Does every 3CNF propositional formula has an equivalent 2CNF propositional formula?Definitions:3CNF propositional formula is conjunctive normal form propositional formula, which is just conjunction $\land$ of clauses, where each clause is disjunction $\lor$ of at most 3 literals, where each literal is just boolean variable either with or without not $\lnot$ operator/connective.2CNF propositional formula is conjunctive normal form propositional formula, which is just conjunction $\land$ of clauses, where each clause is disjunction $\lor$ of at most 2 literals, where each literal is just boolean variable either with or without not $\lnot$ operator/connective.
Does every 3CNF propositional formula has an equivalent 2CNF propositional formula?
logic;first order logic;propositional logic
The single clause 3CNF:$\varphi = a\lor b\lor c$does not have an equivalent 2CNF. Suppose $\psi=\bigwedge\limits_{i=1}^m \psi_i$ is an equivalent 2CNF. Let $\psi_i$ be one of its clauses, and assume it involves the variables $a,b$. An assignment which sets $c=1$, but does not satisfy $\psi_i$, satisfies $\varphi$ but not $\psi$, hence they are not equivalent. 