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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 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.

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.