Understanding the Logic Puzzle

The question presents a statement based on a hypothetical definition and asks whether a conclusion logically follows. Let's break down the statement:

"If cream cracker means computer programming, and it’s not programming, then it’s not cream cracker."

We can analyze this using basic logic. Let's represent the parts of the statement:

• Let P be the statement: "It is cream cracker".
• Let Q be the statement: "It is computer programming".

The first part, "cream cracker means computer programming", can be interpreted in the context of this logical puzzle as setting up a conditional relationship: "If it is cream cracker, then it is computer programming". In logical notation, this is represented as \(P \rightarrow Q\).

The second part of the premise is "and it’s not programming". This is the negation of Q, represented as \(\neg Q\).

The conclusion drawn is "then it’s not cream cracker". This is the negation of P, represented as \(\neg P\).

Analyzing the Logical Structure: Modus Tollens

The statement has the following structure:

If (\(P \rightarrow Q\)) AND (\(\neg Q\)), THEN (\(\neg P\)).

This particular structure is a fundamental rule of inference in logic known as Modus Tollens (or " طريقة الرفع " in Arabic, though not required for the solution context). Modus Tollens states that if a conditional statement (\(P \rightarrow Q\)) is true, and its consequent (\(Q\)) is false, then its antecedent (\(P\)) must also be false.

The structure is:

1. Premise 1: If P, then Q (\(P \rightarrow Q\))
2. Premise 2: Not Q (\(\neg Q\))
3. Conclusion: Therefore, Not P (\(\neg P\))

Modus Tollens is a valid form of argument. This means that if the premises are true, the conclusion must necessarily be true.

Applying Modus Tollens to the Statement

In our case:

• Premise 1: "If it is cream cracker, then it is computer programming" (\(P \rightarrow Q\)) - This is taken as a given condition in the problem statement.
• Premise 2: "It’s not programming" (\(\neg Q\)) - This is explicitly stated in the problem.
• Conclusion: "It’s not cream cracker" (\(\neg P\)) - This is the conclusion drawn in the problem.

Since the structure matches Modus Tollens, and Modus Tollens is a valid argument form, the conclusion (\(\neg P\)) logically follows from the premises (\(P \rightarrow Q\) and \(\neg Q\)).

Therefore, the entire statement "If cream cracker means computer programming, and it’s not programming, then it’s not cream cracker" is logically true because its structure represents a valid logical deduction (Modus Tollens).

Conclusion on the Statement's Truth Value

The statement is a valid deduction based on the initial premise provided. If we accept the premise "cream cracker means computer programming" as establishing the conditional "If cream cracker, then computer programming", then the subsequent inference is logically sound.

Thus, the statement is True.