Question

Only if the electorate is moral and intelligent will a democracy function well. Which one of the following can be logically inferred from the claim above? (A) If the electorate is moral and intelligent, then a democracy will function well. (B) Either a democracy does not function well or else the electorate is not moral or not intelligent. (C) If the electorate is not moral or not intelligent, then a democracy will not function well. (D) If a democracy does not function well, then the electorate is not moral or not intelligent. (E) It cannot, at the same time, be true that the electorate is moral and intelligent and that a democracy will not function well.

Short answer

Options C and E can be logically inferred.

Step-by-step solution

Identify the Logical Form

The statement 'Only if the electorate is moral and intelligent will a democracy function well' can be expressed logically. Here, if a democracy functions well, it implies that the electorate is both moral and intelligent.

Translate to If-Then Statement

The expression 'only if' translates to a conditional statement where the given consequence is necessary. Thus, 'if a democracy functions well, then the electorate is moral and intelligent' can be written as: If D (democracy functions well), then M (electorate is moral) and I (electorate is intelligent): \[ D \rightarrow (M \land I) \]

Find Contrapositive

The contrapositive of a conditional statement is logically equivalent to the original statement. The contrapositive of 'If D, then M and I' is If not (M and I), then not D: \[ eg (M \land I) \rightarrow eg D \] This states: If the electorate is not moral or not intelligent, then a democracy will not function well.

Match Logical Statements with Options

Evaluate options A through E: - **A**: 'If the electorate is moral and intelligent, then a democracy will function well.' This is not directly inferred but often considered in context - **B**: 'Either a democracy does not function well or else the electorate is not moral or not intelligent.' This matches the contrapositive - **C**: 'If the electorate is not moral or not intelligent, then a democracy will not function well.' Directly matches the contrapositive - **D**: 'If a democracy does not function well, then the electorate is not moral or not intelligent.' Is not the contrapositive nor the original - **E**: 'It's impossible for the electorate to be moral and intelligent and for democracy to not function well.' This can be deduced as it is a strong logical implication forward from the original

Select Valid Inferences

From our analysis: - **C**: It is directly inferred from the contrapositive - **E**: Is a forward logical inference fitting the original claim C and E are logical inferences.

Key concepts

Logical Inference

Logical inference is a fundamental concept used in logical reasoning, including LSAT exams. It involves deriving conclusions from given premises or statements. Inferences are drawn by analyzing the structure and content of the premises and identifying what must be true if the premises are true.

In the context of logical reasoning problems, like the one we're examining, the goal is to discern what conclusions logically follow from a statement. Consider the example statement: "Only if the electorate is moral and intelligent will a democracy function well." We start by breaking down the statement into logical parts to see what must be the case.

To identify a logical inference, we ask:

• What conditions make the statement true?
• What are the necessary outcomes or truths based on the initial statement?

In the exercise, options C and E represent valid logical inferences. The correct inference represents a direct consequence of the contrapositive—a widely used method in logical inference. Recognizing such logical translations and their implications is crucial for successfully answering these types of questions.

Conditional Statements

Conditional statements are expressions that establish a condition or premise, and a result or consequence. They are fundamental in logical reasoning because they help us understand the relationship between different statements or events.

A typical conditional statement takes the form "If P, then Q." It suggests that Q happens when P is true. In the exercise, the original statement, "Only if the electorate is moral and intelligent will a democracy function well," can be rephrased into a conditional statement: "If a democracy functions well, then the electorate is moral and intelligent."

This can also be expressed using standard logical notation:

• Let D be "a democracy functions well."
• Let M be "the electorate is moral," and I be "the electorate is intelligent."

Thus, the statement becomes: \[ D \rightarrow (M \land I) \]
Recognizing conditional relationships helps analyze the logical flow of information. Being able to convert between regular speech and conditional statements is an essential skill in logical reasoning, allowing us to evaluate and make inferences correctly.

Contrapositive

The contrapositive is an important logical tool used to derive equivalent statements. Understanding a contrapositive can clarify logical implications and help identify valid conclusions from conditional statements.

The contrapositive of a given conditional statement "If P, then Q" is "If not Q, then not P." Interestingly, a statement and its contrapositive are logically equivalent; if one is true, so is the other. In logical reasoning, forming the contrapositive can help identify conclusions that aren't immediately obvious.

In our specific example, the contrapositive of the original statement \( D \rightarrow (M \land I) \) would be: \[ eg(M \land I) \rightarrow eg D \]
This says, "If the electorate is not moral or not intelligent, then a democracy will not function well." Such a transformation reveals insights not explicitly stated but inherent in the original premise.

Identifying contrapositives can be crucial when evaluating which answers are logically justified quantitatively. In the exercise, option C directly follows this contrapositive transformation. Mastery of contrapositives and the understanding that they offer is vital for anyone aiming to excel in logical reasoning tasks.