The statement $$(p \to (q \to p)) \to (p \to (p \vee q))$$ is:

We have to decide the logical nature of the formula $$S=(p\to(q\to p))\to(p\to(p\vee q)).$$

First, we recall the standard implication-disjunction equivalence:

$$a\to b\;\equiv\;\sim a\;\vee\;b.$$

We apply this rule to the innermost implication $$q\to p.$$

So, $$q\to p\;\equiv\;\sim q\;\vee\;p.$$

Substituting this result back into the left part of $$S$$ we get

$$p\to(q\to p)\;\equiv\;p\to(\sim q\vee p).$$

Once again using the same rule $$a\to b\equiv\sim a\vee b$$ on the implication whose antecedent is $$p$$, we have

$$p\to(\sim q\vee p)\;\equiv\;\sim p\;\vee\;(\sim q\;\vee\;p).$$

Now we re-arrange the disjunction because disjunction is associative and commutative:

$$\sim p\;\vee\;(\sim q\;\vee\;p)\;=\;(\sim p\;\vee\;p)\;\vee\;\sim q.$$

We know that the law of excluded middle tells us $$\sim p\;\vee\;p\equiv \text{True}.$$ Therefore,

$$(\sim p\;\vee\;p)\;\vee\;\sim q\;\equiv\;\text{True}\;\vee\;\sim q\;\equiv\;\text{True}.$$

Hence, the whole sub-formula $$p\to(q\to p)$$ is always true; it is a tautology. We can now write

$$p\to(q\to p)\;\equiv\;\text{True}.$$

Next, we turn to the right part of $$S$$, namely $$p\to(p\vee q).$$ Again we use $$a\to b\equiv\sim a\vee b.$$ So,

$$p\to(p\vee q)\;\equiv\;\sim p\;\vee\;(p\vee q).$$

Associating the disjunctions yields

$$\sim p\;\vee\;p\;\vee\;q.$$

By the law of excluded middle, $$\sim p\;\vee\;p\equiv\text{True},$$ hence

$$\sim p\;\vee\;p\;\vee\;q\;\equiv\;\text{True}\;\vee\;q\;\equiv\;\text{True}.$$

Therefore, the consequent $$(p\to(p\vee q))$$ is also a tautology. We can thus rewrite $$S$$ purely in terms of truth values:

$$S\;\equiv\;(\text{True})\;\to\;(\text{True}).$$

Finally, we know that $$\text{True}\to\text{True}\equiv\text{True},$$ because an implication with a true antecedent and a true consequent is true.

So the entire statement $$S$$ is always true for every possible truth-value assignment of $$p$$ and $$q$$. That is, $$S$$ is a tautology.

Hence, the correct answer is Option 4.