To solve this problem, we need to understand the logical structure described by the statement: "If \( x \Rightarrow y \) and \( y \Rightarrow z \), then \( x \Rightarrow z \)".
Let's break it down:
1. Implication:
- \( x \Rightarrow y \) means "if \( x \) then \( y \)"
- \( y \Rightarrow z \) means "if \( y \) then \( z \)"
2. Transitivity:
- If \( x \) implies \( y \), and \( y \) implies \( z \), then it logically follows that \( x \) implies \( z \).
The structure of this logic is a classic example of a syllogism. A syllogism is a form of reasoning in which a conclusion is drawn from two given or assumed propositions (premises). Each premise shares a common term with the conclusion, which in this case are the logical implications linking \( x \), \( y \), and \( z \).
To clarify why the other options are incorrect:
- Inverse statement: This refers to negating both the hypothesis and the conclusion of an implication (i.e., if \( x \Rightarrow y \), the inverse is \( \neg x \Rightarrow \neg y \)).
- Converse statement: This swaps the hypothesis and conclusion of an implication (i.e., if \( x \Rightarrow y \), the converse is \( y \Rightarrow x \)).
- Contrapositive statement: This refers to negating and swapping the hypothesis and conclusion (i.e., if \( x \Rightarrow y \), the contrapositive is \( \neg y \Rightarrow \neg x \)).
Since none of these terms correctly describe the given logical structure, the best term is indeed "syllogism".
Hence, the correct answer is:
C. A syllogism
