Understanding Logical Entailment in Logic

The question defines logical entailment: Sentence $X$ entails Sentence $Y$ if the truth of $X$ guarantees the truth of $Y$. This means whenever $X$ is TRUE, $Y$ must also be TRUE.

Analyzing the Options

We need to determine which statements accurately represent the condition '$X$ entails $Y$'.

• Option A: $X \Rightarrow Y$

  This is the standard logical notation for implication, meaning 'If $X$, then $Y$'. This perfectly matches the definition of entailment. Therefore, this statement is correct.

• Option B: $X \land \neg Y$ is FALSE

  This statement means that the situation where $X$ is TRUE and $Y$ is FALSE cannot occur ($X \land \neg Y$ is FALSE). If $X$ entails $Y$, then whenever $X$ is TRUE, $Y$ must also be TRUE. This makes the condition '$X$ is TRUE and $Y$ is FALSE' impossible. Thus, this statement is correct.

• Option C: if $X$ then $Y$

  This is the natural language representation of implication. It directly states that if $X$ holds true, then $Y$ must also hold true, which is the definition of entailment. Therefore, this statement is correct.

• Option D: if $Y$ then $X$

  This statement represents the converse implication ($Y \Rightarrow X$). Entailment from $X$ to $Y$ does not guarantee entailment from $Y$ to $X$. For example, 'It is raining' ($X$) entails 'The ground is wet' ($Y$), but 'The ground is wet' ($Y$) does not necessarily mean 'It is raining' ($X$). Therefore, this statement is not always correct.

Conclusion

Based on the definition of entailment, statements A, B, and C accurately describe the relationship. Statement D represents the converse and is not necessarily true.