Unit 2: Syllogism, Time Sequence and Ranking

I. Foundations of Logical and Sequential Reasoning

Syllogism, number tests, ranking tests, and time-sequence tests measure the ability to derive conclusions from stated relationships. Their governing principle is constraint-based reasoning: only the information explicitly given, together with valid logical implications, may be used.

• Deduction: A conclusion is valid when it necessarily follows from the premises; real-world knowledge must not influence the result.
• Possibility: A conclusion is possible when at least one arrangement satisfies both the premises and the proposed conclusion.
• Order relation: Ranking and time problems use relations such as higher/lower, earlier/later, before/after, and older/younger.
• Transitivity: If (A>B) and (B>C), then (A>C), provided all comparisons refer to the same property.
• Inclusiveness: Words such as “between” and “from A to B” may require careful attention to whether endpoints are counted.
• Representation: Venn diagrams, ordered lists, tables, and timelines convert verbal statements into visible constraints.
• Verification: A result should be checked against every original condition, not merely against the condition used in the final step.
• Conventions:
  • “Some” means at least one, not necessarily only one.
  • “All” does not automatically establish that the subject class exists.
  • Rank is normally counted from (1), not (0).
  • Unless stated otherwise, persons or events occupy distinct positions.

II. Categorical Syllogism — Representing Class Relationships

A categorical syllogism contains premises and conclusions concerning classes of objects. Its validity depends on whether the conclusion is forced by the relationships expressed in the premises.

A. Logical Venn diagrams

Logical Venn diagrams represent inclusion, exclusion, and partial membership through circles or regions assigned to classes.

• Universal affirmative: “All (A) are (B)” means (A\subseteq B); the circle for (A) lies wholly inside (B).
  

  TEXT

    A ⊆ B

  • Every member of (A) is a member of (B).
  • The statement does not imply that every (B) is an (A).

• Universal negative: “No (A) is (B)” means the two classes are disjoint.
  

  TEXT

    A ∩ B = ∅

  
  Here, (\cap) denotes intersection and (\varnothing) denotes an empty set.

• Particular affirmative: “Some (A) are (B)” requires at least one member in the overlapping region.
  

  TEXT

    A ∩ B ≠ ∅

• Particular negative: “Some (A) are not (B)” requires at least one member of (A) outside (B).
  

  TEXT

    A − B ≠ ∅

  
  Here, (A-B) is the part of (A) that does not belong to (B).

• Valid conversion:

  • “No (A) is (B)” converts to “No (B) is (A).”
  • “Some (A) are (B)” converts to “Some (B) are (A).”
  • “All (A) are (B)” cannot generally convert to “All (B) are (A).”

• Linked inclusion: If all (A) are (B), and all (B) are (C), then all (A) are (C).
  

  TEXT

    A ⊆ B and B ⊆ C  ⇒  A ⊆ C

• Worked example: Premises: “All poets are writers” and “No writer is illiterate.”

  • Let (P), (W), and (I) denote poets, writers, and illiterate persons.
  • Since (P\subseteq W) and (W\cap I=\varnothing), it follows that (P\cap I=\varnothing).
  • Therefore, “No poet is illiterate” necessarily follows.

• Diagram safeguard: A diagram may display several permissible arrangements; a definite conclusion must hold in every arrangement allowed by the premises.

III. Possibility Syllogism — Testing Consistency Rather Than Necessity

Possibility-based syllogisms ask whether a proposed relationship can exist without contradicting the premises. The task is to construct at least one consistent diagram.

A. Possibility based problems

Possibility based problems distinguish what must be true from what may be true under the given class constraints.

• Possibility criterion: A conclusion is possible if the premises do not force its negation.
  

  TEXT

    Possible(C) ⇔ Premises ∪ {C} are consistent

  
  Here, (C) is the proposed conclusion and “consistent” means free from contradiction.

• Definite versus possible:

  1. Definite conclusion: Must hold in every valid diagram.
  2. Possible conclusion: Holds in at least one valid diagram, even if it fails in others.

• Direct impossibility: If “No (A) is (B)” is given, then “Some (A) being (B) is possible” is impossible because it requires both (A\cap B=\varnothing) and (A\cap B\neq\varnothing).

• Unrestricted overlap: If no premise separates (A) from (B), their overlap may be possible. Absence of a negative relationship permits overlap but does not prove actual overlap.

• Universal possibility: “All (A) being (B) is possible” can hold when placing (A) inside (B) violates no premise. It fails if the premises require some (A) outside (B).

• Existence requirement: A particular claim such as “Some (A) are (B)” introduces existence. A universal statement such as “All (A) are (B)” alone does not necessarily do so.

• Complementary conclusions: For a fixed existing subject class, “Some (A) are (B)” and “Some (A) are not (B)” describe different regions; neither follows merely because the other is absent.

• Worked example: Premises: “All doctors are graduates” and “No artist is a doctor.”

  • Doctors must lie inside graduates and outside artists.
  • Nothing prevents some artists from also being graduates.
  • Therefore, “Some artists being graduates is possible,” but it is not a definite conclusion.

• Checking method:

  • Draw compulsory containment and separation first.
  • Add the proposed possibility.
  • Reject it only if it creates a direct or derived contradiction.

IV. Numerical Reasoning — Detecting Rules and Positional Changes

Numerical reasoning problems test recognition of arithmetic structure, digit properties, and changes created by rearrangement or repeated operations.

A. Number test

A number test identifies a number or position satisfying a stated numerical rule.

• Basic properties:

  • Even numbers are divisible by (2); odd numbers are not.
  • A prime number has exactly two positive factors: (1) and itself.
  • A perfect square has the form (n^2), such as (49=7^2).

• Divisibility checks:

  • By (3): the sum of digits is divisible by (3).
  • By (9): the sum of digits is divisible by (9).
  • By (11): the alternating sum of digits is divisible by (11).
  • Example: (462) is divisible by (3) because (4+6+2=12).

• Series differences: For a sequence (a_1,a_2,\ldots), calculate consecutive differences:
  

  TEXT

    dᵢ = aᵢ₊₁ − aᵢ

  
  Here, (d_i) is the difference between adjacent terms. Constant first differences indicate an arithmetic sequence.

• Higher-order pattern: If first differences vary, inspect second differences, ratios, alternating terms, squares, cubes, or prime numbers.

• Digit operations: Questions may require reversing digits, interchanging positions, adding place values, or counting digits satisfying a condition.

  • In (5724), the place value of (7) is (700), while its face value is (7).

• Position comparison: After rearrangement, compare each digit’s original position with its new position. Record positions in a table to prevent repeated counting.

• Worked example: Consider (3, 8, 15, 24, 35).

  • Differences are (5,7,9,11), which increase by (2).
  • The terms follow (a_n=n^2+2n), where (n) is the term number.
  • Thus the next term is (6^2+2(6)=48).

• Validation: A proposed rule should explain every supplied term; matching only the last two terms is insufficient.

V. Positional Reasoning — Determining Relative and Absolute Place

Ranking problems arrange persons or objects according to height, marks, age, performance, or another common attribute.

A. Ranking test

A ranking test determines a position from one end, both ends, or through comparisons among several individuals.

• Opposite-end rank: If a person’s rank from the top is (T) among (N) persons, rank from the bottom is:
  

  TEXT

    B = N − T + 1

  
  Here, (B) is bottom rank, (N) is total persons, and (T) is top rank.

• Total from two ranks: If the same person is (T)th from the top and (B)th from the bottom:
  

  TEXT

    N = T + B − 1

  
  One is subtracted because the person is counted from both ends.

• Persons between two ranks: For positions (r_1) and (r_2) from the same end:
  

  TEXT

    Between = |r₁ − r₂| − 1

• Relative statements: “A ranks above B” gives (A>B) in performance but does not give the number of positions separating them.

• Immediate position: “A is immediately above B” fixes consecutive ranks; if (B) is seventh, (A) is sixth.

• Ties: Unless explicitly mentioned, ranks are treated as unique. With tied ranks, the stated ranking convention must be followed.

• Worked example: Mira is (18)th from the top and (25)th from the bottom.
  

  TEXT

    N = 18 + 25 − 1 = 42

  
  Therefore, the group contains (42) persons.

• Consistency check: A computed rank must satisfy (1\leq r\leq N); values outside this interval indicate miscounting or incompatible data.

VI. Temporal Reasoning — Ordering Events and Measuring Intervals

Temporal reasoning organizes events according to occurrence, duration, and relative expressions such as before, after, earlier, later, and between.

A. Time sequence test

A time sequence test establishes chronological order or determines an event’s time from stated temporal relationships.

• Timeline method: Represent earlier events on the left and later events on the right.
  

  TEXT

    Earlier  ←  A — B — C  →  Later

• Transitive order: If (A) occurred before (B), and (B) before (C), then (A) occurred before (C).

• Reverse wording: “A happened after B” means (B\rightarrow A); translating every statement into one direction reduces errors.

• Immediate sequence: “Immediately before” or “immediately after” prohibits another event from occupying the intervening position.

• Elapsed time: For times within the same continuous period:
  

  TEXT

    Duration = Ending time − Starting time

  
  When crossing noon, midnight, or a date boundary, divide the interval into parts and add them.

• Inclusive counting: From the 4th to the 9th is (9-4=5) elapsed days, but counting both dated days gives (9-4+1=6) days.

• Calendar sequence: Days repeat in cycles of seven. If an event occurs (d) days later, its weekday shift is:
  

  TEXT

    Shift = d mod 7

  
  Here, “mod” gives the remainder after division by (7).

• Worked example: A meeting occurs three days after Tuesday, and a review occurs two days after the meeting.

  • Three days after Tuesday is Friday.
  • Two days after Friday is Sunday.
  • The order is Tuesday reference (\rightarrow) Friday meeting (\rightarrow) Sunday review.

• Ambiguity control: Terms such as “next,” “previous,” and “within” must be interpreted according to the supplied reference point and whether endpoints are included.