To simplify the expression $\left((P \Rightarrow Q) \Rightarrow P\right) \Rightarrow Q$, we can break it down using logical equivalences step by step.

1. Rewrite implications in terms of disjunctions: The implication $A \Rightarrow B$ is logically equivalent to $\neg A \vee B$. So we can rewrite the expression as:

   $\left((\neg P \vee Q) \Rightarrow P\right) \Rightarrow Q$

2. Apply the implication equivalence to $(\neg P \vee Q) \Rightarrow P$: Now rewrite the inner implication $(\neg P \vee Q) \Rightarrow P$:

   $\neg(\neg P \vee Q) \vee P$

   By De Morgan's Law, $\neg(\neg P \vee Q)$ is equivalent to $\neg \neg P \wedge \neg Q$, which simplifies to $P \wedge \neg Q$. So, the expression becomes:

   $(P \wedge \neg Q) \vee P$

3. Simplify $(P \wedge \neg Q) \vee P$: Using the distributive property, $(P \wedge \neg Q) \vee P$ simplifies to just $P$, because $P \vee (P \wedge \neg Q) = P$ (since $P$ is already included in both terms).

   So, the expression simplifies to:

   $P$

4. Substitute back into the original expression: Now we substitute $P$ into the original outer implication:

   $P \Rightarrow Q$

5. Final simplification: The expression $P \Rightarrow Q$ is already in its simplest form, so the final simplified form of the entire expression is:

   $P \Rightarrow Q$

Final Answer:

$(P \Rightarrow Q) \Rightarrow P \Rightarrow Q \text{ simplifies to } P \Rightarrow Q.$

Would you like more details or have any questions? Here are some related questions:

1. What are some other common logical equivalences used in simplification?
2. How can De Morgan's Laws help in simplifying logical expressions?
3. Can the expression $P \Rightarrow Q$ be expanded using different methods?
4. What role does distributive property play in simplifying logical expressions?
5. How does simplification impact the truth table of a logical statement?

Tip: When simplifying logical expressions, converting implications to disjunctions and using De Morgan's laws can greatly simplify your work.