Categorical Propositions & Immediate Inferences

Categorical Proposition एक ऐसा proposition है जो two classes/categories (Subject और Predicate) के बीच relationship को बिना किसी condition के affirm या deny करता है।

Structure:

• Subject (S): जिसके बारे में कुछ कहा जा रहा है
• Copula: Verb (is/are/am/are not) जो subject और predicate को जोड़ता है
• Predicate (P): जो subject के बारे में कहा जा रहा है

Example:

• All humans are mortal. → Subject = humans, Copula = are, Predicate = mortal
• No cats are dogs. → Subject = cats, Copula = are not, Predicate = dogs

Quantity (मात्रा/परिमाण): Proposition में subject की कितनी portion refer की जा रही है।

• Universal — सार्वभौमिक: Whole subject class (All/No) — All humans are mortal.
• Particular — विशेष: Part of subject class (Some) — Some students are intelligent.
• Singular — एकवचन: Only one member of the class — Socrates is mortal.

Quality (गुण): Proposition affirmative है या negative।

• Affirmative — सकारात्मक: Subject and predicate are connected — All S are P.
• Negative — नकारात्मक: Subject and predicate are separated — No S are P.

Memory Formula (Latin):

• A = Universal Affirmative (Affirmo)
• E = Universal Negative (Nego)
• I = Particular Affirmative (Affirmo)
• O = Particular Negative (Nego)

Distribution का अर्थ है कि कोई term अपने entire class (सभी members) को refer कर रहा है या नहीं।

Rules of Distribution:

• A (All S are P): S is distributed; P is undistributed
• E (No S are P): S is distributed; P is distributed
• I (Some S are P): S is undistributed; P is undistributed
• O (Some S are not P): S is undistributed; P is distributed

Quick Memory:

• A → S distributed, P undistributed
• E → Both distributed
• I → Neither distributed
• O → S undistributed, P distributed

Some propositions need to be translated/standardised into standard categorical form (A, E, I, O) for logical analysis.

Common Standardisations:

• "Only S are P" → All P are S (A proposition with subject = P, predicate = S)
• "None but S are P" → All P are S (Only S can be P)
• "Not all S are P" → Some S are not P (O proposition)
• "S are P" → All S are P (if universal) or Some S are P (if particular, depends on context)

Examples:

• "Only students can enter." → All those who can enter are students. (All entrants are students)
• "None but the brave deserve the fair." → All who deserve the fair are brave.
• "Not all birds can fly." → Some birds are not able to fly. (Some birds cannot fly)

Conversion एक immediate inference है जिसमें subject और predicate को interchange कर दिया जाता है।

Rules:

• A (All S are P) → I (Some P are S) (Conversion by limitation) — Not all A propositions convert validly to A (unless both terms are distributed).
• E (No S are P) → E (No P are S) (Simple conversion) ✅ Valid
• I (Some S are P) → I (Some P are S) (Simple conversion) ✅ Valid
• O (Some S are not P) → Cannot be converted ❌ Invalid

Examples:

• A: All cats are animals → Some animals are cats ✅
• E: No cats are dogs → No dogs are cats ✅
• I: Some students are intelligent → Some intelligent are students ✅
• O: Some students are not intelligent → Cannot convert ❌

Obversion एक immediate inference है जिसमें:

• Quality को बदला जाता है (affirmative ↔ negative)
• Predicate को उसके complement (contradictory) से replace किया जाता है

Rules:

• A (All S are P) → E (No S are non-P) ✅ Valid
• E (No S are P) → A (All S are non-P) ✅ Valid
• I (Some S are P) → O (Some S are not non-P) ✅ Valid
• O (Some S are not P) → I (Some S are non-P) ✅ Valid

Complement (non-P): वह सब कुछ जो P नहीं है।

Examples:

• A: All men are mortal → No men are non-mortal ✅
• E: No cats are dogs → All cats are non-dogs ✅
• I: Some students are intelligent → Some students are not non-intelligent ✅
• O: Some students are not intelligent → Some students are non-intelligent ✅

Contraposition एक immediate inference है जिसमें:

• Subject और Predicate को interchange किया जाता है
• दोनों terms को उनके complement (non-) से replace किया जाता है
• Quality वही रहती है

Rules:

• A (All S are P) → A (All non-P are non-S) ✅ Valid
• E (No S are P) → O (Some non-P are not non-S) (By limitation, not full contraposition)
• I (Some S are P) → Cannot be contraposed ❌ Invalid
• O (Some S are not P) → O (Some non-P are not non-S) ✅ Valid

Examples:

• A: All humans are mortal → All non-mortals are non-humans ✅
• E: No cats are dogs → Some non-dogs are not non-cats (by limitation)
• O: Some students are not intelligent → Some non-intelligent are not non-students ✅

Square of Opposition A, E, I, O propositions के बीच logical relationships को दर्शाता है।

• Contraries — विरोधी: A and E (Cannot both be true, but can both be false)
• Contradictories — विरुद्ध: A and O, E and I (Cannot both be true and cannot both be false)
• Sub-contraries — उपविरोधी: I and O (Cannot both be false, but can both be true)
• Sub-alternation — उपांतरण: A → I, E → O (If universal true → particular true; if particular false → universal false)

• Trap 1: All S are P → All P are S → ❌ A proposition का simple conversion valid नहीं है (only by limitation: Some P are S)
• Trap 2: Some S are not P → No S are P → ❌ O proposition का obversion valid है, लेकिन यह obversion नहीं है
• Trap 3: O proposition cannot be converted → ✅ True, O cannot be converted
• Trap 4: I proposition can be contraposed → ❌ I cannot be contraposed
• Trap 5: Only S are P → All S are P → ❌ Only S are P = All P are S (subject-predicate swap)
• Trap 6: Not all S are P → No S are P → ❌ Not all = Some S are not P (O proposition)
• Trap 7: Distribution: In E proposition, both terms distributed → ✅ True

"No men are perfect" is which type of categorical proposition?

• A. A
• B. E
• C. I
• D. O

Explanation: Universal Negative → E proposition (No S are P).

In the proposition "All students are intelligent", which terms are distributed?

• A. Only Subject
• B. Only Predicate
• C. Both Subject and Predicate
• D. Neither

Explanation: A proposition = Subject distributed, Predicate undistributed.

"Only students can enter the library" is equivalent to:

• A. All students can enter
• B. All who can enter are students
• C. No students can enter
• D. Some students can enter

Explanation: "Only S are P" = All P are S (All entrants are students).

What is the valid conversion of "No cats are dogs"?

• A. Some cats are dogs
• B. All dogs are cats
• C. No dogs are cats
• D. Some dogs are cats

Explanation: E proposition converts simply → No dogs are cats.

What is the obversion of "All humans are mortal"?

• A. No humans are mortal
• B. No humans are non-mortal
• C. All non-humans are mortal
• D. Some humans are mortal

Explanation: A → E with complement predicate: No S are non-P.

What is the contraposition of "All teachers are educated"?

• A. All non-teachers are non-educated
• B. All non-educated are non-teachers
• C. Some non-educated are non-teachers
• D. No non-teachers are educated

Explanation: A → A with both complements: All non-P are non-S.

"Not all birds can fly" is equivalent to:

• A. All birds can fly
• B. No birds can fly
• C. Some birds cannot fly
• D. Some birds can fly

Explanation: "Not all" = Some are not (O proposition).

A, E, I, O:

• A = Universal Affirmative (All S are P) → S-Distributed, P-Undistributed
• E = Universal Negative (No S are P) → Both Distributed
• I = Particular Affirmative (Some S are P) → Neither Distributed
• O = Particular Negative (Some S are not P) → S-Undistributed, P-Distributed

Standardisation:

• Only S are P = All P are S
• None but S are P = All P are S
• Not all S are P = Some S are not P

Immediate Inferences:

• Conversion: Swap S & P → Valid: E, I; A→I (limitation); O: Invalid
• Obversion: Change Quality + Complement P → Valid: All
• Contraposition: Swap S & P + Both Complements → Valid: A, O; E→O (limitation); I: Invalid

Exam Formula:

UGC NET में Categorical Propositions & Immediate Inferences के questions type identification, distribution, और conversion/obversion/contraposition पर based होते हैं।