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Difficulty: Very hardIdentifying Must-Be-True Statements and Valid Deductions

A corporate governance protocol enforces the following rules regarding financial compliance:

1. Every department that incurs an annual operational deficit exceeding 10%10\% undergoes an automatic internal audit.
2. An internal audit takes place if and only if either the division vice president requests it or the central risk oversight committee flags the department's ledger.
3. Whenever the central risk oversight committee flags a department's ledger, that department is prohibited from submitting capital expansion proposals for the following fiscal year.

During the current fiscal year, the Logistics Department was not prohibited from submitting capital expansion proposals for the following fiscal year.

Evaluate the following statement based strictly on the information provided:

Statement: If the Logistics Department incurred an operational deficit exceeding 10%10\% during the current fiscal year, then its division vice president requested an internal audit.

Answer: Answer

Answer

True
The conclusion is logically unavoidable. The fact that Logistics was not prohibited from submitting capital proposals guarantees that the risk committee did not flag its ledger. Therefore, if a deficit over 10%10\% triggered an audit, the audit could only have been requested by the division vice president.

Step-by-Step Solution

1
Formalize the conditional statements into symbolic logic terms.
Let DD = Deficit >10%>10\%, AA = Internal Audit, VV = Vice President requests audit, RR = Risk committee flags ledger, PP = Prohibited from capital proposals.
Premise 1: D→AD \rightarrow A
Premise 2: A↔(V∨R)A \leftrightarrow (V \lor R)
Premise 3: R→PR \rightarrow P
Symbolic representation clarifies the logical relationships and valid inferences.
2
Derive deductions from the given fact using contrapositive logic.
Fact: ¬P\neg P (Not prohibited from submitting proposals).
From R→PR \rightarrow P, taking the contrapositive (¬P→¬R\neg P \rightarrow \neg R) yields ¬R\neg R (The risk committee did not flag the ledger).
A conditional statement and its contrapositive are logically equivalent.
3
Evaluate the antecedent condition of the statement (DD).
Assume DD (Logistics incurred a deficit >10%>10\%). By Modus Ponens on Premise 1 (D→AD \rightarrow A), AA must be true (An internal audit took place).
If the sufficient condition of a valid conditional rule occurs, the necessary condition must also occur.
4
Combine the intermediate deductions to determine the truth of VV.
From Premise 2, A→(V∨R)A \rightarrow (V \lor R). Since AA is true, the disjunction (V∨R)(V \lor R) must be true. Since ¬R\neg R was established in Step 2, disjunctive syllogism forces VV to be true.
If a disjunction 'A or B' is true and 'B' is false, then 'A' must be true.

Key Concept

Multi-step Conditional Deduction via Contrapositive and Disjunctive Syllogism
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Topics

Identifying Must-Be-True Statements and Valid Deductions

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