LOGICAL REASONING
& INDIAN LOGIC (UNIT 6)
Complete Conceptual & Exam Blueprint — Categorical Propositions, Square of Opposition, Syllogisms, Moods & Figures, Fallacies, Pramanas, Vyapti, and Hetvabhasa.
MODULE 01: STRUCTURE OF ARGUMENTS & CATEGORICAL PROPOSITIONS
1.1 What is an Argument?
An Argument is a set of declarative statements (propositions) in which one statement is claimed to follow from the others. It consists of:
- Premises: The supporting evidence, reasons, or assumptions (indicators: Since, Because, For, Given that, As indicated by).
- Conclusion: The statement being established or inferred (indicators: Therefore, Hence, Thus, So, Consequently, It proves that).
1.2 Categorical Propositions (Standard Form)
A categorical proposition asserts or denies that all or some members of one category (Subject term S) are included in another category (Predicate term P). Every standard form has four parts: Quantifier + Subject term + Copula + Predicate term.
| Type | Standard Form | Quantity | Quality | Distribution of Terms |
|---|---|---|---|---|
| A | All S are P | Universal | Affirmative | Subject: Distributed | Predicate: Undistributed |
| E | No S are P | Universal | Negative | Subject: Distributed | Predicate: Distributed |
| I | Some S are P | Particular | Affirmative | Subject: Undistributed | Predicate: Undistributed |
| O | Some S are not P | Particular | Negative | Subject: Undistributed | Predicate: Distributed |
A distributes Subject only.
E distributes Both (Subject & Predicate).
I distributes None (Neither Subject nor Predicate).
O distributes Predicate only.
Short formula: A=S, E=Both, I=None, O=P.
- "Every human is mortal" : All humans are mortal (A type).
- "Only graduates are eligible" : All eligible candidates are graduates (A type — "Only S are P" reverses to "All P are S").
- "Almost all students passed" : Some students passed (I type).
- "Few teachers attended" / "Hardly any" : Some teachers are not attendees (O type — negative connotation!).
- "None but the brave deserve the fair" : All who deserve the fair are brave (A type).
MODULE 02: CLASSICAL SQUARE OF OPPOSITION
The Square of Opposition illustrates the four distinct logical relationships between the four categorical propositions (A, E, I, O) that share the exact same Subject and Predicate terms.
2.1 The 4 Classical Opposition Relations (Strict Rules)
| Relation | Pairs | Truth-Value Golden Rule | Exam Inference Formula |
|---|---|---|---|
| Contradictory | A – O E – I |
Always exact opposites. If one is TRUE, the other is strictly FALSE. If one is FALSE, the other is strictly TRUE. | If A is True, then O is False If I is False, then E is True |
| Contrary | A – E | Both cannot be TRUE together, but both CAN be FALSE at the same time. | If A is True, then E is False If A is False, then E is Undetermined. |
| Sub-Contrary | I – O | Both cannot be FALSE together, but both CAN be TRUE at the same time. | If I is False, then O is True If I is True, then O is Undetermined. |
| Subalternation | A to I E to O |
Truth flows DOWNWARD (from Universal to Particular). Falsehood flows UPWARD (from Particular to Universal). |
If A is True, then I is True If I is False, then A is False |
1. If Universal is TRUE, then the Particular below it is strictly TRUE (A to I, E to O).
2. If Particular is FALSE, then the Universal above it is strictly FALSE (I to A, O to E).
3. If Universal is FALSE, then the subaltern particular is DOUBTFUL / UNDETERMINED.
4. If Particular is TRUE, then the superaltern universal is DOUBTFUL / UNDETERMINED.
MODULE 03: CATEGORICAL SYLLOGISMS, MOODS & FIGURES
3.1 Structure of a Categorical Syllogism
A Standard-form Categorical Syllogism is an argument with exactly two premises, one conclusion, and exactly three distinct terms, each occurring twice in different propositions.
- Major Term (P): The Predicate of the conclusion.
- Minor Term (S): The Subject of the conclusion.
- Middle Term (M): Occurs in both premises, but NEVER in the conclusion. It serves as the logical bridge.
- Major Premise: The premise containing the Major Term (P). Listed FIRST.
- Minor Premise: The premise containing the Minor Term (S). Listed SECOND.
3.2 The Four Syllogistic Figures
The Figure of a syllogism is determined exclusively by the position of the Middle Term (M) in the two premises:
3.3 Mood of a Syllogism
The Mood is the three-letter sequence representing the types of propositions in order: Major Premise + Minor Premise + Conclusion.
Example: If Major Premise is A, Minor Premise is A, and Conclusion is A, the Mood is AAA. Combined with Figure 1, the full identification is AAA-1 (Barbara).
• Total possible Moods = 4 × 4 × 4 = 64 combinations.
• Across 4 Figures = 64 × 4 = 256 standard syllogistic forms.
• Out of 256 forms, only 15 are unconditionally valid in modern Boolean logic (and 24 conditionally valid in traditional Aristotelian logic allowing existential import).
3.4 Six Inviolable Rules of Syllogistic Validity
| # | Rule of Validity | Fallacy Committed if Broken |
|---|---|---|
| 1 | A valid syllogism must contain exactly three terms used in the same sense. | Fallacy of Four Terms (Quaternio Terminorum) |
| 2 | The Middle Term must be distributed at least once in the premises. | Fallacy of Undistributed Middle |
| 3 | If a term is distributed in conclusion, it MUST be distributed in premise. | Illicit Major (if Major term) or Illicit Minor (if Minor term) |
| 4 | Two negative premises yield NO valid conclusion. | Fallacy of Exclusive Premises |
| 5 | If either premise is negative, conclusion MUST be negative (and vice versa). | Fallacy of Drawing an Affirmative Conclusion from a Negative Premise |
| 6 | From two universal premises, a particular conclusion cannot be drawn (Boolean logic). | Existential Fallacy |
MODULE 04: DEDUCTIVE VS INDUCTIVE REASONING & ANALOGIES
4.1 Deductive vs Inductive Reasoning (Key Distinctions)
| Feature | Deductive Reasoning | Inductive Reasoning |
|---|---|---|
| Direction of Thought | General to Particular (Top-Down) | Particular to General (Bottom-Up) |
| Certainty of Conclusion | 100% Necessary / Absolute | Probable / Likelihood (Never absolute) |
| Evaluation Metrics | Valid / Invalid & Sound / Unsound | Strong / Weak & Cogent / Uncogent |
| Information Content | Conclusion contains no new information beyond premises. | Conclusion goes beyond premises (ampliative). |
| Effect of Additional Evidence | Cannot alter validity of a valid argument. | New evidence can strengthen or weaken the conclusion. |
- Validity: Relates strictly to logical structure. An argument is valid if it is impossible for premises to be true and conclusion false simultaneously.
- Truth: Pertains to individual statements/premises matching real-world facts.
- Sound Argument: An argument that is Valid + All premises are FACTUALLY TRUE.
- Takeaway: An argument can be completely Valid even if its premises are blatantly false! (e.g., All elephants have wings; All winged creatures fly; therefore, All elephants fly — this is Valid, but Unsound!).
4.2 Analogical Reasoning
An Argument from Analogy infers that because two things are alike in several known respects (A, B, C), they will also be alike in another further respect (D).
- Number of entities: More observed instances with the resemblance increase strength.
- Number of respecting respects: More shared relevant properties increase probability.
- Relevance of similarities: Shared features must be causally or materially linked to conclusion.
- Disanalogies (Differences): Crucial dissimilarities severely weaken the analogy.
- Modesty of Conclusion: A cautious conclusion is stronger than a bold, sweeping assertion.
MODULE 05: VENN DIAGRAMS & USES OF LANGUAGE
5.1 Venn Diagram Method for Syllogism Validity
Developed by John Venn (1880), Venn diagrams represent categorical propositions using intersecting circles. For a three-term categorical syllogism, three overlapping circles represent S, P, and M.
Test of Validity: Diagram ONLY the two premises. If the conclusion is automatically depicted without drawing anything new, the syllogism is VALID. If not, it is INVALID.
5.2 Connotation and Denotation of Terms
| Dimension | Connotation (Intension) | Denotation (Extension) |
|---|---|---|
| Definition | The set of essential attributes, qualities, or traits shared by all objects to which the term applies. | The collection of all real physical objects or individuals that the term correctly names. |
| Example ("Philosopher") | Rational, reflective, seeker of truth and wisdom. | Socrates, Plato, Aristotle, Shankara, Nagarjuna, Kant... |
| Law of Inverse Variation | As connotation increases (more specific traits added), denotation decreases (fewer members fit). | As denotation increases, connotation decreases. |
1. Animal : Highest Denotation, Lowest Connotation.
2. Mammal : More attributes, fewer animals.
3. Canine : Even more attributes, fewer members.
4. Dog : Specific traits.
5. Golden Retriever : Highest Connotation, Lowest Denotation.
MODULE 06: FORMAL & INFORMAL FALLACIES
6.1 Formal Fallacies (Errors in Logical Form)
| Formal Fallacy | Pattern / Structure | Example |
|---|---|---|
| Affirming the Consequent | If P, then Q. Q. Therefore, P. |
If it rains, roads are wet. Roads are wet. therefore, It rained. (Invalid! Water tanker could cause it). |
| Denying the Antecedent | If P, then Q. Not P. Therefore, Not Q. |
If you are a cricketer, you are an athlete. You are not a cricketer. therefore, You are not an athlete. (Invalid!). |
| Undistributed Middle | All P are M. All S are M. Therefore, All S are P. |
All dogs are mammals. All cats are mammals. therefore, All cats are dogs. |
6.2 Informal Fallacies (Errors in Content & Context)
- Argumentum ad Hominem (Attack on Person): Rejecting an argument by attacking the opponent's background, character, or motive instead of their reasoning. (e.g., "You cannot trust his economic policy because he didn't finish college!").
- Straw Man Fallacy: Distorting, exaggerating, or misrepresenting an opponent's argument to make it easy to demolish.
- Appeal to Inappropriate Authority (Ad Verecundiam): Citing a famous personality outside their area of expertise (e.g., "A film star endorses this medicine, so it must cure asthma!").
- Appeal to Ignorance (Ad Ignorantiam): Arguing that a proposition is true simply because it has not been proven false (or vice versa). (e.g., "No one has proven ghosts don't exist, therefore they do.").
- Appeal to Emotion / Pity (Ad Misericordiam): Exploiting pity or emotional sorrow to force agreement rather than offering proof.
- Appeal to Popularity (Ad Populum / Bandwagon): Arguing a claim is true just because the majority believes it.
- Begging the Question (Petitio Principii / Circular Reasoning): Assuming the truth of the conclusion within the premises themselves. (e.g., "God exists because the Holy Book says so, and the Holy Book is true because God wrote it.").
- Slippery Slope: Claiming without evidence that an initial harmless step will inevitably trigger a disastrous chain reaction of events.
- Fallacy of Composition: Erroneously attributing a property of the individual parts to the whole entity. (e.g., "Every player in this team is world-class, therefore this team is invincible.").
- Fallacy of Division: Erroneously assuming what is true of the whole must be true of every individual part. (e.g., "This company is wealthy, so every employee here must be wealthy.").
MODULE 07: INDIAN LOGIC — PRAMANAS (MEANS OF KNOWLEDGE)
7.1 Epistemological Framework of Indian Philosophy
In Indian philosophy, Prama means valid knowledge; Aprama means invalid knowledge; Pramana is the valid source or instrument of knowledge; Pramata is the knower; and Prameya is the object of knowledge.
7.2 The 6 Classical Pramanas Detailed
| Pramana | English Term | Definition & Mechanism | Classic Example |
|---|---|---|---|
| 1. Pratyaksha | Perception | Direct cognition arising from contact of sense organs (Indriyas) with objects (Artha).
• Nirvikalpaka (Indeterminate / raw) • Savikalpaka (Determinate / recognized with name & attributes) |
Seeing a blue lotus or hearing a temple bell directly. |
| 2. Anumana | Inference | Knowledge derived after perception based on invariant relation (Vyapti) between a mark (Hetu) and the unperceived fact (Sadhya). | "The hill has fire because it has smoke, like a kitchen." |
| 3. Upamana | Comparison / Analogy | Knowledge of the relation between a word and the object denoted by it through similarity with a previously known object. | Recognizing a wild cow (Gavaya) in the forest based on an elder's description: "A Gavaya resembles a domestic cow." |
| 4. Shabda | Verbal Testimony | Statement of a trustworthy expert (Aptavakya) who knows the truth and communicates it correctly.
• Vaidika (Scriptural / infallible) • Laukika (Secular / human speech) |
Directions from a reliable doctor; scientific encyclopedias. |
| 5. Arthapatti | Implication / Postulation | Supposition of an unperceived fact to resolve an apparent contradiction between two known facts. | "Devadatta is fat, yet he does not eat during the day." Hence, we postulate that he eats at night! |
| 6. Anupalabdhi | Non-Apprehension | Perception of the non-existence (Abhava) of an object where it would have been perceived if present. | "There is no jar on this table" — we perceive the absence of the jar directly. |
7.3 Schools of Indian Philosophy & Their Accepted Pramanas
| School of Philosophy | Number | Accepted Pramanas |
|---|---|---|
| Charvaka (Materialist) | 1 | Pratyaksha only |
| Buddhism & Vaisheshika | 2 | Pratyaksha, Anumana |
| Sankhya & Yoga | 3 | Pratyaksha, Anumana, Shabda |
| Nyaya (Gautama) | 4 | Pratyaksha, Anumana, Upamana, Shabda |
| Prabhakara Mimamsa | 5 | Pratyaksha, Anumana, Upamana, Shabda, Arthapatti |
| Advaita Vedanta & Bhatta Mimamsa | 6 | All 6 Pramanas (Pratyaksha, Anumana, Upamana, Shabda, Arthapatti, Anupalabdhi) |
MODULE 08: STRUCTURE OF ANUMANA, VYAPTI & HETVABHASA
8.1 Terms of Anumana (Nyaya Syllogism)
| Sanskrit Term | Western Logic Equivalent | Role in Inference | Example ("Hill has fire because of smoke") |
|---|---|---|---|
| Paksha | Minor Term | The subject / location where the inference is made | The Hill (Parvata) |
| Sadhya | Major Term | The object of inference / property to be established | Fire (Agni) |
| Hetu / Linga / Sadhana | Middle Term | The ground, sign, mark, or reason that indicates Sadhya | Smoke (Dhuma) |
8.2 The 5-Member Nyaya Syllogism (Pancavayava)
Unlike the Western Aristotelian syllogism (3 propositions), the Nyaya school specifies a 5-step structured syllogism for proving an inference to others (Pararthanumana):
| Step | Sanskrit Name | Significance | Standard Formula Example |
|---|---|---|---|
| 1 | Pratijna | Proposition to be proved | The hill has fire. |
| 2 | Hetu | Reason / Ground for statement | Because it has smoke. |
| 3 | Udaharana | Universal relation with universal Example | Wherever there is smoke, there is fire, as in a kitchen hearth. |
| 4 | Upanaya | Application of universal relation to the case | This hill has smoke, which is invariably associated with fire. |
| 5 | Nigamana | Conclusion proved | Therefore, this hill has fire. |
Pratijna — Hetu — Udaharana — Upanaya — Nigamana.
8.3 What is Vyapti?
Vyapti is the invariable, unconditional, and universal concomitance between the Hetu (middle term) and Sadhya (major term). Without Vyapti, no inference is possible.
• Vyapya: The pervaded term (e.g., Smoke).
• Vyapaka: The pervading term (e.g., Fire).
• Anupadhika Sambandha: A relation free from all adventitious conditions (Upadhis).
8.4 Hetvabhasa (Fallacies of Inference)
In Nyaya logic, Hetvabhasa literally means "a semblance of reason" — a middle term that appears valid but is defective. Nyaya classifies 5 primary forms:
| Hetvabhasa | Nature of Defect | English Translation | Classic Exam Example |
|---|---|---|---|
| 1. Savyabhichara | Irregular / inconstant middle term; Hetu exists where Sadhya is absent. | Fallacy of Irregular Middle | "All bipeds are rational; Swans are bipeds; therefore, Swans are rational." |
| 2. Viruddha | Contradictory middle term; Hetu disproves the very Sadhya it seeks to prove. | Fallacy of Contradictory Middle | "Sound is eternal, because it is produced." (Produced things are non-eternal!). |
| 3. Satpratipaksha | Counterbalanced middle term; An equally strong counter-reason exists to prove the opposite. | Fallacy of Counterbalanced Reason | "Sound is eternal because it is audible" countered by "Sound is non-eternal because it is an effect." |
| 4. Asiddha (Sadhyasama) | Unproved / non-existent middle term.
• Ashrayasiddha: Substratum is fictitious (e.g., "Sky-lotus has fragrance"). • Svarupasiddha: Hetu cannot exist in Paksha. • Vyapyatvasiddha: Concomitance is conditional/unproved. |
Fallacy of Unproved Middle | "Sky-lotus is fragrant because it is a lotus." (Sky-lotus does not exist!). |
| 5. Badhita | Contradicted by another stronger Pramana (e.g., direct perception). | Fallacy of Non-Inferentially Contradicted Middle | "Fire is cold, because it is a substance." (Direct touch proves fire is hot, nullifying inference!). |
MODULE 09: QUICK REVISION — ONE-LINERS & HIGH-YIELD PYQ CARDS
- Contradictories: A and O are opposites; E and I are opposites. Truth of one guarantees falsehood of other.
- Contraries (A & E): Cannot both be True; Can both be False.
- Sub-contraries (I & O): Cannot both be False; Can both be True.
- Subalternation: Truth runs downward (A to I, E to O); Falsehood runs upward (I to A, O to E).
- Distributed Terms: A distributes Subject; E distributes Both; I distributes Neither; O distributes Predicate.
- Deductive Validity: Truth of premises logically guarantees truth of conclusion.
- Syllogism Terms: Major term = Predicate of conclusion; Minor term = Subject of conclusion; Middle term = appears in both premises, never in conclusion.
- Arthapatti vs Anumana: Arthapatti is accepted as independent only by Mimamsa and Advaita; Nyaya subsumes it under Anumana.
- Anupalabdhi: Distinct means to apprehend Abhava (absence). Accepted as independent by Kumarila Bhatta and Shankaracharya.
- Vyapti: Universal, unconditional concomitance between Hetu and Sadhya.
- Badhita: When the middle term is contradicted by direct perception (e.g., "Fire is cold").
- Viruddha: The middle term proves the exact contradictory of what it was supposed to prove.
Pattern 1: "If 'All men are honest' is TRUE, what is the status of 'Some men are not honest'?"
Answer: Strictly FALSE (Contradictory relationship A and O).
Pattern 2: "If 'Some flowers are red' is FALSE, what is the status of 'Some flowers are not red'?"
Answer: Strictly TRUE (Sub-contrary relationship I and O cannot both be false).
Pattern 3: "Fat Devadatta does not eat food during the day, yet he continues to grow fatter."
Answer: Arthapatti (Implication / Postulation).
Pattern 4: "The hill has fire because it has smoke" — here the Hill is?
Answer: Paksha (Minor term).
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