Propositional logic DNF when formula is unsatisfiable

Question

I understand that every formula in propositional logic can be represented in the form of a DNF, and such DNF can be constructed either by extracting true propositions from a truth table or using logical equivalencies, cancelling out propositions which are false. My question is then, if a formula was unsatisfiable and hence had no true propositions to extract from a truth table, what would it’s DNF be? Nothing? 0?

Thanks!

Answer 1

Notice that this method gives you a generalized disjunction, because the disjunction you come up with can have any number of disjuncts. And so, yes, you could treat the case where a statement is unsatisfiable (i.e. it is a contradiction) as a generalized disjunction with $0$ disjuncts ... which (as I will demonstrate below) is a contradiction.

Now, one problem is, as you ask: how do you write that?

One suggestion is: $\bigvee \emptyset$

Indeed, we can treat generalized disjunction as statements of the form $\bigvee S$, with $S$ being some set of statements. For example, $\bigvee \{ P,Q,R\land S \}$ would be $P \lor Q \lor (R\land S)$

And now notice:

As a general rule, we of course want that (for any sets $R$ and $S$):

$$\bigvee R \lor \bigvee S = \bigvee R \cup S$$

So, in particular we have that for any $R$:

$$\bigvee R \lor \bigvee \emptyset = \bigvee R \cup \emptyset = \bigvee R$$

Now let's say that $R = \{ \bot \}$

Then

$$\bigvee R = \bigvee \{ \bot \} = \bot$$

So, since $\bigvee R \lor \bigvee \emptyset = \bigvee R$

we get that

$$ \bot \lor \bigvee \emptyset = \bot$$

... which holds iff $\bigvee \emptyset = \bot$

And so yes, we indeed have that $\bigvee \emptyset = \bot$