The problem asks to find both the Principal Disjunctive Normal Form (PDNF) and the Principal Conjunctive Normal Form (PCNF) of the logical expression:

$p \lor (\neg p \rightarrow (q \lor (\neg q \rightarrow r))).$

Step 1: Simplifying the expression

Start with the inner parts of the expression and use logical equivalences to simplify it.

1. Implication equivalence: Recall that $A \rightarrow B$ is logically equivalent to $\neg A \lor B$.

   • $\neg p \rightarrow (q \lor (\neg q \rightarrow r))$ becomes: $\neg (\neg p) \lor (q \lor (\neg q \rightarrow r)) \quad \text{(using implication equivalence)}$ which simplifies to: $p \lor (q \lor (\neg q \rightarrow r)).$

2. Handling the next implication: Again, use the equivalence for implication on $\neg q \rightarrow r$, which becomes: $\neg (\neg q) \lor r = q \lor r.$

   So, the expression now becomes: $p \lor (q \lor (q \lor r)).$ This simplifies further to: $p \lor q \lor r.$

Step 2: PDNF and PCNF

Now that we have the simplified expression $p \lor q \lor r$, we can easily write its PDNF and PCNF.

• Principal Disjunctive Normal Form (PDNF): The PDNF is simply the expression in its current form because it is already in disjunctive normal form: $p \lor q \lor r.$

• Principal Conjunctive Normal Form (PCNF): The PCNF would involve expressing the same logic in terms of conjunctions of disjunctions. Using De Morgan's law, the negation of the expression would be: $\neg(p \lor q \lor r) = \neg p \land \neg q \land \neg r.$ By negating again to restore the original expression, we get the PCNF: $(p \lor q \lor r).$

Conclusion

Both the PDNF and PCNF of the expression are the same: $p \lor q \lor r.$

Would you like further details on any of the steps?

Here are 5 additional questions for further exploration:

1. What are other methods to convert a logical expression into normal forms?
2. How do you derive a truth table for this expression?
3. Can you generalize the transformation of implications in logical statements?
4. How would the normal forms change for more complex nested logical expressions?
5. What role do De Morgan’s laws play in converting between PDNF and PCNF?

Tip: To simplify complex logical expressions, always tackle implications and negations first, using known equivalences.