Question

Given: If X, then Y. If Y, then Z. Z is false. So:

Answer 1

The question is a logical deduction problem in mathematics. By applying modus tollens, if 'Z is false,' it necessarily follows that 'Y is false,' and subsequently, 'X must also be false' due to the conditional relationships given.

Step-by-step explanation:

The question deals with a logical argument and is best categorized under Mathematics, specifically within the realm of logical reasoning or deductive logic used commonly in high school curriculum. The original statement involves a series of conditionals: 'If X, then Y' and 'If Y, then Z.' These are followed by the declaration that 'Z is false.' Using deductive logic, particularly modus tollens, we can analyze the validity of inferences drawn from these statements.

Modus tollens is a valid form of deductive argument which can be summarized as:

1. If P, then Q.
2. Q is false.
3. Therefore, P is also false.

Applying this structure to the question at hand:

1. If X, then Y (given).
2. If Y, then Z (given).
3. Z is false (given).
4. Therefore, Y must also be false. This follows because Y is a necessary condition for Z.
5. Since Y is false and Y is necessary for X, we can deduce that X must also be false.

This logical progression showcases how the conclusion that X is false follows necessarily from the premises provided. This argument structure ensures that the conclusion is valid as long as the premises are true. Hence, when Z is false, it guarantees that both Y and X are also false, because the truth of Y and X are dependent on the truth of Z, as per the given conditionals.