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Difficulty: MediumQuantifier Logic (Some, Most, All)

Botanical surveys of the lower river valley reveal several consistent patterns. All native shrubs in the valley require alkaline soil to thrive. Furthermore, most of these native shrubs bloom in late spring. Finally, none of the plants that bloom in late spring are frost-resistant. Which one of the following conclusions must logically follow from the survey results described above?

  1. At least some plants that require alkaline soil are not frost-resistant.Answer
  2. B
    Most plants in the valley that require alkaline soil bloom in late spring.
  3. C
    Any plant in the valley that requires alkaline soil is a native shrub.
  4. D
    If a plant in the valley does not bloom in late spring, it must be frost-resistant.
  5. E
    Most of the plants in the valley that are not frost-resistant are native shrubs.

Answer

At least some plants that require alkaline soil are not frost-resistant.
The correct answer accurately synthesizes the overlapping sets presented in the premises. Since most native shrubs bloom in late spring, and no late-blooming plants are frost-resistant, it must be true that most native shrubs are not frost-resistant. Since every native shrub requires alkaline soil, those specific non-frost-resistant shrubs must also have alkaline soil, proving that at least some plants requiring alkaline soil are not frost-resistant.

Step-by-Step Solution

1
Diagram the first premise to establish a universal relationship.
Native Shrubs (NS) -> Alkaline Soil (AS)
This establishes that anything classified as a native shrub possesses the trait of requiring alkaline soil.
2
Diagram the remaining premises to identify characteristics applying to the majority.
Native Shrubs (NS) -most- Bloom Late Spring (BLS). Bloom Late Spring (BLS) -> NOT Frost-Resistant (~FR). Therefore, Native Shrubs (NS) -most- NOT Frost-Resistant (~FR).
Chaining a 'most' statement into a universal 'none/not' statement yields a valid 'most' deduction.
3
Synthesize the overlapping traits to find a guaranteed deduction.
Since Native Shrubs have the universal trait of needing alkaline soil, and most of them are not frost-resistant, the groups 'Alkaline Soil' and 'NOT Frost-Resistant' must overlap.
Whenever a single group (Native Shrubs) entirely belongs to Group X and mostly belongs to Group Y, Groups X and Y must share at least some members.

Key Concept

Quantifier overlapping and valid deduction from combinations of 'all', 'most', and 'none'.
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Topics

Quantifier Logic (Some, Most, All)

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