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Difficulty: MediumDiagramming Conditional Statements

A pharmaceutical regulatory agency uses the following propositional variables to evaluate clinical trials:

MM: Marketing is approved
TT: Toxicity is low
RR: Regulatory audit is passed
CC: Control group improves

Match each English policy statement with its logically equivalent formal diagram. Note that the correct match may represent either the direct translation or its valid contrapositive.

  • Marketing is approved only if toxicity is low.∼T→∼M\sim T \rightarrow \sim M
  • Toxicity is low if the regulatory audit is passed.∼T→∼R\sim T \rightarrow \sim R
  • The control group will not improve unless toxicity is low.C→TC \rightarrow T
  • Passing the regulatory audit requires that the control group improves.∼C→∼R\sim C \rightarrow \sim R

Answer

Statement 1 matches ∼T→∼M\sim T \rightarrow \sim M; Statement 2 matches ∼T→∼R\sim T \rightarrow \sim R; Statement 3 matches C→TC \rightarrow T; Statement 4 matches ∼C→∼R\sim C \rightarrow \sim R.
Each English statement translates to a specific formal logic relationship based on its indicator words. Finding the correct match requires establishing the base conditional statement and, if the direct translation is not an option, taking its valid contrapositive (reversing and negating both terms).

Step-by-Step Solution

1
Analyze 'Marketing is approved only if toxicity is low.'
Translates to M→TM \rightarrow T. Its contrapositive is ∼T→∼M\sim T \rightarrow \sim M.
The phrase 'only if' always modifies the necessary condition, which belongs on the right side of the arrow.
2
Analyze 'Toxicity is low if the regulatory audit is passed.'
Translates to R→TR \rightarrow T. Its contrapositive is ∼T→∼R\sim T \rightarrow \sim R.
When 'if' is used alone in the middle of a sentence, it modifies the sufficient condition, which belongs on the left side of the arrow.
3
Analyze 'The control group will not improve unless toxicity is low.'
Translates to ∼T→∼C\sim T \rightarrow \sim C. Its contrapositive is C→TC \rightarrow T.
The term 'unless' means 'if not'. Applying this gives: If toxicity is not low (∼T\sim T), the control group will not improve (∼C\sim C).
4
Analyze 'Passing the regulatory audit requires that the control group improves.'
Translates to R→CR \rightarrow C. Its contrapositive is ∼C→∼R\sim C \rightarrow \sim R.
The word 'requires' indicates that the subsequent concept is mandatory, making it the necessary condition (right side of the arrow).

Key Concept

Translating conditional indicator words ('only if', 'if', 'unless', 'requires') into formal logic and deriving their contrapositives.
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Topics

Diagramming Conditional Statements

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