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Difficulty: HardDiagramming Conditional Statements

A civil aviation authority is formalizing its autonomous vehicle protocols into a logical database. The fundamental propositions are defined as follows:

AA: The airspace is restricted
BB: The beacon is activated
CC: The clearance is revoked
DD: The drone is grounded

Match each complex English regulatory rule to a logically equivalent formal representation. (Note: ∼\sim denotes logical negation, and →\rightarrow denotes implication. The correct formal representations may be in their contrapositive forms.)

  • The airspace is restricted unless the beacon is activated.∼A→B\sim A \rightarrow B
  • The drone is grounded only if the airspace is restricted and the clearance is revoked.(∼A or ∼C)→∼D(\sim A \text{ or } \sim C) \rightarrow \sim D
  • The only drones that are grounded are those for which the beacon is not activated.B→∼DB \rightarrow \sim D
  • Activating the beacon without revoking the clearance ensures that the drone is grounded.∼D→(∼B or C)\sim D \rightarrow (\sim B \text{ or } C)

Answer

The correct pairings are: Left 1 with Right 2, Left 2 with Right 4, Left 3 with Right 1, and Left 4 with Right 3.
Each correct pair logically links a complex English conditional statement to its precise formal contrapositive. Success requires two distinct analytical actions: accurately diagramming the initial statement using syntactic indicator words, and then strictly applying logical negation rules (including De Morgan's Laws for compound conditions) to locate the equivalent formal expression.

Step-by-Step Solution

1
Translate the 'unless' statement (Left 1).
Yields ∼B→A\sim B \rightarrow A. Then form the contrapositive to find the match: ∼A→B\sim A \rightarrow B (Right 2).
Statements using 'unless' require treating the modified term as necessary, and negating the remaining term as sufficient. Since the direct translation is absent, the contrapositive is required.
2
Translate the 'only if' statement containing a compound necessary condition (Left 2).
Yields D→(A and C)D \rightarrow (A \text{ and } C). Applying the contrapositive and De Morgan's Laws yields (∼A or ∼C)→∼D(\sim A \text{ or } \sim C) \rightarrow \sim D (Right 4).
Indicator 'only if' designates the necessary condition. When negating a compound 'and' statement during contraposition, it must flip to an 'or'.
3
Translate 'The only' statement (Left 3).
Yields D→∼BD \rightarrow \sim B. Its contrapositive is B→∼DB \rightarrow \sim D (Right 1).
The phrase 'The only' identifies the sufficient condition. Negating both sides and reversing gives the proper contrapositive.
4
Translate the 'ensures' statement with a 'without' modifier (Left 4).
Yields (B and ∼C)→D(B \text{ and } \sim C) \rightarrow D. The contrapositive is ∼D→(∼B or C)\sim D \rightarrow (\sim B \text{ or } C) (Right 3).
'Ensures' functions as an 'if' (sufficient). 'Without' acts as an 'and not'. Applying De Morgan's Laws to the contrapositive flips the 'and' back to an 'or'.

Key Concept

Advanced Conditional Logic: Translating complex English indicators ('unless', 'only if', 'the only') and applying De Morgan's Laws to form contrapositives of compound statements.
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Topics

Diagramming Conditional Statements

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