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Difficulty: MediumQuantifier Logic (Some, Most, All)

In formal logic, specific quantifier combinations yield strictly valid deductions. Analyze the four multi-premise setups below, each from a different domain. Match each setup to the abstract description of its logically valid deduction.

  • Ecology: All wetland ecosystems support amphibian populations, and most wetland ecosystems are highly sensitive to pollution.Therefore, some members of the broader category must possess the trait shared by the majority of the subset.
  • Astronomy: Most exoplanets in the catalog orbit red dwarf stars, and every exoplanet that orbits a red dwarf star is tidally locked.Therefore, the majority of the initial group must also be members of the final enclosing category.
  • Archaeology: All artifacts found in the lower stratum predate the Bronze Age, yet some artifacts found in the lower stratum are not made of stone.Therefore, some members of the enclosing category must lack the specific characteristic.
  • Policy: Most of the university's board members voted for the expansion, and most of the university's board members are alumni.Therefore, the two traits must overlap for at least some members of the parent group.

Answer

Ecology matches the 'some members of the broader category...' deduction; Astronomy matches the 'majority of the initial group...' deduction; Archaeology matches the 'some members of the enclosing category must lack...' deduction; Policy matches the 'two traits must overlap...' deduction.
To solve quantifier logic setups, you must track the formal relationships without being distracted by the subject matter. 1. Ecology (All A are B + Most A are C) proves that the trait 'C' transfers to the broader category 'B'. 2. Astronomy (Most A are B + All B are C) proves that the majority of 'A' is enclosed in 'C'. 3. Archaeology (All A are B + Some A are not C) proves that the 'not C' trait transfers to the enclosing category 'B'. 4. Policy (Most A are B + Most A are C) proves that traits 'B' and 'C' must overlap.

Step-by-Step Solution

1
Diagram the Ecology setup.
All W -> A (W is a subset of A). Most W -> S. The W's that are S are also A's, yielding the deduction: Some A are S.
To identify the abstract logical result, we must map the specific categories (W, A, S) to their roles: subset, broader category, and trait.
2
Diagram the Astronomy setup.
Most E -> R. All R -> T. Because the 'Most E' group is entirely contained within R, and R is entirely within T, that 'Most E' group must be within T. Deduction: Most E are T.
This establishes a transitive 'Most' relationship passing through an 'All' relationship.
3
Diagram the Archaeology setup.
All L -> P (L is a subset of P). Some L -> ~S. Those specific L's that are not S must still be inside P. Deduction: Some P are ~S.
This confirms how a negative 'Some' statement interacts with an enclosing 'All' statement.
4
Diagram the Policy setup.
Most B -> V. Most B -> A. Because both represent more than 50% of the exact same parent group (B), they must mathematically intersect. Deduction: Some V are A.
This tests the classic 'Most + Most = Some' overlap rule.

Key Concept

Formal deduction using combinations of 'All', 'Most', and 'Some' quantifiers, specifically recognizing transitive properties and mathematical overlaps of majorities.
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Topics

Quantifier Logic (Some, Most, All)

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