5.1 Syllogisms (Only a few, Some not & Reverse Syllogism)

Modern Syllogisms for SBI PO

Syllogism questions in the State Bank of India Probationary Officer (SBI PO) examination have evolved significantly beyond traditional categorical logic. Contemporary Prelims and Mains papers feature specialized quantifiers such as 'Only', 'Only a few', 'Can never be', and 'Reverse Syllogism'. Solving these questions requires strict adherence to formal set theory rules and Euler/Venn diagram representation without relying on intuitive or real-world assumptions.

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Standard vs. Exclusive Quantifiers

To master modern syllogisms, you must understand the precise set-theoretic translations of standard and exclusive statements.

Standard Quantifiers

1. All A are B: The set $A$ is a subset of set $B$ ($A \subseteq B$).
2. Some A are B: The intersection of set $A$ and set $B$ is non-empty ($A \cap B \neq \emptyset$).
3. No A is B: Set $A$ and set $B$ are completely disjoint ($A \cap B = \emptyset$).
4. Some A are not B: At least one element of set $A$ lies outside set $B$ ($A \setminus B \neq \emptyset$).

Modern Exclusive Quantifiers

The 'Only' Quantifier

A statement of the form 'Only A is B' translates to 'All B are A' with a critical, non-negotiable exclusivity constraint: B can NEVER have any positive overlap with any other element except A.

• Example Statement: "Only Bankers are Managers."
• Translation: All Managers are Bankers. Furthermore, No Manager can be an Analyst, Director, or any third variable mentioned in the question.

The 'Only a few' Quantifier

A statement of the form 'Only a few A are B' is a compound quantifier that asserts two definite facts simultaneously:

1. Some A are B (Positive Definite relation)
2. Some A are not B (Negative Definite relation)

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Rules for Definite Conclusions and Possibilities

A conclusion in syllogism can fall into one of three truth states:

1. Definitely True: Supported by all valid minimum-overlapping Venn diagrams.
2. Definitely False: Directly contradicts at least one given premise.
3. Undetermined (Can't Say): Neither definitely true nor definitely false based on the minimum-overlapping model.

Possibility Rule Matrix

A conclusion phrased as a possibility (e.g., "All A being B is a possibility") is evaluated based on the baseline truth value of the definite statement:

• Definitely True + Possibility = FALSE (If it is already certain, stating it as a possibility is logically redundant and invalid).
• Definitely False + Possibility = FALSE (If it violates a premise, it can never occur).
• Undetermined (Can't Say) + Possibility = TRUE (If no direct boundary prevents it, the scenario is logically plausible).

'Can Never Be' Statements

The phrase 'All A can never be B' is NOT a possibility conclusion; it is a definite negative conclusion. It translates directly to: 'Some A are not B' (or in certain contexts, 'No A is B'). If you can prove that even a small portion of set $A$ is restricted from entering set $B$, then "All A can never be B" is definitely true.

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Complimentary Pairs and Either-Or Rules

When two conclusions are individually undetermined (Can't Say), they may form an 'Either-Or' relationship if they represent mutually exhaustive scenarios. For an Either-Or relationship to hold, all three of the following conditions must be satisfied:

1. Same Elements: Both conclusions must contain the exact same subject and predicate (e.g., $A$ and $B$).
2. Both Undetermined: Both conclusions must be individually 'Can't Say' (neither definitely true nor definitely false).
3. Valid Complimentary Pair: The conclusions must belong to one of two valid complimentary pair combinations:
   • Pair 1: Some A are B + No A is B
   • Pair 2: All A are B + Some A are not B

[!WARNING] Common Trap: The combination All A are B + No A is B does NOT form an Either-Or pair because both statements can be false simultaneously if Some A are B and Some A are not B.

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Reverse Syllogism Strategy

In Reverse Syllogism, the conclusions are given in the question stem, and you must select the set of statements from the options that yields those conclusions.

4-Step Elimination Protocol

1. Analyze Conclusion Polarity: If all given conclusions are positive, any option containing negative statements (No, Some not, Only a few) is highly suspicious or invalid unless compensated.
2. Identify Cross-Restrictions: If a conclusion states "No A is C", look for a statement set that provides a direct or indirect negative link between $A$ and $C$. Eliminate option sets where $A$ and $C$ are only connected via positive 'Some' relations.
3. Scan for Exclusive Quantifiers: If a conclusion states "Some A are not B", look for 'Only a few A are B' or 'No A is B' in the statements.
4. Draw Minimum Overlap Diagrams for Shortlisted Options: Never draw Venn diagrams for all 5 options. Use steps 1–3 to eliminate 3 options, then draw diagrams only for the remaining 2 candidate sets.

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Step-by-Step Worked Example

Premises:

1. Only a few Laptops are Desktops.
2. All Desktops are Monitors.
3. Only Monitors are Servers.
4. No Desktop is a Printer.

Conclusions to Evaluate:

• Conclusion I: All Laptops being Desktops is a possibility.
• Conclusion II: No Server is a Desktop.
• Conclusion III: Some Monitors are not Printers.

Step-by-Step Logical Trace

1. Decode Statement 1: 'Only a few Laptops are Desktops' $\implies$ Some Laptops are Desktops AND Some Laptops are not Desktops.
2. Decode Statement 3: 'Only Monitors are Servers' $\implies$ All Servers are Monitors AND Servers cannot overlap with Laptops, Desktops, or Printers.
3. Evaluate Conclusion I: Statement 1 mandates that a portion of Laptops can NEVER be Desktops. Therefore, putting ALL Laptops inside Desktops violates Statement 1. Result: FALSE.
4. Evaluate Conclusion II: Because Servers are exclusive to Monitors (Statement 3), Servers cannot be Desktops. Result: TRUE.
5. Evaluate Conclusion III: All Desktops are inside Monitors (Statement 2), and No Desktop is a Printer (Statement 4). The portion of Monitors occupied by Desktops can never be Printers. Therefore, at least those Monitors are not Printers. Result: TRUE.

Final Answer: Conclusions II and III follow.