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Question

Difficulty: HardForming Contrapositives

According to the newly ratified Outer Space Mining Treaty, securing a commercial extraction license on the lunar surface requires that a corporation either establish an autonomous life-support habitat or formally partner with a state-sponsored space agency.

Which one of the following statements logically follows from the treaty's requirement?

  1. A
    If a corporation either fails to establish an autonomous life-support habitat or does not formally partner with a state-sponsored space agency, it will be unable to secure a commercial extraction license on the lunar surface.
  2. B
    Any corporation that either establishes an autonomous life-support habitat or formally partners with a state-sponsored space agency will secure a commercial extraction license on the lunar surface.
  3. If a corporation neither establishes an autonomous life-support habitat nor formally partners with a state-sponsored space agency, it will not secure a commercial extraction license on the lunar surface.Answer
  4. D
    If a corporation does not secure a commercial extraction license on the lunar surface, it must be true that the corporation neither established an autonomous life-support habitat nor formally partnered with a state-sponsored space agency.
  5. E
    Establishing an autonomous life-support habitat is a sufficient condition to ensure that a corporation will secure a commercial extraction license on the lunar surface.

Answer

The statement that logically follows is: 'If a corporation neither establishes an autonomous life-support habitat nor formally partners with a state-sponsored space agency, it will not secure a commercial extraction license on the lunar surface.'
The correct answer accurately applies formal logic and De Morgan's laws to create a valid contrapositive. The original premise dictates: License → (Habitat OR Partner). The contrapositive requires reversing the direction and negating both sides. Crucially, negating an 'OR' statement transforms it into an 'AND' statement. Therefore, the valid inference is: If NOT Habitat AND NOT Partner (phrased as 'neither... nor'), then NOT License.

Step-by-Step Solution

1
Identify the logical structure of the original rule in the stimulus.
Sufficient condition: Securing a commercial extraction license. Necessary condition: Establishing a habitat OR partnering with a space agency.
Mapping the exact formal logic relationship is required before evaluating conditional deductions.
2
Formulate the logical contrapositive of the rule.
To find the contrapositive, reverse the terms and negate both sides. Negating the 'OR' statement requires De Morgan's laws: 'NOT (Habitat OR Partner)' becomes 'NOT Habitat AND NOT Partner'.
The only statement that definitively must be true based on a conditional premise is its logically valid contrapositive.
3
Match the formulated contrapositive to the answer choices.
The correct deduction is: If a corporation does NOT establish a habitat AND does NOT partner with an agency (neither... nor), then it CANNOT secure a license.
This eliminates choices that contain formal logical fallacies such as mistaken reversals, mistaken negations, or incomplete applications of De Morgan's laws.

Key Concept

Forming valid contrapositives from compound conditional statements using De Morgan's laws.
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Topics

Forming Contrapositives

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