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Difficulty: MediumChaining Conditional Statements

Any painting restored by Dr. Vance will be included in the upcoming exhibition. If a painting is included in the exhibition, the museum must insure it for over a million dollars. Furthermore, the museum will not insure any painting for over a million dollars unless it has been appraised by an independent panel. Any painting belonging to the Renaissance period has been appraised by an independent panel. Finally, no painting featuring religious motifs was restored by Dr. Vance.

If the statements above are true, which one of the following must also be true?

  1. A
    Any painting appraised by an independent panel belongs to the Renaissance period.
  2. If a painting has not been appraised by an independent panel, it will not be included in the upcoming exhibition.Answer
  3. C
    If a painting was not restored by Dr. Vance, it will not be included in the upcoming exhibition.
  4. D
    Being appraised by an independent panel is sufficient to guarantee that a painting will be insured for over a million dollars.
  5. E
    Most of the paintings that have been appraised by an independent panel were restored by Dr. Vance.

Answer

If a painting has not been appraised by an independent panel, it will not be included in the upcoming exhibition.
The correct answer properly chains two conditional statements and takes the contrapositive. The stimulus states: 'Included in exhibition -> Insured >1M' and 'Insured >1M -> Appraised by independent panel'. Chaining these gives: 'Included in exhibition -> Appraised by independent panel'. The logical contrapositive of this new statement is: 'NOT Appraised by independent panel -> NOT Included in exhibition', which perfectly matches this option.

Step-by-Step Solution

1
Translate the statements into formal conditional logic.
1: Restored by Vance -> Included in exhibition. 2: Included in exhibition -> Insured >1M. 3: Insured >1M -> Appraised by independent panel. 4: Renaissance -> Appraised by independent panel. 5: Religious motifs -> NOT Restored by Vance.
Mapping the logical relationships is necessary to see how the rules connect.
2
Identify connections between the conditional statements to form a chain.
Rules 1, 2, and 3 link together perfectly: Restored by Vance -> Included in exhibition -> Insured >1M -> Appraised by independent panel.
Chaining conditionals reveals deductive inferences that are not explicitly stated.
3
Derive the valid contrapositives of the logical chain.
The chain 'Included in exhibition -> Appraised by independent panel' yields the contrapositive: 'NOT Appraised by independent panel -> NOT Included in exhibition'.
Correct answers in Must Be True questions frequently rely on the contrapositive of a chained conditional statement.

Key Concept

Chaining Conditional Statements
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Topics

Chaining Conditional Statements

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