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Zorluk: ZorQuantifier Logic (Some, Most, All)

Consider the following premises regarding staff at the Central Research Institute:

All research fellows at the institute possess a doctoral degree.
Most research fellows at the institute are currently directing a grant-funded project.
Most research fellows at the institute have served on the ethics board.

Based on these premises, the following deduction must be true: At least some individuals who are directing a grant-funded project and have served on the ethics board possess a doctoral degree.

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True. The formal logic premises guarantee that at least one individual satisfies all three conditions.
The conclusion logically follows from the premises. The intersection of two 'most' statements concerning the same base group (research fellows) proves that 'some' research fellows have both characteristics. Applying the universal 'all' condition to these specific fellows proves they must also hold a doctoral degree.

Adım Adım Çözüm

1
Analyze the relationship between the two 'most' premises applied to the same group.
Identify that because 'most' means strictly more than half, the group of research fellows directing projects and the group serving on the board must overlap by at least one person.
Two 'most' statements applied to the exact same finite group mathematically require an intersection (yielding a 'some' statement).
2
Apply the 'all' premise to the overlapping group identified in Step 1.
Conclude that since all research fellows possess a doctoral degree, the specific research fellow(s) who both direct a project and serve on the board must inherently possess a doctoral degree.
A universal 'all' statement applies to every single member of the group, which inherently includes any and all subsets.
3
Evaluate the target statement against the deductions.
The target statement claims 'some individuals' (meaning at least one) have all three characteristics, which perfectly matches the deduction.
To confirm if the statement logically must be true based strictly on the formal logic of the provided premises.

Anahtar Kavram

Quantifier Overlap (Most + Most = Some)
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Konular

Quantifier Logic (Some, Most, All)

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