Categorical Syllogisms & Overlapping Sets Past Papers

Concept: Syllogistic Deduction.

Formula: Intersection of subset collections.

Solution: Since all spheres belong to the set of round things, and some round things overlap with mesmerizing items, it logically follows that some mesmerizing items share space with spheres (Conclusion III).

Why other options are incorrect:

• Options B, C, and D: Assume universal inclusion ('All balls are mesmerizing') which is not guaranteed.

Concept: Syllogism Conversions.

Formula: \(\text{Some } A \text{ are } B \iff \text{Some } B \text{ are } A\)

Solution: 'Some animals are herbivores' converts directly to 'Some herbivores are animals' (Conclusion III).

Why other options are incorrect:

• Options A, C, and D: Include unproven assumptions regarding cats or goats.

Concept: Syllogistic Deduction Sets.

Formula: Disjoint subset relationships.

Solution:

• Conclusion I: Conversion of 'Some carnivores are humans' is true.



• Conclusion II: Omnivores are inside Herbivores; Herbivores share zero overlap with Carnivores. Thus, no carnivores are omnivores (True).

Why other options are incorrect:

• Options B and C: Include Conclusion III which directly contradicts premise 2.

Concept: Syllogistic Quantifier Scope.

Formula: N/A

Solution: Premises only establish that some kings are hardworking (those who are men). Neither universal conclusion follows.

Why other options are incorrect:

• Option B: Assumes non-male kings are also hardworking, which is unstated.

Concept: Syllogism Inclusions.

Formula: N/A

Solution: 'Some tools are useless' + 'All useless are trash' \(\implies\) those useless tools are trash (II is true). No link binds hammers directly to trash.

Why other options are incorrect:

• Options A, B, and C: Claim unverified links for hammer subsets.

Concept: Venn Diagram Deductions.

Formula: N/A

Solution:

• I is valid: 'All children are silly' converts to 'Some silly people are children'.



• III is valid: Silly people who are rich are automatically big shots.

Why other options are incorrect:

• Options B, C, and D: Include II which lacks mandatory subset overlap.

Concept: Syllogistic Overlaps.

Formula: N/A

Solution: Since all spheres are round and some round things are mesmerizing, the subset of mesmerizing round items includes spheres.

Why other options are incorrect:

• Options B and D: Assume an absolute connection between balls and mesmerizing things where none is guaranteed.

Concept: Venn Diagram Subset Inclusions.

Formula: N/A

Solution:

• II is true: Teachers who are students are girls.



• III is true: Conversion of 'All students are girls'.

Why other options are incorrect:

• Option A: Conclusion I is invalid because no student is a boy.

Concept: Categorical Syllogisms.

Formula: N/A

Solution: Glass is entirely enclosed within Wall. Since Pens overlap with Glass, Pens must overlap with Wall (I is true). Also, since all Glass is Wall, some Wall is Glass (II is true).

Why other options are incorrect:

• Options A and B: Are incomplete.

Concept: Negative Premise Inferences.

Formula: Two negative premises yield no definitive affirmative conclusion.

Solution: Conclusion I contradicts Premise 1 directly. Conclusion II claims a relation between villages and cities that is not guaranteed by the premises.

Why other options are incorrect:

• Options A, B, and C: Violate syllogistic rules for double negative premises.

Concept: Universal Affirmative Conversions.

Formula: \(\text{All } A \text{ are } B \implies \text{Some } B \text{ are } A\)

Solution: 'All cups are pencils' converts directly to 'Some pencils are cups' (I is valid).

Why other options are incorrect:

• Option B: Cups and pens do not have a guaranteed overlap.

Concept: Overlapping Set Syllogisms.

Formula: N/A

Solution: Grapes overlap with fruits, and all fruits are leaves; thus grapes must overlap with leaves (I is true).

Why other options are incorrect:

• Option B: Premise states 'Some fruits are grapes', not 'All grapes are fruits'.

Concept: Disjoint Group Constraints.

Formula: N/A

Solution: Oranges that are mangoes cannot be bananas because no mango is a banana. Thus, 'All oranges can never be bananas' is true.

Why other options are incorrect:

• Option B: Contradicts the disjoint set between apples (inside bananas) and mangoes.

Concept: Syllogistic Deduction Sets.

Formula: N/A

Solution:

• I is valid: Direct conversion of 'All children are silly'.



• III is valid: Silly people who are rich are automatically big shots.

Why other options are incorrect:

• Option B: Conclusion II is unproven.

Concept: Categorical Overlaps.

Formula: N/A

Solution: Conclusion III ('Some mesmerizing things are spheres') is the only logically necessary deduction.

Why other options are incorrect:

• Options B, C, and D: Assert unproven universal conclusions.

Concept: Syllogistic Deduction Sets.

Formula: N/A

Solution: I and III follow logically from the subset constraints.

Why other options are incorrect:

• Option B: II lacks mandatory set inclusion.

Concept: Syllogism Overlaps.

Formula: N/A

Solution: Conclusion II ('Some tools are trash') is the only guaranteed deduction.

Why other options are incorrect:

• Options A, B, and C: Claim unverified hammer/trash connections.

Concept: Syllogistic Inclusion Sets.

Formula: N/A

Solution: Bags that are pouches are automatically inside Cases because all pouches are cases (Conclusion II is true). No purses can be pouches because purses share zero overlap with cases.

Why other options are incorrect:

• Options A and B: Conclusion I contradicts premise 3.

Concept: Feature Matching Logic.

Formula: N/A

Solution: Drake's unique distinguishing feature is non-black (brown) shoes combined with black jeans. Ahmad's description explicitly specifies wearing non-black shoes and black jeans, matching Drake's specific outfit combination.

Why other options are incorrect:

• Option B: John wore all-black footwear.