Proving negation is not expressible in language $\mathcal{L}(\rightarrow)$ of propositional logic.

Question

Consider language $\mathcal{L}(\rightarrow)$ of propositional logic, where $\rightarrow$ is interpreted as implication. Let $\psi \in \mathcal{L}$ (i.e. sentence made of just variables and implications), then how to prove there is no sentence $\phi$, such that $\phi \equiv \neg \psi$?

Answer 1

HINT

By induction you should be able to show that if you put any statement constructed using just variables and implications on a truth-table, you'll get more $T$'s than $F$'s, no matter how complex it is.