$2 \uparrow^n 2 = 4$ and the magnificence of $2$

Question

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because of the binary nature of binary operations. I require a formal way to prove that $2 \uparrow^n 2 = 4$
Please, can you help?

Answer 1

Well from the definition $a\uparrow^n2=a\uparrow^{n-1}a$ so for $a=2$ you get $2\uparrow^n2=2\uparrow^{n-1}2=\ldots=2\uparrow2=4$.

Answer 2

Hint: Use induction.

If you have the luxury of defining your operation, you should define it recursively. If not, and your definition isn't recursive, then wow that's cool but you should probably tell us about it.