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Difficulty: Very hardParallel Conditional Logic

Match each natural language argument below to its exact underlying formal symbolic structure. Use the variables XX, YY, and ZZ to represent the distinct propositions in the order they logically arise in the text.

  • Failure to ratify the treaty by the deadline guarantees that economic sanctions will be imposed. However, economic sanctions are imposed against a nation only if it possesses significant natural resources. Consequently, any nation lacking significant natural resources must have ratified the treaty by the deadline.∼X→Y, Y→Z ∴ ∼Z→X\sim X \rightarrow Y, \ Y \rightarrow Z \ \therefore \ \sim Z \rightarrow X
  • Unless a corporation undergoes rigorous internal auditing, its stock valuation will not stabilize. Yet, a corporation's stock valuation stabilizes only if investor confidence remains high. Therefore, if a corporation undergoes rigorous internal auditing, its investor confidence will remain high.∼X→∼Y, Y→Z ∴ X→Z\sim X \rightarrow \sim Y, \ Y \rightarrow Z \ \therefore \ X \rightarrow Z
  • A scientific theory achieves mainstream acceptance only if it flawlessly explains existing empirical anomalies. Furthermore, to flawlessly explain existing empirical anomalies, a theory must introduce novel mathematical frameworks. Thus, any theory that does not introduce novel mathematical frameworks cannot achieve mainstream acceptance.X→Y, Y→Z ∴ ∼Z→∼XX \rightarrow Y, \ Y \rightarrow Z \ \therefore \ \sim Z \rightarrow \sim X
  • No comprehensive infrastructure plan is considered viable if it relies on unchecked foreign debt. Moreover, any initiative that relies on unchecked foreign debt is universally opposed by the conservative coalition. Hence, all comprehensive infrastructure plans considered viable are supported by the conservative coalition.X→∼Y, X→Z ∴ Y→∼ZX \rightarrow \sim Y, \ X \rightarrow Z \ \therefore \ Y \rightarrow \sim Z

Answer

The arguments map to the symbolic structures by isolating conditional indicators. The treaty argument translates to the valid negated chain. The auditing argument matches the flawed chain. The scientific theory argument matches the valid transitive chain. The infrastructure argument matches the flawed sufficient deduction.
Matching each natural language argument to its symbolic skeleton requires isolating the conditional indicators ('unless', 'only if', 'guarantees', 'no X if Y') and mapping them accurately to antecedents and consequents. The treaty and theory arguments employ valid reasoning frameworks (contrapositives and transitive chains). The auditing argument commits an illegal reversal flaw, and the infrastructure argument improperly links two outcomes of a shared sufficient condition.

Step-by-Step Solution

1
Translate the treaty argument into formal logic.
The argument becomes ∼X→Y\sim X \rightarrow Y and Y→ZY \rightarrow Z, concluding ∼Z→X\sim Z \rightarrow X.
This establishes the exact structural skeleton for the first stimulus, recognizing 'guarantees' as a sufficient indicator and 'only if' as a necessary indicator.
2
Translate the auditing argument into formal logic.
The argument becomes ∼X→∼Y\sim X \rightarrow \sim Y and Y→ZY \rightarrow Z, concluding X→ZX \rightarrow Z.
This isolates the flawed conditional logic by correctly mapping 'unless' as 'if not'.
3
Translate the scientific theory argument into formal logic.
The argument becomes X→YX \rightarrow Y and Y→ZY \rightarrow Z, concluding ∼Z→∼X\sim Z \rightarrow \sim X.
This isolates the valid transitive reasoning and contrapositive application in the third stimulus.
4
Translate the infrastructure argument into formal logic.
The argument becomes X→∼YX \rightarrow \sim Y and X→ZX \rightarrow Z, concluding Y→∼ZY \rightarrow \sim Z.
This exposes the flawed structural mapping of the fourth stimulus, noting that 'no A is B if C' translates to 'If C, then not B'.

Key Concept

Parallel Conditional Logic
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Topics

Parallel Conditional Logic

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