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Difficulty: HardChaining Conditional Statements

A municipal legal analyst is constructing a formal proof to demonstrate that if the city's Annual Arts Festival takes place this year, then the Mayor must not have signed the controversial Riverfront Zoning Act. The analyst relies on the following established municipal facts:

- The Mayor signs the Riverfront Zoning Act only if the Riverside Corporation initiates the condo project.
- The Riverside Corporation initiates the condo project only if they receive a property tax exemption.
- Any property tax exemption granted to developers requires the city to reduce its discretionary spending.
- Whenever the city reduces its discretionary spending, the Parks Maintenance Budget is eliminated.
- The Annual Arts Festival takes place only if the Parks Maintenance Budget is not eliminated.

Arrange the following intermediate deductions in the logically valid sequence that bridges the given premise 'The Annual Arts Festival is taking place' to the final conclusion 'The Mayor did not sign the Riverfront Zoning Act'.

  1. 1Because the Annual Arts Festival is taking place, the necessary condition for it must be met, meaning the Parks Maintenance Budget remains intact.
  2. 2Given that the Parks Maintenance Budget remains intact, the sufficient condition for its elimination cannot have occurred, so the city did not reduce its discretionary spending.
  3. 3Since the city avoided reducing its discretionary spending, the prerequisite for granting developers a property tax exemption was not fulfilled.
  4. 4The absence of a property tax exemption guarantees that the necessary requirement for the Riverside Corporation's condo project was not met, meaning the project is not initiated.
  5. 5Without the condo project being initiated, the condition required for the Mayor to sign the Riverfront Zoning Act is absent, proving the Act was not signed.

Answer

The logically valid sequence begins with the occurrence of the festival and proceeds backwards through a chain of contrapositives: deducing the budget is intact, leading to no reduction in spending, meaning no tax exemption, resulting in no condo project, and finally proving the Mayor did not sign the Act.
The correct sequence forms a continuous chain of valid deductive inferences using the contrapositive of each conditional premise. Starting with the given fact (Festival), we apply Festival -> ~Eliminated, then ~Eliminated -> ~Reduced, then ~Reduced -> ~Exemption, then ~Exemption -> ~Project, and finally ~Project -> ~Mayor. Each step correctly identifies the logical relationship (sufficient vs. necessary) required to bridge the previous deduction to the next.

Step-by-Step Solution

1
Identify the initial given fact and apply the first relevant conditional statement.
Deduce that the Parks Maintenance Budget remains intact.
The festival is taking place, and it taking place requires the budget to not be eliminated. Symbolically: Festival -> ~Eliminated.
2
Apply the derived fact (~Eliminated) to the next conditional statement in the chain.
Deduce that the city did not reduce its discretionary spending.
Reducing spending is sufficient for eliminating the budget (Reduced -> Eliminated). By the contrapositive (~Eliminated -> ~Reduced), spending was not reduced.
3
Continue the contrapositive chain using the absence of reduced spending.
Deduce that no property tax exemption was granted.
Exemptions require reduced spending (Exemption -> Reduced). By the contrapositive (~Reduced -> ~Exemption), no exemption was granted.
4
Apply the absence of the tax exemption to the condo project condition.
Deduce that the condo project is not initiated.
The project requires an exemption (Project -> Exemption). By the contrapositive (~Exemption -> ~Project), it is not initiated.
5
Apply the absence of the condo project to the final conditional statement.
Conclude that the Mayor did not sign the Act.
The Mayor signing requires the project to be initiated (Mayor -> Project). By the contrapositive (~Project -> ~Mayor), the Mayor did not sign the Act.

Key Concept

Chaining Conditional Statements via Contrapositives
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Topics

Chaining Conditional Statements

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