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

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 $$\phi \equiv \neg \psi$$?

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.