# Negation normal form with cancellations barred

Language: consider propositional logic over the connectives $$\land$$, $$\lor$$, and $$\lnot$$.

1. Consider a propositional formula $$\phi$$ which contains at least one negation symbol and is neither a contradiction nor a tautology. We are also given that no negation symbols can be cancelled in $$\phi$$ by any valid biconditional such as $$\alpha \land (\lnot \alpha \lor \beta) \iff \alpha \land \beta$$ $$\lnot \lnot \alpha \iff \alpha$$

2. Now let's push negation inward to reduce $$\phi$$ to negation normal form and call that normal form $$\phi'$$. Just to be clear, $$\phi \iff \phi'$$.

3. Can we conclude that $$\phi'$$ also contains at least one negation symbol? If not, please provide a counterexample.

EDIT: The allowable biconditionals were clarified to include any valid biconditional that would cancel a negation. The intention is to capture the notion that $$\phi$$ contains no cancellable negations.

I mean, $$A \land ( B \lor \neg B)$$ is equivalent to $$A$$, so it is neither a tautology nor a contradiction ... but clearly one has to use an equivalence that 'cancels a negation'. So, does this equivalence fall under 'the like'? When, indeed, would some equivalence not fall under 'the like'? What, in sum, is one allowed to use and what not?
• @ShyPerson If $\phi$ 'contains no cancellable negations'.... then the negations cannot be cancelled ... Sep 26 '19 at 1:06