Topics Covered
Structure of Arguments
Categorical Propositions
Mood and Figure
Formal and Informal Fallacies
Connotation and Denotation
Square of Opposition
Deduction and Induction
Analogies
Venn Diagrams
Pramanas
Anumana and Vyapti
Hetvabhasas
Contents
- 1. The Structure of Arguments and Argument Forms
- 2. Categorical Propositions, Mood and Figure
- 3. Formal and Informal Fallacies
- 4. Uses of Language; Connotation and Denotation
- 5. The Classical Square of Opposition
- 6. Deductive and Inductive Reasoning
- 7. Analogies
- 8. Venn Diagrams and Validity
- 9. Indian Logic: Means of Knowledge
- 10. The Pramanas
- 11. Anumana, Vyapti and the Hetvabhasas
Topic Overview — What & Why
Unit VI joins two traditions: the Western logic of arguments, categorical propositions and fallacies, and the Indian logic of the Nyaya school and its rivals. The Western half is mechanical once the rules are known; a Venn diagram settles any syllogism. The Indian half is mostly terms and their meanings, and a few counts that are asked again and again (how many pramanas each school accepts; the five members of the inference; the five fallacies).
- An argument is premises offered in support of a conclusion; it is valid if the conclusion must be true when the premises are.
- Categorical propositions are A, E, I and O; a syllogism's mood lists them, its figure places the middle term.
- The square of opposition relates A, E, I and O: contradictory, contrary, subcontrary, subaltern.
- Pramanas are the means of valid knowledge: perception, inference, comparison, testimony, postulation and non-apprehension.
1. The Structure of Arguments and Argument Forms
An argument is a set of statements in which one, the conclusion, is claimed to follow from the others, the premises. Words such as since, because, for mark premises; therefore, hence, so, it follows that mark the conclusion.
- A deductive argument is valid when it is impossible for the premises to be true and the conclusion false. It is sound when it is valid and its premises are true. Validity is a matter of form; truth is a matter of fact.
- A statement is true or false; an argument is valid or invalid. "A valid statement" and "a true argument" are both category mistakes.
Argument forms
| Form | Pattern | Valid? |
|---|---|---|
| Modus ponens (affirming the antecedent) | If P then Q; P; therefore Q | Valid |
| Modus tollens (denying the consequent) | If P then Q; not Q; therefore not P | Valid |
| Hypothetical syllogism | If P then Q; if Q then R; therefore if P then R | Valid |
| Disjunctive syllogism | P or Q; not P; therefore Q | Valid |
| Affirming the consequent | If P then Q; Q; therefore P | Invalid (a formal fallacy) |
| Denying the antecedent | If P then Q; not P; therefore not Q | Invalid (a formal fallacy) |
EXAMPLE 1
"If it rains, the match is cancelled. The match was not cancelled. So it did not rain." Modus tollens: valid.
EXAMPLE 2
"If he is a doctor, he has a degree. He has a degree. So he is a doctor." Affirming the consequent: invalid, since many people with degrees are not doctors.
2. Categorical Propositions, Mood and Figure
| Type | Form | Quality, quantity | Distributes |
|---|---|---|---|
| A | All S is P | Universal affirmative | Subject only |
| E | No S is P | Universal negative | Both terms |
| I | Some S is P | Particular affirmative | Neither term |
| O | Some S is not P | Particular negative | Predicate only |
The letters come from the Latin AffIrmo (I affirm) and nEgO (I deny). A term is distributed when the proposition says something about every member of its class.
The syllogism
A categorical syllogism has three propositions and three terms: the major term (the predicate of the conclusion), the minor term (its subject) and the middle term (in both premises, not in the conclusion).
- Mood: the types of the major premise, the minor premise and the conclusion, in that order: AAA, EAE, AII, and so on.
- Figure: where the middle term (M) stands. Figure 1: M–P, S–M. Figure 2: P–M, S–M. Figure 3: M–P, M–S. Figure 4: P–M, M–S.
The rules of a valid syllogism
- The middle term must be distributed at least once (else: undistributed middle).
- A term distributed in the conclusion must be distributed in its premise (else: illicit major or illicit minor).
- No conclusion follows from two negative premises (exclusive premises).
- If a premise is negative, the conclusion must be negative, and the reverse.
- No conclusion follows from two particular premises.
- There must be exactly three terms, used in the same sense (else: four terms).
EXAMPLE 1
"All mammals are animals; all dogs are mammals; so all dogs are animals." Mood AAA, figure 1 (M–P, S–M): the valid form traditionally called Barbara.
EXAMPLE 2
"All cats are animals; all dogs are animals; so all dogs are cats." The middle term, animals, is the predicate of two A propositions and is never distributed: undistributed middle.
3. Formal and Informal Fallacies
A formal fallacy is an error in the form of an argument (undistributed middle, illicit major or minor, affirming the consequent, denying the antecedent, four terms). An informal fallacy is an error in content or language: the form may look fine, but the premises do not really support the conclusion.
| Informal fallacy | What it does |
|---|---|
| Ad hominem | Attacks the person instead of the argument |
| Ad populum | Appeals to popularity: many believe it, so it is true |
| Ad baculum | Appeals to force or threat |
| Ad verecundiam | Appeals to an authority outside their field |
| Ad ignorantiam | It has not been proved false, so it is true (or the reverse) |
| Ad misericordiam | Appeals to pity |
| Straw man | Attacks a distorted version of the opponent's view |
| Red herring | Changes the subject to a different issue |
| Petitio principii (begging the question) | Assumes the conclusion among the premises |
| Hasty generalisation | Generalises from too few cases |
| Post hoc ergo propter hoc | After this, therefore because of this |
| False dilemma | Offers two options when there are more |
| Slippery slope | Claims one step must lead to a chain of bad ones, without showing why |
| Equivocation | Uses one word in two senses |
| Composition and division | What is true of the parts must be true of the whole, and the reverse |
EXAMPLE 1
"Ravi's argument for the new syllabus is worthless; he has never taught a day in his life." Ad hominem.
EXAMPLE 2
"The crime rate fell after the new commissioner took charge, so she cut crime." Post hoc ergo propter hoc: the fall came after, which does not show it came because of.
4. Uses of Language; Connotation and Denotation
- Uses of language (as in Irving Copi's account): informative (to state facts, true or false), expressive (to express or arouse feeling), and directive (to cause or prevent action: commands, requests). Some add ceremonial (greetings, rituals) and performative ("I promise", "I name this ship").
- Denotation (extension) of a term is the class of things it applies to. Connotation (intension) is the set of attributes that something must have for the term to apply.
- As connotation increases, denotation decreases (or stays the same): animal → mammal → dog adds attributes and narrows the class. This is the law of inverse variation.
EXAMPLE 1
"Please submit the form by Friday" is directive; "The form was due on Friday" is informative; "What a relief that it's done!" is expressive.
EXAMPLE 2
The denotation of planet of the solar system is Mercury, Venus, Earth and the rest; its connotation is "a body that orbits the Sun, is nearly round, and has cleared its orbit".
5. The Classical Square of Opposition
The traditional square assumes that the subject class has members (existential import). On that reading:
| Relation | Between | Rule |
|---|---|---|
| Contradictories | A and O; E and I | Exactly one is true: they cannot both be true or both be false |
| Contraries | A and E | They cannot both be true; they can both be false |
| Subcontraries | I and O | They cannot both be false; they can both be true |
| Subalternation | A to I; E to O | If the universal is true, the particular is true; if the particular is false, the universal is false |
In modern (Boolean) logic, which drops existential import, only the contradictories survive. Questions in this paper follow the traditional square unless they say otherwise.
EXAMPLE 1
"All students are present" (A) is given as true. Then "Some students are present" (I) is true, "No students are present" (E) is false, and "Some students are not present" (O) is false.
EXAMPLE 2
"Some politicians are honest" (I) is given as false. Then its contradictory "No politicians are honest" (E) is true; "All politicians are honest" (A) is false; and "Some politicians are not honest" (O) is true, since I and O cannot both be false.
6. Deductive and Inductive Reasoning
| Deductive | Inductive | |
|---|---|---|
| Claim | The conclusion follows necessarily | The conclusion follows with probability |
| Judged as | Valid or invalid; sound or unsound | Strong or weak; cogent or not |
| New premises | Cannot make a valid argument invalid | Can strengthen or weaken it |
| Content | The conclusion makes explicit what the premises contain | The conclusion goes beyond the premises |
Forms of induction: enumeration (many cases, one generalisation), analogy, causal reasoning by John Stuart Mill's methods (agreement, difference, the joint method, residues, concomitant variation), and statistical generalisation from a sample.
EXAMPLE 1
"Every sample of this alloy we have heated melted at 660 °C, so all samples will." Induction by enumeration; strong, but not certain.
EXAMPLE 2
Two groups of plants are alike except that one is given a fertiliser; only that group grows faster. Mill's method of difference.
7. Analogies
In an analogy question, A : B :: C : ?, find the relation between A and B and apply the same to C. Common relations: worker and tool, product and raw material, part and whole, cause and effect, degree (warm : hot), opposite, function, and the class a thing belongs to.
An argument from analogy is stronger when the things compared share more relevant features and the differences are irrelevant, and weaker when the conclusion claims too much.
EXAMPLE 1
Author : Book :: Composer : ? — an author creates a book, so a composer creates a symphony (not a piano, which is an instrument).
EXAMPLE 2
Thermometer : Temperature :: Barometer : ? — an instrument and what it measures: atmospheric pressure.
8. Venn Diagrams and Validity
Draw three overlapping circles for the three terms. For each premise, shade the regions a universal proposition says are empty, and put an X in the region a particular proposition says is occupied. Draw the universal premises first. Then look: if the conclusion is already drawn, the syllogism is valid.
- A (All S is P): shade the part of S outside P.
- E (No S is P): shade the overlap of S and P.
- I (Some S is P): an X in the overlap of S and P.
- O (Some S is not P): an X in S outside P.
In the common question "Statements … Conclusions I and II … which follow?", draw the least picture the statements allow. A conclusion follows only if it holds in every picture that fits the statements, not just the obvious one. This page reads the statements in the traditional way: each class named has at least one member.
EXAMPLE 1
Statements: All pens are books. Some books are chairs. Conclusions: I. Some pens are chairs. II. Some chairs are books. — I does not follow: the books that are chairs may all be outside the pens. II follows: "Some books are chairs" converts to "Some chairs are books". Only II follows.
EXAMPLE 2
Statements: No cat is a dog. All dogs are animals. Conclusions: I. Some animals are not cats. II. No animal is a cat. — Every dog is an animal and no dog is a cat, and dogs exist, so some animals are not cats: I follows. II does not: there may be animals, outside the dogs, that are cats. Only I follows.
9. Indian Logic: Means of Knowledge
- Prama is valid knowledge; pramana is the means of obtaining it; prameya is the object known; pramata is the knower.
- Knowledge that is not valid is aprama: doubt (samshaya), error (viparyaya) and hypothetical reasoning (tarka) are its kinds in Nyaya.
- The schools differ on how many pramanas there are. Each accepts the ones before it in the list below and adds its own.
| School | Number | Pramanas accepted |
|---|---|---|
| Charvaka (Lokayata) | 1 | Pratyaksha |
| Buddhism; Vaisheshika | 2 | Pratyaksha, anumana |
| Sankhya; Yoga | 3 | Pratyaksha, anumana, shabda |
| Nyaya | 4 | Pratyaksha, anumana, upamana, shabda |
| Prabhakara Mimamsa | 5 | The four, and arthapatti |
| Bhatta Mimamsa; Advaita Vedanta | 6 | The five, and anupalabdhi |
Jainism classifies knowledge differently, into the direct (pratyaksha) and the indirect (paroksha), and is usually left out of this count.
EXAMPLE 1
A question asks which school accepts perception alone. The answer is Charvaka; it rejects even inference, since no universal connection can be perceived.
EXAMPLE 2
A question asks which pramana Nyaya accepts but Sankhya does not. Upamana (comparison).
10. The Pramanas
| Pramana | Meaning | Classic example |
|---|---|---|
| Pratyaksha (perception) | Knowledge from the contact of a sense with its object | Seeing a pot in front of you |
| Anumana (inference) | Knowledge of something through a mark that is invariably connected with it | Inferring fire on the hill from the smoke rising from it |
| Upamana (comparison) | Knowledge of what a word names, from a described likeness | Told that a gavaya (wild ox) looks like a cow, a person recognises one in the forest |
| Shabda (verbal testimony) | Knowledge from the words of a trustworthy person (apta) or from scripture | Learning a country's capital from a reliable teacher |
| Arthapatti (postulation, implication) | Supposing what alone explains an otherwise conflicting fact | Devadatta grows fat though he never eats by day, so he must eat at night |
| Anupalabdhi (non-apprehension) | Knowing an absence from not perceiving what would be perceived if present | Knowing there is no pot on the floor because none is seen there |
- Perception is nirvikalpaka (indeterminate: bare awareness, before naming) or savikalpaka (determinate: "this is a pot"). Nyaya also speaks of ordinary (laukika) and extraordinary (alaukika) perception.
EXAMPLE 1
"The room is dark and quiet, so the children are not at home." The absence is known from not perceiving them where they would be perceived: anupalabdhi.
EXAMPLE 2
"He is alive, and he is not at home, so he must be somewhere outside." The only supposition that reconciles the two facts: arthapatti.
11. Anumana, Vyapti and the Hetvabhasas
The terms
| Nyaya term | Meaning | In "the hill has fire because it has smoke" | Western equivalent |
|---|---|---|---|
| Paksha | The subject of the inference | The hill | Minor term |
| Sadhya | What is to be proved of it | Fire | Major term |
| Hetu (linga) | The reason, the mark | Smoke | Middle term |
Vyapti is the invariable relation between the hetu and the sadhya: wherever there is smoke, there is fire. It is the ground of every inference. Paksadharmata is the presence of the hetu in the paksha (the smoke is on the hill).
The structure: the five members of the Nyaya inference
- Pratijna (proposition): The hill has fire.
- Hetu (reason): Because it has smoke.
- Udaharana (example): Whatever has smoke has fire, as in a kitchen.
- Upanaya (application): The hill has smoke, which is always accompanied by fire.
- Nigamana (conclusion): Therefore the hill has fire.
Kinds of anumana
- Svarthanumana, inference for oneself, and pararthanumana, inference for others, set out in the five members.
- In the Nyaya Sutra: purvavat (from cause to effect), sheshavat (from effect to cause) and samanyatodrishta (from what is commonly seen together).
- By the kind of vyapti: kevalanvayi (only agreement in presence), kevalavyatireki (only agreement in absence), anvayavyatireki (both).
The five hetvabhasas (fallacies of inference)
| Hetvabhasa | The fault in the reason | Example |
|---|---|---|
| Savyabhichara (anaikantika) | Irregular: it goes with both the sadhya and its absence | "The hill has fire because it is knowable" (the lake is knowable too) |
| Viruddha | Contradictory: it proves the opposite of the sadhya | "Sound is eternal because it is produced" (whatever is produced is non-eternal) |
| Satpratipaksha | Counter-balanced: another reason proves the opposite | "Sound is eternal because it is audible" against "sound is non-eternal because it is produced" |
| Asiddha | Unproved: the reason itself is not established (in the paksha, or at all) | "The sky-lotus is fragrant because it is a lotus" (there is no sky-lotus) |
| Badhita | Contradicted (sublated): the conclusion is refuted by another pramana | "Fire is cold because it is a substance" (perception shows fire is hot) |
EXAMPLE 1
"The lake has fire because it has vapour." Vapour is mistaken for smoke; the reason does not really belong to the paksha. Asiddha.
EXAMPLE 2
A question lists the five members and asks which one states the universal rule with an example. Udaharana.
Exam pointers for Unit VI.
- Valid: modus ponens, modus tollens, hypothetical and disjunctive syllogism. Invalid: affirming the consequent, denying the antecedent.
- A distributes S; E both; I neither; O P.
- Contradictories A–O and E–I; contraries A–E; subcontraries I–O; subalterns A→I, E→O.
- Connotation up, denotation down.
- Pramanas: Charvaka 1, Buddhism and Vaisheshika 2, Sankhya and Yoga 3, Nyaya 4, Prabhakara 5, Bhatta and Advaita 6.
- Paksha = minor, sadhya = major, hetu = middle; vyapti = invariable concomitance.
- Five members: pratijna, hetu, udaharana, upanaya, nigamana. Five hetvabhasas: savyabhichara, viruddha, satpratipaksha, asiddha, badhita.
Final Revision Checklist
- Premises and conclusion; validity and soundness; the six argument forms.
- A, E, I, O and distribution; mood and the four figures; the six rules.
- Formal against informal fallacies, and the fifteen informal ones above.
- Uses of language; connotation, denotation and inverse variation.
- The square of opposition, traditional reading.
- Deduction against induction; Mill's methods; analogies.
- Venn diagrams for syllogisms and for "which conclusions follow".
- Prama and pramana; the schools and their counts; the six pramanas with examples; anumana, vyapti, the five members and the five hetvabhasas.
Sources
- I. M. Copi, C. Cohen and K. McMahon, Introduction to Logic (fallacies, uses of language, the square of opposition, Venn diagrams).
- J. S. Mill, A System of Logic (1843) (the methods of induction).
- S. Chatterjee and D. Datta, An Introduction to Indian Philosophy (the pramanas of each school, anumana and the hetvabhasas).
- The Nyaya Sutra of Gautama, with Vatsyayana's commentary (the five members; purvavat, sheshavat, samanyatodrishta).