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Zorluk: OrtaQuantifier Logic (Some, Most, All)

A recent audit of the city's public transit network revealed several operational details. Most of the transit routes in the northern sector currently operate at a financial loss. Furthermore, all transit routes that operate at a financial loss are heavily subsidized by the municipal government. Interestingly, the audit also showed that most of the transit routes in the northern sector are serviced exclusively by hybrid vehicles.

Which one of the following must be true based on the information above?

  1. At least some transit routes serviced exclusively by hybrid vehicles are heavily subsidized by the municipal government.Cevap
  2. B
    All transit routes that are heavily subsidized by the municipal government operate at a financial loss.
  3. C
    All of the transit routes in the northern sector are heavily subsidized by the municipal government.
  4. D
    Transit routes that do not operate at a financial loss are not heavily subsidized by the municipal government.
  5. E
    The exclusive use of hybrid vehicles on most northern sector transit routes is the primary cause of their financial loss.

Cevap

At least some transit routes serviced exclusively by hybrid vehicles are heavily subsidized by the municipal government.
The correct answer is derived by combining three premises. First, because most northern routes operate at a loss and all loss-making routes are subsidized, we can definitively deduce that most northern routes are subsidized. Second, we are explicitly told that most northern routes are serviced by hybrid vehicles. Because more than half of the northern routes are subsidized and more than half are serviced by hybrid vehicles, there must be an overlap between these two groups. Therefore, it is absolutely true that at least some routes with hybrid vehicles are subsidized.

Adım Adım Çözüm

1
Chain the first two premises to form a new deduction.
Deduction: Most of the transit routes in the northern sector are heavily subsidized.
If most northern routes operate at a loss, and every route operating at a loss is subsidized, then logically, most northern routes must be subsidized.
2
Analyze the final premise in relation to the new deduction.
We now have two facts about the northern sector: Most of its routes are subsidized, and most of its routes are serviced exclusively by hybrid vehicles.
Identifying multiple 'most' statements about the exact same subject group (northern sector routes) is the key to unlocking the final inference.
3
Apply the overlapping quantifiers rule.
Conclusion: At least some routes serviced by hybrid vehicles are subsidized.
If a majority of a defined group possesses trait X, and a majority of that same group possesses trait Y, there is a mathematical necessity that at least one member of the group possesses both trait X and trait Y.

Anahtar Kavram

Formal logic deductions using overlapping quantifiers, specifically deriving a 'some' statement from two intersecting 'most' statements applied to the same group.
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Konular

Quantifier Logic (Some, Most, All)

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