To evaluate the logical expression ~(p → q) → p, let's first break it down step by step:

Step 1: Understand the Components

1. p → q means "if p, then q" (the implication).

   • This can be rewritten as ~p ∨ q using the implication equivalence.

2. ~(p → q) means the negation of p → q.

   • So, ~(p → q) becomes ~(~p ∨ q) (negating the implication).

3. Apply De Morgan's law to ~(~p ∨ q):

   • By De Morgan's law, ~(~p ∨ q) is equivalent to p ∧ ~q.

Step 2: Rewrite the Expression

• Now, the expression becomes (p ∧ ~q) → p.

Step 3: Evaluate the Implication

• The implication (p ∧ ~q) → p is always true whenever p is true.
• If p is false, the implication is still true, since an implication with a false antecedent is always true.

Conclusion:

• The expression ~(p → q) → p is a tautology, meaning it is always true regardless of the truth values of p and q.

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the truth table for ~(p → q) → p?
2. How does De Morgan's law apply to logical negations?
3. What are the equivalence rules for implications in logic?
4. How does an implication behave when the antecedent is false?
5. Can you explain how tautologies work in formal logic?

Tip: In logic, negating implications can be simplified using basic equivalence rules like p → q being equivalent to ~p ∨ q.