The logical expression \[(\sim p \Rightarrow q) \Rightarrow (\sim p \Rightarrow q)\] can be analyzed step by step to check for its validity.

### Step-by-step breakdown:
1. The expression \(\sim p \Rightarrow q\) means "if not \(p\), then \(q\)." 
   - This is logically equivalent to \(p \lor q\), since \(\sim p \Rightarrow q\) is true unless \(\sim p\) is true and \(q\) is false.

2. The entire expression is \((\sim p \Rightarrow q) \Rightarrow (\sim p \Rightarrow q)\), meaning "if \(\sim p \Rightarrow q\) holds, then \(\sim p \Rightarrow q\) also holds." 
   - Since the statement on both sides of the implication is identical, this is trivially true. It is a tautology because any statement implies itself.

### Conclusion:
The expression is always true, as it represents a tautology.

Would you like to go into further detail or have any questions?

### Related Questions:
1. What is a tautology in logic?
2. How can we simplify logical implications using truth tables?
3. What is the relationship between implications and disjunctions in logic?
4. How can we represent logical statements using truth functions?
5. How do logical equivalences help in simplifying complex statements?

### Tip:
When working with logical implications, converting them to their disjunctive forms (using \(\sim p \lor q\)) can often make simplifications easier!

Explore the tautology (~p ⟹ q) ⟹ (~p ⟹ q) in propositional logic, understanding the meaning of implication and how self-referential statements lead to tautologies. This solution breaks down key concepts such as implication and tautology with relevant formulas and theorems. Suitable for advanced high school students and undergraduates.

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