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Difficulty: HardParallel Conditional Logic

Analyze the following natural language arguments concerning interstellar mission protocols. Match each argument to its exact underlying formal symbolic conditional structure. Let the variables XX, YY, and ZZ represent the distinct conditions or events in the order they logically appear.

  • A spacecraft can initiate the landing sequence only if it has cleared the debris field. The debris field will not be cleared unless the shield generator is fully operational. Thus, if a spacecraft lacks a fully operational shield generator, it cannot initiate the landing sequence.Premises: X→Y,∼Z→∼YX \rightarrow Y, \sim Z \rightarrow \sim Y. Conclusion: ∼Z→∼X\sim Z \rightarrow \sim X
  • Approval from the science directorate is required for commencing the drilling operation. Approval from the science directorate requires submission of a hazard report. Therefore, if a hazard report has been submitted, the drilling operation will commence.Premises: X→Y,Y→ZX \rightarrow Y, Y \rightarrow Z. Conclusion: Z→XZ \rightarrow X
  • The crew will experience optimal gravity only if the rotation module is engaged. The rotation module will be engaged only if the primary reactor is functioning. The primary reactor is currently offline. Thus, the crew will not experience optimal gravity.Premises: X→Y,Y→Z,∼ZX \rightarrow Y, Y \rightarrow Z, \sim Z. Conclusion: ∼X\sim X
  • If the navigation sensors detect an anomaly, the main thrusters will automatically shut down. The main thrusters shutting down will cause the auxiliary beacons to activate. The navigation sensors did not detect an anomaly. Thus, the auxiliary beacons will not activate.Premises: X→Y,Y→Z,∼XX \rightarrow Y, Y \rightarrow Z, \sim X. Conclusion: ∼Z\sim Z

Answer

The arguments logically map as follows: the landing sequence argument features a contrapositive chain; the drilling operation argument features an illegal reversal flaw; the gravity argument features a valid modus tollens deduction; and the navigation sensors argument features an illegal negation flaw.
Each natural language argument uniquely translates into one of the provided formal skeletons based on strict syntactic indicators ('only if', 'unless', 'requires', 'causes'). Carefully mapping the sufficient and necessary conditions reveals two flawed arguments (one illegal reversal, one illegal negation) and two valid deductions (one contrapositive, one modus tollens chain).

Step-by-Step Solution

1
Translate the 'landing sequence' argument into formal logic.
X -> Y, ~Z -> ~Y. Conclusion: ~Z -> ~X.
Understanding 'only if' as introducing a necessary condition, and 'unless' as 'if not' reveals the valid conditional chain.
2
Translate the 'drilling operation' argument into formal logic.
X -> Y, Y -> Z. Conclusion: Z -> X.
Identifying 'is required for' and 'requires' helps spot the illegal reversal flaw where the conclusion reverses the established sufficient-necessary relationship.
3
Translate the 'optimal gravity' argument into formal logic.
X -> Y, Y -> Z, ~Z. Conclusion: ~X.
Chaining the two 'only if' statements creates a transitive path. Denying the final necessary condition (~Z) validly denies the initial sufficient condition (~X).
4
Translate the 'navigation sensors' argument into formal logic.
X -> Y, Y -> Z, ~X. Conclusion: ~Z.
The chain is established by 'if' and 'cause'. The premise denies the initial sufficient condition (~X) to wrongly deny the final necessary condition (~Z), which is an illegal negation.

Key Concept

Parallel Conditional Logic
Estimated Time:2m 30s
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Topics

Parallel Conditional Logic

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