# How to write a formula, $S$, so that the set $P_1, \cdots, P_k$ is not consistent iff $S$ is valid?

A set of propositional formulas $$P_1, \cdots, P_k$$ is consistent iff there is an environment in which they are all true.

Write a formula, $$S$$, so that the set $$P_1, \cdots, P_k$$ is not consistent iff $$S$$ is valid.

Sorry for asking this homework question. I really don't know how to solve it.

Let $$S := \lnot (P_1 \land \ldots \land P_k)$$.

If $$\{ P_1, \ldots, P_k \}$$ is inconsistent (i.e. unsatisfiable), then $$(P_1 \land \ldots \land P_k)$$ is always false, and thus its negation is a tautology (i.e. valid).

If $$S$$ is valid, then $$(P_1 \land \ldots \land P_k)$$ is always false (a contradiction) and thus there is no truth assignment that can simultaneously satisfy all the $$P_i$$'s, i.e. $$\{ P_1, \ldots, P_k \}$$ is unsatisfiable (i.e. inconsistent).

a set $$F$$ of formulas is consistent iff there is a truth-assignment such that all the formulas of $$F$$ are true.
• The proof is pretty good and clear. But I don't understand what the '(i.e. unsatisfiable)' means? $\{P_1, \cdots, P_k\}$ is a set. A set does not poess the property of satisfiability which is the quality of a proposition formula. – 王文军 or Wenjun Wang Mar 14 at 6:41