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Difficulty: HardChaining Conditional Statements

The Board of Antiquities has issued the following directives for affiliated institutions: Any institution that acquires a Class I historical artifact must possess a reinforced subterranean vault. An institution is eligible for the federal preservation grant only if it acquires a Class I historical artifact. Furthermore, unless an institution maintains a specialized restoration laboratory, it cannot possess a reinforced subterranean vault. Finally, an institution can host the biennial symposium only if it is eligible for the federal preservation grant.

Based on the directives above, which of the following must be true?

  1. A
    Any institution that maintains a specialized restoration laboratory is eligible to host the biennial symposium.
  2. B
    If an institution does not acquire a Class I historical artifact, it cannot maintain a specialized restoration laboratory.
  3. If an institution does not maintain a specialized restoration laboratory, it cannot host the biennial symposium.Answer
  4. D
    An institution is eligible for the federal preservation grant if it possesses a reinforced subterranean vault.
  5. E
    Most institutions that possess a reinforced subterranean vault eventually host the biennial symposium.

Answer

If an institution does not maintain a specialized restoration laboratory, it cannot host the biennial symposium.
Based on the formal logic chain derived from the directives, hosting the biennial symposium requires being eligible for the grant, which requires a Class I artifact, which requires a vault, which requires a restoration laboratory (HBS -> SRL). Therefore, by taking the contrapositive, if an institution lacks a restoration laboratory, it cannot host the biennial symposium (~SRL -> ~HBS).

Step-by-Step Solution

1
Translate the first premise into formal logic
Acquires Class I (C1) -> Subterranean Vault (RSV)
The word 'Any' introduces a sufficient condition.
2
Translate the second premise into formal logic
Preservation Grant (FPG) -> Acquires Class I (C1)
The phrase 'only if' introduces a necessary condition.
3
Translate the third premise into formal logic
Subterranean Vault (RSV) -> Restoration Lab (SRL)
The word 'unless' means 'if not'. 'Unless SRL, ~RSV' translates to 'If RSV, then SRL'.
4
Translate the fourth premise into formal logic
Biennial Symposium (HBS) -> Preservation Grant (FPG)
The phrase 'only if' introduces a necessary condition.
5
Chain the statements and find the contrapositive
Chain: HBS -> FPG -> C1 -> RSV -> SRL. Contrapositive: ~SRL -> ~HBS
Connecting the necessary condition of one statement to the sufficient condition of another reveals the full logical chain. The contrapositive of this full chain must also be true.

Key Concept

Chaining conditional statements with various indicator words (any, only if, unless) and forming contrapositives.
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Topics

Chaining Conditional Statements

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