How to express other logical operations via Pierce's arrow?

Question

x↑y, x⇒y, and x⇔y. So I have really given my best, but all I could do is express the conjunction, disjunction, negation, and impilcation.

Answer 1

A complement to the comments above.

It suffices to note that:

$\neg$:

$\ $ $\ $ $\ $ $\ $$\neg \varphi \Leftrightarrow \varphi \uparrow \varphi$

$\wedge$:

$\ $ $\ $ $\ $ $\ $$\varphi \wedge \psi \Leftrightarrow (\varphi \uparrow \varphi) \uparrow (\psi \uparrow \psi)$

Now how to proceed to the others connectives? Since we already have $\neg$ and $\wedge$ we can proceed by translating them in some of the well-knwon standard equivalences:

$\vee$:

$\ $ $\ $ $\ $ $\ $$\varphi \vee \psi \Leftrightarrow \neg(\neg \varphi \wedge \neg \psi) \Leftrightarrow$ (and so on)

Answer 2

Write the other operations in terms of implication, conjunction, and disjunction, and then replace the implications, conjunctions, and disjunctions with formulas you already have that involve only the arrow operator.