Categorical Logic: Propositions, Syllogisms & Square of Opposition

Core Concepts: Categorical Propositions

Categorical propositions are statements that relate two classes (terms) using quantifiers ("all," "some," "none") and copulas ("are," "are not"). They form the foundation of traditional logic and are classified into four types based on quantity (universal/particular) and quality (affirmative/negative):

Types of Categorical Propositions

Key Terms:

• Subject (S): The term being described (e.g., "Nepali citizens").
• Predicate (P): The term describing the subject (e.g., "voters").
• Distribution: A term is distributed if the proposition makes a claim about all members of that term.
  • A (All S are P): S is distributed; P is not.
  • E (No S are P): Both S and P are distributed.
  • I (Some S are P): Neither S nor P is distributed.
  • O (Some S are not P): P is distributed; S is not.

Visualizing Distribution

classDiagram
    class Proposition {
        +Type: A/E/I/O
        +Quantity: Universal/Particular
        +Quality: Affirmative/Negative
        +Distributes S?
        +Distributes P?
    }
    Proposition <|-- A : "S distributed, P not"
    Proposition <|-- E : "S and P distributed"
    Proposition <|-- I : "Neither distributed"
    Proposition <|-- O : "P distributed, S not"

Example:

• "All TU students are tech-savvy." (A)
  • S (TU students): Distributed (all are claimed).
  • P (tech-savvy): Not distributed (not all tech-savvy people are TU students).
• "Some Pathao riders ignore traffic rules." (I)
  • Neither term is distributed.

The Square of Opposition

The Square of Opposition shows the logical relationships between the four types of propositions about the same subject (S) and predicate (P). It applies only to universal propositions (A and E).

![square of opposition labelled diagram](/media/5bf0e9044e21a0eee33a.png "A classic Venn diagram showing the relationships between A, E, I, and O propositions. (Image: WatchduckYou can name the author as 'T. Piesk', 'Tilman Pies, Public domain, via Wikimedia Commons)")

Relationships:

1. Contradiction (A ↔ O; E ↔ I):

   • If one is true, the other is false, and vice versa.
   • Example:
     • A: All Ncell users pay on time. (True)
     • O: Some Ncell users do not pay on time. (False)
     • If A is false, O must be true.

2. Contrariety (A vs. E):

   • Cannot both be true, but can both be false.
   • Example:
     • A: All Daraz deliveries are on time. (False)
     • E: No Daraz deliveries are on time. (Also false)

3. Subcontrariety (I vs. O):

   • Can both be true, but not both false.
   • Example:
     • I: Some eSewa users forget passwords. (True)
     • O: Some eSewa users do not forget passwords. (Also true)

4. Subalternation (A → I; E → O):

   • If the universal is true, the particular must be true.
   • Example:
     • If A (All TU students pass logic) is true, then I (Some TU students pass logic) is automatically true.

Immediate Inferences

These are direct logical transformations of propositions without adding new information. They help simplify or rephrase arguments.

1. Conversion

Swap the subject and predicate. Only valid for E and I propositions.

Example (Valid Conversion):

• E: No NEPSE stocks are risk-free. → E: No risk-free investments are NEPSE stocks.

2. Obversion

Change the quality (affirmative ↔ negative) and replace the predicate with its complement (e.g., "P" → "non-P").

3. Contraposition

Swap and negate both subject and predicate. Valid only for A and O propositions.

Example (Valid Contraposition):

• A: All logic students solve puzzles. → A: All non-puzzle-solvers are non-logic students.

Categorical Syllogisms

A syllogism is a deductive argument with two premises and a conclusion, containing three terms:

1. Major term (P): Predicate of the conclusion.
2. Minor term (S): Subject of the conclusion.
3. Middle term (M): Appears in both premises but not in the conclusion.

Structure:

Premise 1: M → P (Major premise)
Premise 2: S → M (Minor premise)
Conclusion: S → P

Rules for Validity

For a syllogism to be valid, it must follow these 7 rules:

1. Three terms: Only three distinct terms (M, S, P) can appear.
2. Middle term distributed: The middle term (M) must be distributed at least once.
3. No term distributed in the conclusion unless distributed in a premise.
4. No two negative premises.
5. If a premise is negative, the conclusion must be negative.
6. If the conclusion is particular, at least one premise must be particular.
7. No illicit major/minor: The major term (P) or minor term (S) cannot be distributed in the premise where they do not appear in the conclusion.

Mood and Figure

• Mood: The types of propositions (A, E, I, O) in the order: major premise → minor premise → conclusion. Example: AEE (All M are P; No S are M; Therefore, No S are P).
• Figure: The position of the middle term (M) in the premises. There are four figures:

Worked Example: Syllogism in Real Life

Scenario: NTC (Nepal Telecommunications Corporation) wants to ensure all its employees who use company laptops also attend cybersecurity training. They announce:

1. All employees with company laptops must attend training. (A: All M are P)
2. All NTC managers have company laptops. (A: All S are M)
3. Therefore, all NTC managers must attend training. (A: All S are P)

Analysis:

• Terms:
  • M = Employees with company laptops
  • S = NTC managers
  • P = Employees who attend training
• Mood: AAA (All, All, All)
• Figure: 1 (M in subject of major premise, predicate of minor premise)
• Validity Check:
  1. Three terms? ✅ (M, S, P)
  2. Middle term (M) distributed? ✅ (in both premises)
  3. No term distributed in conclusion unless in premise? ✅ (P is not distributed in conclusion)
  4. No negative premises. ✅
  5. No illicit major/minor. ✅
• Conclusion: Valid syllogism.

In the Real World

1. eSewa and Khalti (Digital Payments):

   • Categorical Propositions in Fraud Detection:
     • A: All eSewa transactions with amounts > Rs. 50,000 require OTP verification.
     • O: Some Khalti users bypass OTP verification.
     • Implication: If a transaction is flagged as high-risk (M), and it lacks OTP (S), then it is fraudulent (P). This mirrors Figure 2 syllogisms used in fraud algorithms.

2. Pathao (Ride-Hailing App):

   • Syllogism in Driver Allocation:
     • Premise 1 (A): All available Pathao drivers in Kathmandu are registered. (All M are P)
     • Premise 2 (A): All riders requesting a ride are in Kathmandu. (All S are M)
     • Conclusion (A): All riders get a registered driver. (All S are P)
     • Real-World Twist: If a rider is outside Kathmandu (O: Some S are not M), the conclusion fails (O: Some riders do not get a registered driver). This aligns with Rule 6 (particular conclusion requires a particular premise).

3. NEPSE (Stock Exchange):

   • Square of Opposition in Investment Advice:
     • A: All blue-chip stocks are low-risk. (True for most investors)
     • O: Some blue-chip stocks are high-risk. (Contradictory; if A is true, O is false)
     • Investor Dilemma: If an advisor claims E (No blue-chip stocks are risky), but later data shows I (Some blue-chip stocks have volatility), it creates a contrariety conflict. Investors must resolve this using obversion (e.g., "All risky stocks are not blue-chip" → "Some risky stocks are not blue-chip").

Common Fallacies in Categorical Logic

Even valid syllogisms can lead to logical fallacies if misapplied. Two key ones:

1. Illicit Major/Minor:

   • Violates Rule 7 by distributing a term in the conclusion without distributing it in a premise.
   • Example:
     • Premise 1: All Ncell users are loyal. (A: S distributed)
     • Premise 2: Some TU students are Ncell users. (I: S not distributed)
     • Conclusion: Some TU students are loyal. (I: But S was not distributed in premise 2 → Illicit Minor)

2. Undistributed Middle:

   • Violates Rule 2 by not distributing the middle term.
   • Example:
     • Premise 1: Some Daraz sellers are honest. (I: M not distributed)
     • Premise 2: All honest sellers get bonuses. (A: P distributed)
     • Conclusion: Some Daraz sellers get bonuses. (I)
     • Problem: Middle term ("honest sellers") is not distributed in either premise → invalid.

Exam Tip

How This Unit is Examined (TU/PU/NEB Pattern)

1. Short Questions (2–5 marks):

   • Define categorical proposition, distribution, or Square of Opposition.
   • Convert/obvert/contrapose a given proposition.
   • Identify the mood and figure of a syllogism.
   • Example Question: "Obvert the following proposition: 'Some TU students fail logic.' Write the obverted form and state its type."

2. Long Questions (10–15 marks):

   • Analyze a real-world syllogism (e.g., from NTC, banks, or eSewa) and check its validity using all 7 rules.
   • Construct a syllogism given a conclusion (e.g., "Therefore, all Khalti users are secure").
   • Apply the Square of Opposition to show contradictions/contrarieties between propositions.
   • Example Question: "Ncell claims: 'All postpaid users get unlimited calls.' A customer says, 'Some postpaid users still face call drops.' a) Identify the types of these propositions. b) Show their relationship using the Square of Opposition. c) If Ncell’s claim is false, what must be true about the customer’s statement?"

3. Problem-Solving (5–10 marks):

   • Spot fallacies in given syllogisms (e.g., undistributed middle, illicit major).
   • Reconstruct invalid arguments to make them valid by adjusting premises.

Marks Distribution (Typical TU Exam):

Top 3 Exam Strategies:

1. Memorize the Square of Opposition as a table and practice contradictions/contrarieties with Nepali examples (e.g., NEPSE stocks, NTC services).
2. For syllogisms:
   • Always label M, S, P and draw a quick figure diagram.
   • Check Rule 2 (middle term distributed) first—it’s the most common mistake.
3. Use real-world analogies in long answers. For example:
   • Compare eSewa fraud detection to EIO contraposition.
   • Relate Pathao driver allocation to AAA-1 syllogisms.

Quick Revision Checklist

• Can I classify any statement as A, E, I, or O?
• Do I know which terms are distributed in each type?
• Can I draw the Square of Opposition and explain its relationships?
• Can I perform conversion, obversion, and contraposition correctly?
• Can I identify the mood and figure of a syllogism?
• Can I check a syllogism’s validity using all 7 rules?
• Can I spot fallacies like undistributed middle or illicit major?