# Disjunctive Normal Form for Fuzzy Logic

I'm asked to prove that every propositional assertion in Fuzzy Logic, expressed using the standard propositional connectives $\{\land, \lor,\lnot, \rightarrow, \leftrightarrow\}$ can be expressed in Disjunctive Normal Form (DNF).

I want to prove by induction on formulas, using 'if $\phi$ and $\psi$ are propositional variables, then they are already in DNF' as a base case, and showing that applying either $\lnot$ or $\lor$ preserves DNF, taking $\{\lnot, \lor \}$ to be a complete set of connectives in this case. (Alternatively, I could take the set of $\{\lnot, \rightarrow\}$, or $\{\lnot, \land\}$)

First, eliminate $\leftrightarrow$ by $\phi \leftrightarrow \psi \equiv (\phi \rightarrow \psi) \land (\psi \rightarrow \phi)$, eliminate $\rightarrow$ by $(\phi\rightarrow\psi)\equiv \lnot\phi \lor \psi$, and eliminate $\land$ by $\phi\land\psi \equiv\lnot(\lnot\phi\lor\lnot\psi)$. We then have $\lnot$ and $\lor$.

This seems to be the right track, since these are the connectives used in DNF, but where do I go from here?

NOTE: An earlier version of this question simply asked how to eliminate $\land$.

$\phi \land \psi \equiv \lnot (\lnot \phi \lor \lnot \psi) \equiv \lnot (\phi \implies \lnot \psi)$