusion decides which is which.

• Major term: predicate of the conclusion.
• Minor term: subject of the conclusion.
• Middle term: in both premises, never in the conclusion.
• The premise with the major term is the major premise.

In 'No criminals are gentlemen', gentlemen is the major term and criminals the minor. In 'No musicians are Japanese; all barbers are musicians', musicians is the middle term: subject of the first premise, predicate of the second.

Mood and figure

Mood is the letters of the three propositions. Figure is the position of the middle term.

• Write major premise, minor premise, conclusion in that order.
• Figure I: middle is subject then predicate.
• Figure II: middle is the predicate of both.
• Figure III: middle is the subject of both. Figure IV: predicate then subject.

'All actors are athletes. Some actors are comedians. Therefore some comedians are athletes' is A, I, I, so the mood is AII. The middle term actors is the subject of both premises, so it is the third figure. 'All sodium salts are water-soluble; all soaps are sodium salts' is AAA in the first figure.

Rules of the syllogism

Each rule has a matching fallacy.

• Exactly three terms, or the fallacy of four terms.
• Middle term distributed at least once, or undistributed middle.
• A term distributed in the conclusion must be distributed in its premise, or illicit process.
• Two negative premises give nothing. A negative premise needs a negative conclusion.

Rules are asked as 'which is correct'. Wrong options reverse one rule. Examples: 'two negative premises give an affirmative conclusion', or 'middle term distributed in both premises'. Two particular premises also yield no conclusion.

Valid conclusions from two premises

To draw conclusions, combine the premises through the middle term, then try conversion.

• All cats are dogs; all dogs are cows gives All cats are cows.
• An A statement also gives its converse by limitation: Some cows are cats.
• All diaries are copies; no copy is a book gives No diary is a book.
• Two particular premises give no conclusion.

Do not reverse an A premise: 'All dogs are cats' does not follow. When the middle term is undistributed in both premises, nothing links the other two terms. Conclusions taken singly from one premise by conversion may still be valid.

Syllogistic fallacies by name

Name the fallacy by which rule the argument breaks.

• All criminals are human; all saints are human; so all saints are criminals: undistributed middle.
• Premise 'Some snakes are poisonous' to conclusion 'all poisonous creatures': illicit minor.
• Two negative premises: exclusive premises.
• Affirmative conclusion from a negative premise.

The existential fallacy draws a particular conclusion from two universal premises when the class may be empty, such as unicorns. Illicit minor appears when the conclusion is universal about the minor term but the premise is an I proposition.

Venn diagrams

Venn pictures of A, E, I and O

Draw two overlapping circles S and P. Shade to deny, put an x to assert.

• A: shade the part of S outside P.
• E: shade the overlap.
• I: put an x in the overlap.
• O: put an x in S outside P.

A universal premise shades regions out. A particular premise puts an x. For three-term tests you draw three overlapping circles and shade the regions a premise denies, then check whether the conclusion is already drawn.

Everyday class diagrams

Pick the diagram that shows how the classes relate.

• Wholly inside: concentric circles. New Delhi, India, World.
• Outside the class: a separate circle. Domestic help and family.
• Partial overlap: overlapping circles.

Take the classes in pairs. Family member lies wholly within family, so nested. Domestic help is not a member of the family, so it lies outside. Combine the pairwise answers into one picture.

Two-set counting

With two sets, use total equals first plus second minus both.

• If all are in some set, those outside the first are only in the second.
• 40 people, 25 speak Hindi, 20 speak both: only Urdu is 40 - 25 = 15.
• Only Hindi is 25 - 20 = 5.

Check by adding: only Hindi 5, both 20, only Urdu 15 gives 40. A Venn sketch makes this quick.

Deduction and argument forms

Validity is about form

A deductive argument is valid when true premises cannot give a false conclusion.

• Invalid means premises can be true while the conclusion is false.
• A valid argument can have false premises and a false conclusion.
• All four-legged things have wings; spiders have four legs; so spiders have wings: valid, false content.

Test the form first, then the content separately. That spider argument is Barbara, a valid form, with false premises and a false conclusion. Soundness is validity plus true premises.

Patterns of deduction

Know the standard deductive patterns.

• Modus ponens: if raining, the ground is wet; it is raining; so the ground is wet.
• Affirming the consequent and denying the antecedent are invalid.
• Chain argument: if A then B, if B then C, so if A then C.
• Arguments from mathematics and from definition are deductive.

'Because x = 3 and y = 5, then x + y = 8' is a deductive argument based on mathematics. A predictive argument is not a pattern of deduction because prediction goes beyond the premises.

Steps to evaluate a deductive argument

Evaluate in order: identify, test form, check premises, decide soundness.

• Identify premises and conclusion first.
• Test the logical structure for validity.
• Assess the truth of the premises.
• Determine soundness last.

Nothing can be assessed until the roles are clear, so identification comes first. Soundness needs both a valid form and true premises, so it must come last.

Premise and conclusion, deduction against induction, analogy, and the line between truth and validity.

Structure of an argument

Premise, conclusion and indicator words

An argument is a group of statements where some, the premises, are offered in support of one, the conclusion.

• Premise indicators: because, for, since, in as much as.
• Conclusion indicators: therefore, accordingly, hence, thus.
• A statement is a sentence that can be true or false.
• A premise gives a reason; a conclusion is what the reasons support.

The word after the indicator tells you which part you are reading. 'Every law is an evil, for every law is an infraction of liberty' is an argument, with 'for' introducing the premise. In 'Accordingly, science sometimes requires courage', the claim after 'accordingly' is the conclusion.

Argument or explanation?

An argument supports a claim in doubt. An explanation tells why an accepted fact happened.

• 'I ate because I was hungry' contains 'because' but is an explanation.
• Nobody doubts the fact explained.
• The word 'therefore' can be narrative, not logical.

Ask whether the main statement needs proof. If it is already accepted and the sentence tells why, it is an explanation, not an argument. The Tower of Babel passage accounts for the name, so it is not an argument.

Strong and weak arguments

In a strong argument the consideration is relevant and has real weight.

• A strong argument addresses the question directly.
• A weak argument dismisses a cost without weighing it or is irrelevant.
• A premise that counts against the conclusion is negatively relevant.

Accuracy and efficiency are real benefits for the question 'Should India go in for e-governance', so that argument is strong. Being two years old is negatively relevant to going to college.

Deduction, induction and analogy

Deductive or inductive?

Test whether the conclusion follows with strict necessity.

• Deductive: if the premises are true, the conclusion must be true.
• Inductive: the conclusion is only probable.
• 'Alan is a father, therefore Alan is male' passes the strict necessity test.
• Indicator words and common patterns are weaker tests.

Inductive arguments go from particulars to universals: this crow is black, that crow is black, so all crows are black. They add new knowledge and are the leap from the known to the unknown. Deduction only makes explicit what is already in the premises.

Vocabulary: valid, sound, strong, cogent

Each word belongs to one kind of argument.

• Deductive: valid or invalid.
• Valid with all true premises: sound.
• Inductive: strong or weak. Validity does not apply.
• Strong with all true premises: cogent.

Wrong options swap the vocabulary, such as 'deductive arguments are weak or strong' or 'validity applies to inductive arguments'. Arguments by analogy are inductive, so their conclusions are probable and are assessed as strong or weak.

Analogy and argument from analogy

An analogy compares two things. An argument from analogy draws a conclusion from the likeness.

• A descriptive analogy infers nothing: 'Abhay's mind is a cave'.
• 'A and B share X and Y; A has Z; so B has Z' is analogy.
• Analogical arguments are inductive and probable.

A sentence with a comparison but no conclusion is an analogy only, so validity and soundness do not arise. 'Habits are like a cable' carries a conclusion across, so it is an argument from analogy.

Building and judging an analogy

A good analogy has shared attributes, similar structure and a clear relation to the conclusion.

• Build: known case, target case, shared attribute, draw analogy, evaluate.
• Evaluate: source and target, common features, relevance, conclusion, strength.
• Dissimilar subject is not a mark of a good analogy.

In both sequences the evaluation comes last. The more relevant and extensive the similarities, the stronger the analogy. Analogical relation questions ask the relation between the first pair and apply it: sing is to song as act is to play.

Truth versus validity

Truth belongs to statements, validity to arguments

The rule asked again and again: truth and falsity attach to propositions; validity and invalidity attach to arguments.

• An argument is never true or false.
• A single proposition is never valid or invalid.
• Falsity is not an attribute of an argument.
• Validity needs a relation between premises and conclusion.

Wrong options cross the vocabularies, such as 'Validity applies to any single proposition' or 'Truth and falsehood apply to arguments'. Answers often say 'Both statements are correct' when both halves of the rule are stated.

The four combinations

Validity does not guarantee true content. Check each combination.

• Valid with true premises means true conclusion.
• A valid argument can have false premises and a true conclusion, or a false conclusion.
• An invalid argument can have any combination of truth values.
• A valid argument cannot have true premises and a false conclusion.

'All dogs are cats; all cats are whales; therefore all whales are dogs' is invalid, with false premises and a false conclusion. Valid with false premises and false conclusion is possible, as in the cats, birds and fish example.

Spot the faulty reasoning: formal fallacies in the form, informal fallacies in the content, and how they are classified.

Informal fallacies

Begging the question

The conclusion is assumed in a premise. Other names: petitio principii and circular argument.

• Premise restates the conclusion.
• 'People who lack arrogance are intelligent because intelligent people do not have arrogance.'
• 'The world was created by an intelligent God; look at its organisation.'
• Ignoratio elenchi is a different fallacy: irrelevant conclusion.

Ask whether the premise could be accepted by someone who doubts the conclusion. If not, the argument begs the question. The design argument treats the world's order as proof of a designer, which is the very point at issue.

Appeal to ignorance, majority and emotion

Fallacies of relevance replace evidence with something else.

• Ignorance: true because not proved false, or false because not proved true.
• Majority (ad populum): true because everyone says so, or polls show support.
• Emotion: feelings in place of logical connection.
• Ad hominem attacks the person; ad baculum appeals to force.

'No one has shown aliens did not make crop circles, so they did' is appeal to ignorance. So is 'astrology is unproved, so it is nonsense'. Popular support does not make a policy effective, which is appeal to majority. Red herring brings in a distraction.

Composition and division

Composition goes from parts to whole. Division goes from whole to parts.

• 'The universe is spherical because its parts are spherical': composition.
• 'American Indians are disappearing; that man is one; so he is disappearing': division.
• 'Big poet because he came from a big country': division.
• 'A pot has mass, so its parts have mass' is NOT a fallacy.

Both depend on whether the property is shared between part and whole. Mass is distributive: a pot has mass because its parts do. Length is not: points have none, yet a line has length. Two aspirins double the protection is composition.

Hasty generalisation, false cause, false dichotomy

Three more common informal fallacies.

• Hasty generalisation: a tiny sample, a sweeping claim. 'Ford cars are lemons, I owned two.'
• False cause: correlation read as causation. Banana eaters, more laws and more crime.
• False dichotomy: only two options offered when more exist.

In the Las Vegas example the only options are a holiday or lifelong misery, so the claim ignores holidaying elsewhere or later. For false cause, the causation may run the other way or another factor may drive both.

Ambiguity, accident and straw man

Know which fallacies belong to ambiguity.

• Fallacies of ambiguity: equivocation, amphiboly, accent.
• Accident is a fallacy of presumption, not ambiguity.
• Composition and division are classed with ambiguity in the item asked.
• Straw man misrepresents an opponent's position as more extreme.

Equivocation uses one word with two meanings. Straw man depicts an opponent's position as more extreme or unreasonable than what was said, then attacks that version. Accident applies a general rule to an exceptional case.

Classifying informal fallacies by ground

Informal fallacies are grouped by the ground of the fault.

• Relevance: appeal to emotion, ad hominem.
• Defective induction: argument from ignorance, false cause.
• Ambiguity: composition, division, equivocation.
• Presumption: complex question, accident.

The match-the-list item asks which GROUND each fallacy falls under, not what the fallacy is. On 30 August 2024 the pairs were false cause with defective induction, emotion with relevance, and composition with ambiguity.

Formal fallacies

Affirming the consequent and denying the antecedent

Two invalid forms of the conditional argument.

• If P then Q; Q; so P: affirming the consequent.
• If P then Q; not P; so not Q: denying the antecedent.
• Valid forms: affirm the antecedent, or deny the consequent.
• Both fallacies are deductive, formal fallacies.

'If it rains the drought will end; the drought ended; so it rained' affirms the consequent. 'If your pet is a cat it has a tail; it is not a cat; so no tail' denies the antecedent. Other things also produce the consequent.

Syllogism fallacies: existential, exclusive, affirmative from negative

Three formal syllogism fallacies repeat in the papers.

• Exclusive premises: both premises negative.
• Affirmative conclusion from a negative premise.
• Existential fallacy: particular conclusion from two universals when a class may be empty.
• Undistributed middle: all Russians were revolutionaries, all anarchists were revolutionaries.

'No tragic actors are idiots; some comedians are not idiots' has two negative premises. 'No poets are scientists; some artists are poets; so some artists are scientists' has an affirmative conclusion from a negative premise. Unicorns do not exist, which signals the existential fallacy.

Nyaya inference: the five members, vyapti, the fallacies of the middle term, the means of knowledge, and the schools of Indian philosophy.

Nyaya inference

Five members and three terms

The Nyaya syllogism has five members (propositions) but only three terms.

• Pratijna: the thesis. Hetu: the reason.
• Udaharana: the universal rule with an example.
• Upanaya: the application to this case. Nigamana: the conclusion.
• Three terms: paksa (minor), sadhya (major), hetu or linga (middle).

The Aristotelian syllogism has three propositions and three terms. A common trap says Nyaya has three propositions or five terms. The Nyaya form is both inductive and deductive, and both formal and material. Upanaya for the hill example reads 'The hill has smoke which is invariably associated with fire'.

Vyapti and upadhi

Vyapti is invariable concomitance between the hetu and the sadhya.

• Wherever there is smoke, there is fire.
• It needs no qualifying condition (upadhi).
• It holds between hetu and sadhya, not between hetu and paksa.
• Wherever there is fire, there is smoke is conditional (sopadhika).

Fire gives smoke only if the fuel is wet. Red-hot iron has fire without smoke. So that universal is false, and an inference built on it is fallacious. The genuine vyapti runs from smoke to fire.

Pramanas and perception

Know the means of knowledge, their names and the kinds of perception.

• Nyaya pramanas: perception, inference (anumana), comparison, testimony.
• Arthapatti is presumption; Devadatta grows fat though he does not eat by day.
• Savikalpaka perception is determinate; nirvikalpaka is indeterminate.
• Laukika is ordinary perception; alaukika is extraordinary; yogaja is yogic.

Devadatta eats at night is arthapatti, called implication or postulation. It is not inference from a sign. Anupalabdhi is non-perception, the means for knowing absence.

Forms of non-existence

Four kinds of absence (abhava) in Nyaya-Vaisesika.

• Pragabhava: before a thing is produced. 'The pot will exist.'
• Pradhvamsabhava: after destruction.
• Atyantabhava: absolute, never anywhere. Hare's horn; air has no colour.
• Anyonyabhava: mutual, 'this is not that'. The pot is not cloth.

One item asked which pairing is INCORRECT. 'The pot is not cloth' is anyonyabhava, not pradhvamsabhava, so that was the incorrect one.

Hetvabhasa: fallacies of the reason

The named fallacies

Match each fallacious reason to its name.

• Viruddha: the reason proves the opposite. Sound is eternal because it is produced.
• Badhita: contradicted by another pramana. Fire is cold because it is a substance.
• Savyabhicara (anaikantika): irregular middle, found with and without the major term.
• Asiddha: unproved reason. Satpratipaksa: balanced by an opposite reason.

'Produced' and 'caused' mean the same, so 'Sound is eternal because it is caused' is viruddha too. Perception shows fire is hot, so 'cold fire' is badhita. 'All bipeds are rational' with swans is savyabhicara.

Asadharana, kevalanvayi and non-existent subject

Special fallacies when the middle term is too narrow or too wide.

• Asadharana: the middle term belongs only to the subject. Sound is permanent because it is audible.
• 'Anything thinkable is nameable because it is thinkable': asadharana.
• All-pervasive major and middle terms leave no negative instance: kevalanvayi.
• Sky-lotus has no locus, so the middle term has no place to stand.

Audibility belongs only to sound, so it cannot support the inference either way. For the sky-lotus, the real defect is that the subject does not exist, which differs from the too-wide middle.

Reading a Nyaya fallacy question

Recognise the definition in the stem and match it to the example given.

• Pervaded by the absence of the major term: viruddha.
• Contradicted by non-inferential knowledge: badhita.
• Instead of proving presence, proves non-existence: viruddha.
• Fallacy of irregular middle: choose the reasons that go both ways.

The stems restate the same definition in longer words. In August 2024, 'Sound is eternal because it is produced' answered both viruddha stems. The June 2023 irregular-middle item was keyed to 'audible' and 'knowable', so follow the key of the paper in front of you.

Indian philosophy

Schools and texts

Match schools with doctrines and authors with their texts.

• Gautama: Nyaya Sutra. Kanada: Vaisesika Sutra.
• Patanjali: Yoga Sutra. Kautilya: Arthashastra.
• Atomism: Vaisesika.
• Mimamsa is one of the six astika (orthodox) schools, centred on Vedic ritual and sabda.

Astika schools accept the authority of the Vedas. The Four Noble Truths of Buddhism run in order: dukkha, samudaya, nirodha, marga.

Vedanta on manas and perception

In Vedanta, manas is not an independent substance. It is a function of the antahkarana.

• Manas is not an indriya.
• In perception, manas goes out to the object through the senses.
• The antahkarana takes the form of the object.

These items were multi-statement questions, so check each statement against these three facts. Wrong options make the antahkarana the channel or make manas fundamental.

Coding-decoding, letter and alphanumeric series, blood relations, direction sense, ordering and small logic puzzles.

Coding and decoding

Sum of letter positions

Many codes add the alphabet positions of the letters, sometimes halved.

• A = 1, B = 2 ... Z = 26.
• CONSTABLE = 91 and STABLE = 59, so PORTABLE = 89.
• DELHI = 19 and CHENNAI = 27: half the sum. GUWAHATI = 45.
• Test the rule on the second example before using it.

PRIME = 61, SIGHT = 63, QUITE = 72 and RHYME = 69 by plain sum. A shortcut: CONSTABLE minus STABLE leaves C, O, N, which sum to 32, the gap between 91 and 59.

Shift codes

Each letter moves forward or backward by a fixed or alternating amount.

• SOUND to VLXKG: +3, -3, +3, -3, +3. Then TRAIN gives WODFQ.
• FLOUT to IIRRW follows the same rule; CLING gives FILKJ.
• PRIME to NPGKC: each letter back two.
• Write the shift under each letter first.

The same four words came with a number code and a letter code, so check for a sum, then a shift. A reflected alphabet cipher maps A to Z, B to Y, so position p becomes 27 minus p. PERFECT then codes to KVIUVXG.

Substitution across sentences

Match words to codes by what the sentences share.

• Find the word common to two or three sentences.
• Find the code common to the same sentences.
• That word and code are a pair; remove it and repeat.
• Do not match by position.

In 'Indian cuisines are good', Indian is the only word in sentences 1, 2 and 3. MI is the only common code, so Indian = MI. Then solve the rest by intersection.

Series

Letter series and alphanumeric series

Split the term into columns and treat each as its own number series.

• ALY, CJW, EHU: first +2, second -2, third -2, so GFS.
• BFD, ELG, HRJ: +3, +6, +3, so KXM.
• AM, GJ, MG, SD: +6 and -3, so YA.
• 2Z5, 7Y7, 14X9: numbers +5, +7, +9, letters fall one, last numbers +2, so 47U15.

Convert letters to positions. B, D, G, K, P, V has gaps +2, +3, +4, +5, +6 and wraps to K after Z. The sequence B, B, C, D, F, I, N, V has Fibonacci gaps. AaC, BbD, DdF, HhJ doubles the first letter's position each time, so PpR.

Relations, direction and order

Blood relations

Build the tree from the statements and read off the required relation.

• Madhu's husband is Apoorav's father; his brother is Apoorav's uncle.
• Aunt's son is a cousin.
• The only daughter of my maternal grandmother is my mother.
• 'His wife's father is my mother's only daughter's husband': mother-in-law.

Start with the clearest relation and add one person at a time. Mark gender where a word gives it, such as husband, sister or daughter.

Direction sense

Track position and facing direction step by step.

• Take the start as the origin.
• A left turn from north faces west; a left from west faces south.
• 12 km south, 7 km north, 5 km left gives 5 root 2 km south-west.
• Rank from the bottom equals total minus rank from the top, plus one.

For angle questions, work in degrees clockwise from north. Anticlockwise turns subtract. Walk 70 m south, 30 m east, 40 m north: you are 30 m east and 30 m south, so return north-west.

Ordering and small puzzles

Chain the comparisons, then read the answer.

• Ram > Vineet > Monika > Jacob > Dalip: the tallest is Ram.
• Bangalore > Jhansi > Rajgarh > Satpura > Chittor: the smallest is Chittor.
• Seating: facing the centre, a person's left hand points clockwise.
• Teachers and subjects: collect each person's subjects from every statement.

Combine statements into one line before answering. For the flight crew and the golf players, use each clue to eliminate options until one remains.