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Difficulty: Very hardDiagramming Conditional Statements

According to the guild's charter, no scholar is appointed to the High Council without possessing both a doctorate in epistemology and a mastery of ancient syntax. Furthermore, mastery of ancient syntax requires a scholar to either have studied under a Grandmaster or have deciphered the Lumina Texts. Any scholar who has deciphered the Lumina Texts is immediately granted university tenure. However, receiving university tenure completely disqualifies a scholar from being appointed to the High Council.

Which of the following conclusions can be logically drawn from the statements above?

  1. Any scholar appointed to the High Council must have studied under a Grandmaster.Answer
  2. B
    If a scholar lacks a doctorate in epistemology, they cannot have studied under a Grandmaster.
  3. C
    Anyone who has studied under a Grandmaster and holds a doctorate in epistemology will be appointed to the High Council.
  4. D
    A scholar who has not deciphered the Lumina Texts and has not studied under a Grandmaster cannot possess a doctorate in epistemology.
  5. E
    Most scholars who possess a doctorate in epistemology and a mastery of ancient syntax are appointed to the High Council.

Answer

The correct answer states that any scholar appointed to the High Council must have studied under a Grandmaster.
The stimulus establishes a chain of necessary conditions. Appointment to the High Council (HC) requires both a doctorate and mastery of syntax (AS). Mastery of syntax, in turn, requires either studying under a Grandmaster (GM) or deciphering the Lumina Texts (LT). However, deciphering the Lumina Texts leads to university tenure, which strictly disqualifies one from the High Council. Therefore, anyone on the High Council cannot have deciphered the Lumina Texts (otherwise they would have tenure and be disqualified). Because they possess mastery of syntax but have not deciphered the Lumina Texts, they must have fulfilled the only remaining valid pathway to mastery: studying under a Grandmaster.

Step-by-Step Solution

1
Diagram the requirements for High Council appointment.
High Council (HC) → Doctorate (DE) AND Syntax Mastery (AS)
Translates the rule: 'no scholar is appointed... without possessing both'.
2
Diagram the requirements for mastery of ancient syntax.
Syntax Mastery (AS) → Grandmaster (GM) OR Lumina Texts (LT)
Translates the rule: 'mastery... requires a scholar to either... or'.
3
Diagram the rules regarding university tenure and the High Council.
Lumina Texts (LT) → Tenure (UT); and Tenure (UT) → NOT High Council (~HC)
Translates the rules: 'Any scholar who has deciphered... is granted tenure' and 'tenure completely disqualifies...'.
4
Form the contrapositive chain linking the High Council to the Lumina Texts.
If HC is true, then ~UT must be true. If ~UT is true, then ~LT must be true. Therefore, HC → ~LT.
To be on the High Council, one cannot have tenure. To not have tenure, one cannot have deciphered the Lumina Texts.
5
Synthesize the findings to resolve the disjunction for syntax mastery.
Since HC → AS, a High Council member must satisfy (GM OR LT). Because we proved HC → ~LT, the LT condition is impossible for a High Council member. Therefore, they must satisfy GM.
Applies disjunctive syllogism within a conditional chain: if one required alternative is eliminated, the other must be true. Thus, HC → GM.

Key Concept

Complex Conditional Chaining and Disjunctive Syllogism
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Topics

Diagramming Conditional Statements

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