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Zorluk: ZorChaining Conditional Statements

A city planner is reviewing new zoning regulations for a commercial district, which state the following:

* A building will not be granted a Class A permit unless it features a subterranean parking garage.
* Any building featuring a subterranean parking garage must be evaluated by a structural engineer.
* Buildings are exempt from the deep excavation tax only if they are not evaluated by a structural engineer.
* No property is subject to the deep excavation tax unless it requires specialized foundation reinforcements.

Arrange the following logical derivations in the correct sequential order to form a valid deductive chain proving the conclusion: 'If a building is granted a Class A permit, it must require specialized foundation reinforcements.'

  1. 1Translate the first 'unless' condition to establish that obtaining a Class A permit requires a subterranean parking garage (CA→SPGCA \rightarrow SPG).
  2. 2Apply the direct conditional statement to deduce that having a subterranean parking garage necessitates a structural engineer's evaluation (SPG→SESPG \rightarrow SE).
  3. 3Take the contrapositive of the 'only if' statement to establish that receiving a structural engineer's evaluation subjects a building to the deep excavation tax (SE→DETSE \rightarrow DET).
  4. 4Take the contrapositive of the second 'unless' condition to show that being subject to the deep excavation tax requires specialized foundation reinforcements (DET→SFRDET \rightarrow SFR).
  5. 5Transitively link the complete sequence of necessary conditions to establish the final conclusion (CA→SFRCA \rightarrow SFR).

Cevap

The correct logical sequence begins by translating the initial 'unless' condition for a Class A permit, directly linking it to the structural engineer evaluation, contraposing the 'only if' statement to link to the deep excavation tax, contraposing the final 'unless' statement for foundation reinforcements, and finally chaining them all together.
The correct deductive sequence must build a valid transitive chain (A→BA \rightarrow B, B→CB \rightarrow C, C→DC \rightarrow D, D→ED \rightarrow E). We begin with the sufficient condition of the desired conclusion (Class A permit) and correctly translate each rule—using contrapositives for 'unless' and 'only if' statements—to link the necessary conditions sequentially until we arrive at the final necessary condition (specialized foundation reinforcements).

Adım Adım Çözüm

1
Identify the initial trigger and necessary condition from the first premise.
CA→SPGCA \rightarrow SPG
The first premise uses 'unless'. 'Not A unless B' translates to ∼B→∼A\sim B \rightarrow \sim A, which contraposes to A→BA \rightarrow B. Thus, a Class A Permit (CACA) requires a Subterranean Parking Garage (SPGSPG).
2
Link the first necessary condition to the second premise.
SPG→SESPG \rightarrow SE
The second premise is a direct conditional ('Any A must B'), translating directly to SPG→SESPG \rightarrow SE (Subterranean Parking Garage requires a Structural Engineer).
3
Translate and contrapose the 'only if' statement to continue the chain.
SE→DETSE \rightarrow DET
'A only if B' means A→BA \rightarrow B. 'Exempt only if not evaluated' translates to Exempt→∼SEExempt \rightarrow \sim SE. The contrapositive is SE→∼ExemptSE \rightarrow \sim Exempt, meaning evaluated buildings are subject to the Deep Excavation Tax (DETDET).
4
Translate and contrapose the final 'unless' statement.
DET→SFRDET \rightarrow SFR
'No A unless B' translates to ∼B→∼A\sim B \rightarrow \sim A. The contrapositive is A→BA \rightarrow B. Thus, being subject to the Deep Excavation Tax (DETDET) requires Specialized Foundation Reinforcements (SFRSFR).
5
Transitively chain all conditional statements together.
CA→SFRCA \rightarrow SFR
By linking CA→SPGCA \rightarrow SPG, SPG→SESPG \rightarrow SE, SE→DETSE \rightarrow DET, and DET→SFRDET \rightarrow SFR, we prove the transitive conclusion CA→SFRCA \rightarrow SFR.

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Chaining Conditional Statements
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Chaining Conditional Statements

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