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Zorluk: OrtaQuantifier Logic (Some, Most, All)

At the city's architectural preservation site, most of the historic buildings were constructed before the year 1900. Furthermore, most of the historic buildings at the site feature masonry facades. All buildings constructed before 1900 at the site are protected by the heritage registry.

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

  1. At least some buildings protected by the heritage registry feature masonry facades.Cevap
  2. B
    Most of the historic buildings at the site feature both masonry facades and protection by the heritage registry.
  3. C
    Any building at the site that is protected by the heritage registry was constructed before the year 1900.
  4. D
    Any building at the site that was not constructed before 1900 is not protected by the heritage registry.
  5. E
    At least some buildings featuring masonry facades at the site were constructed after the year 1900.

Cevap

The correct answer is the statement that at least some buildings protected by the heritage registry feature masonry facades.
The premises establish that 'most' historic buildings are pre-1900 and 'most' feature masonry facades. Because both statements apply to the majority of the exact same group, there must be a mathematical overlap: at least some historic buildings are both pre-1900 and feature masonry. The stimulus then states that all pre-1900 buildings are protected by the heritage registry. Since we already proved that some of those pre-1900 buildings have masonry facades, those specific overlapping buildings must also be protected. Therefore, it is definitively true that at least some protected buildings feature masonry facades.

Adım Adım Çözüm

1
Identify and map the formal quantifier relationships provided in the premises.
1. Most Historic -> Pre-1900
2. Most Historic -> Masonry
3. All Pre-1900 -> Protected
Deconstructing the stimulus into formal logical notation is the first step to making rigorous deductions.
2
Combine the two 'most' statements concerning the same subject group (Historic buildings).
Since 'most' indicates more than 50%, the two groups (Pre-1900 and Masonry) must overlap. Therefore, at least some Pre-1900 buildings feature Masonry.
In quantifier logic, any two 'most' statements applied to the exact same population logically guarantee an intersection (a 'some' statement).
3
Chain the resulting 'some' statement with the 'all' statement.
Since 'Some Pre-1900 -> Masonry' and 'All Pre-1900 -> Protected', it logically follows that 'Some Protected -> Masonry'.
A 'some' statement can be chained with an 'all' statement if the 'all' condition applies to the shared term, transferring the trait to the new group.

Anahtar Kavram

Deducing overlaps from multiple 'most' statements and chaining them with 'all' statements.
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Konular

Quantifier Logic (Some, Most, All)

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