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Difficulty: MediumStatement and Conclusions or Inferences

Analyze the following extract from a regional infrastructure guideline and determine which of the given conclusions logically follows.

Statement: Under the new urban mobility framework, any metropolitan sector exceeding a population density of 8,000 residents per square kilometer must dedicate a minimum of 15% of its annual transit budget to pedestrian walkways.

Conclusion I: A metropolitan sector with a population density of 6,000 residents per square kilometer is not permitted to dedicate 15% of its transit budget to pedestrian walkways.
Conclusion II: If a metropolitan sector dedicates only 5% of its annual transit budget to pedestrian walkways, its population density does not exceed 8,000 residents per square kilometer.

Which of the conclusions logically follows from the given statement?

  1. A
    Only conclusion I follows
  2. Only conclusion II followsAnswer
  3. C
    Both conclusions I and II follow
  4. D
    Neither conclusion I nor II follows

Answer

Only conclusion II follows.
The correct answer accurately applies the rule of logical contraposition. If a sector exceeding the density threshold must allocate at least 15%, then any sector allocating less than 15% (such as the 5% mentioned in conclusion II) cannot possibly exceed that density threshold. Conclusion I is logically invalid because it incorrectly assumes the inverse of the statement.

Step-by-Step Solution

1
Analyze the formal logical structure of the original statement.
The statement can be expressed conditionally: If Density > 8,000, then Pedestrian Budget >= 15%.
Identifying the formal conditional relationship prevents assumption-based errors and clarifies the direction of the rule.
2
Evaluate Conclusion I against the formal conditional statement.
Conclusion I assumes: If Density <= 8,000, then Pedestrian Budget cannot be 15%. This logic does not hold.
A conditional statement establishes a requirement for a specific group; it does not forbid other groups from voluntarily meeting that same criteria (the inverse fallacy).
3
Evaluate Conclusion II against the formal conditional statement.
Conclusion II states: If Pedestrian Budget is 5% (which is < 15%), then Density <= 8,000. This is logically valid.
This represents the contrapositive of the original statement. If the necessary condition (Budget >= 15%) is not met, the sufficient condition (Density > 8,000) cannot possibly be true.

Key Concept

Logical Deduction, Modus Tollens, and Conditional Statements
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Topics

Statement and Conclusions or Inferences

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