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Difficulty: Very hardForming Contrapositives

Match each conditional statement regarding corporate merger regulations on the left with its logically valid contrapositive on the right.

  • The merger will be approved only if the antitrust division grants clearance and the competing firm withdraws its lawsuit.If the antitrust division does not grant clearance or the competing firm does not withdraw its lawsuit, the merger will not be approved.
  • Approval of the merger requires that the antitrust division grants clearance or the competing firm withdraws its lawsuit.If the antitrust division fails to grant clearance and the competing firm refuses to withdraw its lawsuit, the merger will not be approved.
  • The merger is guaranteed to be approved if the antitrust division grants clearance and the competing firm withdraws its lawsuit.If the merger is not approved, then either the antitrust division did not grant clearance or the competing firm did not withdraw its lawsuit.
  • The antitrust division granting clearance or the competing firm withdrawing its lawsuit is sufficient for the merger to be approved.If the merger is not approved, it must be the case that the antitrust division failed to grant clearance and the competing firm failed to withdraw its lawsuit.

Answer

The logically valid pairs map combinations of necessary/sufficient conditions with their De Morgan's Law contrapositives. 'Approved only if... and...' pairs with 'If not... or not..., not approved'. 'Requires that... or...' pairs with 'If fails to... and refuses to..., not approved'. 'Guaranteed to be approved if... and...' pairs with 'If not approved, then either not... or not...'. '...or... is sufficient for' pairs with 'If not approved, it must be that failed to... and failed to...'.
Each correct pair properly executes the two mandatory steps of forming a complex contrapositive: (1) flipping the necessary and sufficient conditions while negating both sides, and (2) applying De Morgan's Laws to switch any negated 'AND' operator to an 'OR' operator, and vice versa.

Step-by-Step Solution

1
Define variables for the three propositions to systematically evaluate the logical structures without relying on English syntax.
Let M = 'Merger is approved'. Let C = 'Antitrust division grants clearance'. Let W = 'Competing firm withdraws lawsuit'.
Using symbolic logic prevents confusion caused by complex phrasing and varied conditional trigger words (only if, requires, sufficient, guaranteed).
2
Translate the four left-side conditional statements into formal logic.
Statement 1: M → (C and W). Statement 2: M → (C or W). Statement 3: (C and W) → M. Statement 4: (C or W) → M.
Accurately identifying necessary versus sufficient conditions is mandatory before forming a contrapositive.
3
Form the logically valid contrapositive for each translated statement by reversing the arrow and negating both sides.
Contrapositive 1: ~(C and W) → ~M. Contrapositive 2: ~(C or W) → ~M. Contrapositive 3: ~M → ~(C and W). Contrapositive 4: ~M → ~(C or W).
The contrapositive is the only inherently valid inference that can be drawn from a standard conditional statement.
4
Apply De Morgan's Laws to distribute the negations across the compound 'AND' / 'OR' conditions.
Contrapositive 1: (~C or ~W) → ~M. Contrapositive 2: (~C and ~W) → ~M. Contrapositive 3: ~M → (~C or ~W). Contrapositive 4: ~M → (~C and ~W). These directly map to the four statements on the right.
Negating a conjunction (AND) always yields a disjunction (OR), and negating a disjunction yields a conjunction.

Key Concept

Forming Complex Contrapositives using De Morgan's Laws
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Topics

Forming Contrapositives

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