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Difficulty: Very hardQuantifier Logic (Some, Most, All)

Analyze the four formal logic setups involving professional and social categories. Match each premise setup to the precise logical deduction or scope limit that uniquely follows from it.

  • All judges are lawyers. Most judges are politicians. Most politicians are wealthy.It must be true that some lawyers are politicians, but it is not necessarily true that any lawyers are wealthy.
  • Most judges are lawyers. Most judges are politicians. All politicians are wealthy.It must be true that most judges are wealthy and some judges are lawyers, but it is not necessarily true that all politicians are lawyers.
  • All judges are politicians. Most politicians are lawyers. Most politicians are wealthy.It must be true that some lawyers are wealthy, but it is not necessarily true that any judges are lawyers.
  • Most judges are politicians. All politicians are lawyers. All lawyers are wealthy.It must be true that all politicians are both lawyers and wealthy, but it is not necessarily true that most lawyers are politicians.

Answer

Setup 1 matches the deduction that some lawyers are politicians but none are necessarily wealthy. Setup 2 matches the deduction that most judges are wealthy but not all politicians are lawyers. Setup 3 matches the deduction that some lawyers are wealthy but no judges are necessarily lawyers. Setup 4 matches the deduction that all politicians are lawyers and wealthy, but most lawyers are not necessarily politicians.
Each setup uniquely allows for precise deductions based on formal logic rules: universal chaining (All A are B + All B are C -> All A are C), majority overlaps within the same group (Most A are B + Most A are C -> Some B are C), and universal/majority combinations (Most A are B + All B are C -> Most A are C).

Step-by-Step Solution

1
Analyze Setup 1: All judges are lawyers. Most judges are politicians. Most politicians are wealthy.
Deduce that since judges are a subset of lawyers, and most judges are politicians, there is an overlap guaranteeing some lawyers are politicians. Note that 'Most politicians are wealthy' cannot be chained back.
To identify that while 'some lawyers are politicians' must be true, we lack evidence to prove any lawyers are wealthy.
2
Analyze Setup 2: Most judges are lawyers. Most judges are politicians. All politicians are wealthy.
Deduce that combining 'Most judges are politicians' and 'All politicians are wealthy' yields 'Most judges are wealthy'. We cannot prove all politicians are lawyers.
To find the unique matching deduction regarding what must be true (most judges are wealthy) versus what is unsupported.
3
Analyze Setup 3: All judges are politicians. Most politicians are lawyers. Most politicians are wealthy.
Deduce that two majorities of the same group (politicians) must intersect, proving some lawyers are wealthy. However, the subset 'judges' might not intersect with the 'lawyers' majority.
To confirm that while some lawyers must be wealthy, we cannot guarantee any judges are lawyers.
4
Analyze Setup 4: Most judges are politicians. All politicians are lawyers. All lawyers are wealthy.
Deduce through universal chaining that all politicians are wealthy (and lawyers). However, 'All politicians are lawyers' does not mean 'Most lawyers are politicians'.
To verify that all politicians are both lawyers and wealthy, but it is not necessarily true that most lawyers are politicians.

Key Concept

Quantifier Logic (Some, Most, All) Valid Inferences and Scope Limitations
Estimated Time:3m 0s
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Topics

Quantifier Logic (Some, Most, All)

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