91Ó°ÊÓ

If retail stores experience a decrease in revenues during this holiday season, then either attitudes toward extravagant gift-giving have changed or prices have risen beyond the level most people can afford. If attitudes have changed, then we all have something to celebrate this season. If prices have risen beyond the level most people can afford. then it must be that salaries have not kept pace with rising prices during the past year. Assume the premises above to be true. If salaries have kept pace with rising prices during the past year, which one of the following must be true? (A) Attitudes toward extravagant gift-giving have changed. (B) Retail stores will not experience a decrease in retail sales during this holiday season. (C) Prices in retail stores have not risen beyond the level that most people can afford during this holiday season. (D) Attitudes toward extravagant gift-giving have not changed, and stores will not experience a decrease in revenues during this holiday season. (E) Either attitudes toward extravagant gift-giving have changed or prices have risen beyond the level that most people can afford during this holiday season.

Step by step solution

01

Understanding the Problem

We need to determine the logical conclusion that follows from the given premises if salaries have kept pace with rising prices during the past year.

02

Translate the Statements into Logic Expressions

Define statements: \( P \): Prices have risen beyond the level most people can afford. \( A \): Attitudes have changed. \( R \): Retail stores experience a decrease in revenues. \( S \): Salaries have not kept pace with rising prices. The premises are: \( R \to (A \lor P) \), \( A \to \text{celebration} \), and \( P \to S \).

03

Analyze the Given Assumption

We are given that salaries have kept pace with rising prices. Therefore, \( eg S \) is true. This implies \( eg P \) because \( P \to S \) and \( S \) is false. Hence, \( P \) must be false.

04

Determine the Consequence of \( \neg P \)

Since \( P \) is false, if \( R \) is true, then \( A \) must be true due to \( R \to (A \lor P) \). However, if \( A \) is also false, then \( R \) must be false as well because both \( A \) and \( P \) can't be false when \( R \) is true.

05

Conclude from \( \neg R \)

\( eg R \) tells us retail stores will not experience a decrease in revenues during this holiday season. \( P \) is false, as concluded, meaning prices have not risen beyond what most can afford.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Premise Analysis

When analyzing a problem involving logical reasoning, it is crucial to begin with a thorough understanding of the premises. In this exercise, we are dealing with a set of conditions or premises that outline specific circumstances. Taken together, they form a foundation for drawing conclusions. Here's a simple breakdown of the approach:

• Identify each premise: Break each statement into logical expressions. For example, one premise is that if retail revenues drop, either gift-giving attitudes have changed, or prices have become unaffordable.
• Determine relationships: Notice how premises are often interrelated. For instance, prices rising beyond affordability is directly linked to whether salaries have kept pace with them.
• Examine assumptions: The problem asks us to start from a particular assumption — that salaries have kept up with prices — and this directly influences the validity of other premises.

By analyzing premises in this way, we establish a clear understanding of the scenario described in the exercise, which is essential for making accurate logical deductions.

Logical Conclusions

Drawing logical conclusions is the process of using given information to arrive at a new statement that must be true. In this exercise, once we assume salaries have kept up with prices, we can start identifying what else must logically follow from this assumption. This involves:

• Eliminating impossibilities: Since salaries have kept pace, the idea that prices have risen beyond what people can afford must be false, based on the premise linking prices to salaries.
• Subsequent deductions: With prices not risen beyond affordability, if there were a decline in retail, it must hinge on changing attitudes, as all other parts of the premise are accounted for.
• Consistent reasoning: The logical consistency helps us conclude that since prices remain affordable, retail will not see a revenue decrease due to pricing issues.

Logical conclusions help us use premises effectively, leading to the final right answer, which in this case happens to be that stores won't suffer revenue drops based on prices.

Conditional Statements

Conditional statements are at the heart of logical reasoning and often present themselves in 'if-then' formats. In this exercise:

• Understanding implications: The main conditional statement given is "If retail stores experience a decrease, then attitudes or prices must have changed." This helps us set boundaries for what outcomes are possible based on initial conditions.
• Interpreting conditions: When we learn that salaries have kept pace, any consequence tied to a price rise (like a decline in retail for that reason) can't be true, given the problem's structure.
• Breaking down 'if-then': With such statements, always ask what 'then' implies when 'if' conditions are altered. Here, if prices haven't risen alarmingly, then attitudes must negatively affect retail (if there's any effect).

Understanding conditional statements allows us to see the chain of events and dependencies, offering clarity on how one part of a problem is reliant on another, ultimately guiding us to correct conclusions consistent with provided premises.