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Difficulty: MediumDiagramming Conditional Statements

In formal logic, accurately diagramming conditional statements is crucial for deductive reasoning. Let MM represent the statement 'The motion is granted' and SS represent the statement 'A settlement is reached'. Match each legal rule on the left with its correct propositional logic translation on the right.

  • The motion will be granted only if a settlement is reached.M→SM \rightarrow S
  • Reaching a settlement ensures that the motion will be granted.S→MS \rightarrow M
  • A settlement will be reached unless the motion is granted.∼M→S\sim M \rightarrow S
  • The motion will not be granted if a settlement is reached.S→∼MS \rightarrow \sim M

Answer

The correct pairings are: 'only if' maps to M→SM \rightarrow S; 'ensures' maps to S→MS \rightarrow M; 'unless' maps to ∼M→S\sim M \rightarrow S; and 'not... if' maps to S→∼MS \rightarrow \sim M.
Each legal rule features a distinct logical indicator that dictates the direction of the conditional relationship. 'Only if' denotes a necessary condition (M→SM \rightarrow S). 'Ensures' acts identically to 'if', marking a sufficient condition (S→MS \rightarrow M). 'Unless' translates to 'if not', negating one term to form the sufficient condition (∼M→S\sim M \rightarrow S). Finally, an 'if' clause in the middle of a sentence still acts as the sufficient condition, and the negated main clause becomes the necessary condition (S→∼MS \rightarrow \sim M).

Step-by-Step Solution

1
Identify the sufficient and necessary conditions for the 'only if' statement.
The motion is granted (MM) is sufficient, and a settlement is reached (SS) is necessary.
The phrase 'only if' immediately precedes the necessary condition.
2
Analyze the statement using 'ensures'.
Reaching a settlement (SS) guarantees the motion (MM), giving S→MS \rightarrow M.
Synonyms for 'if' like 'ensures' or 'guarantees' introduce the sufficient condition.
3
Translate the 'unless' statement into an 'if not' conditional.
The statement becomes 'If the motion is not granted, a settlement will be reached' (∼M→S\sim M \rightarrow S).
The logical equivalent of 'A unless B' is 'If not B, then A'.
4
Determine the trigger and outcome for the 'if' statement containing a negation.
The sufficient condition is 'a settlement is reached' (SS) and the necessary condition is 'the motion will not be granted' (∼M\sim M), yielding S→∼MS \rightarrow \sim M.
The word 'if' introduces the sufficient condition, regardless of where it appears in the sentence.

Key Concept

Diagramming Conditional Statements
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Topics

Diagramming Conditional Statements

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