While reading through Artificial Intelligence: A Modern Approach by Stuart Russell and Peter Norvig, I came upon the following question:
Any propositional logic sentence is logically equivalent to the
assertion that each possible world in which it would be false is not
the case. From this observation, prove that any sentence can be
written in CNF.
Does anyone have a reference or a proof that Every sentence in propositional logic can also be written in Conjunctive Normal Form?
My (lousy) attempt was that since CNF requires $\forall x\ \mathit{Clauses}(x)$, and propositional logic implies that all sentences it declares must be true for the statement to hold true, then one implies you can convert it?
Best Answer
Wikipedia's article on conjunctive normal form gives an idea of why a sentence is logically equivalent to its conjunctive normal form:
I think that what Russell and Norvig are getting at (the question is sort of vague, in my opinion) is that if you have a sentence like
$$ ((A \lor B) \to C) \land \lnot(D \land E) \tag{1}$$
you could then ask "in what cases would this sentence be false?" In this case, it's when
From those you can read off a disjunctive normal form:
$$ (A \land \lnot C) \lor (B \land \lnot C) \lor (D \land E) \tag{2} $$
Those are all the cases where $(1)$ is false. If we negate $(2)$, it is like the "assertion that each possible world in which it would be false is not the case," and we get
$$ \lnot( (A \land \lnot C) \lor (B \land \lnot C) \lor (D \land E) ) \tag{3} $$
which by De Morgan's laws is
$$ \lnot(A \land \lnot C) \land \lnot(B \land \lnot C) \land \lnot(D \land E) \tag{4} $$
and then
$$ (\lnot A \lor C) \land (\lnot B \lor C) \land (\lnot D \lor \lnot E) \tag{6} $$
which is conjunctive normal form for $(1)$. So, by enumerating all the ways that a sentence could be false, turning that into a single sentence, negating it, and applying De Morgan's laws, we have a logically equivalent sentence in conjunctive normal form.