Semantic Consequence Union Negation

Theorem
Let $U$ be a set of logical formulas.

Let $P$ be a logical formula.

Let $U \models P$ denote that $U$ is a logical consequence $P$.

Then:
 * $U \models P$

iff:
 * $U \cup \left\{{\neg P}\right\}$ has no models.

Sufficient Condition
Suppose that $U \models P$.

Let $\mathcal M$ be a model such that:
 * $\mathcal M \models U \cup \left\{{\neg P}\right\}$

Then from Logical Consequence with Union, we have that:
 * $\mathcal M \models U$

and by definition of logical consequence:
 * $\mathcal M \models P$

So we have that $\mathcal M \models P$ and $\mathcal M \models \neg P$

By definition of contradiction, it follows that there can be no such model.

So $U \models P$ is a sufficient condition for $U \cup \left\{{\neg P}\right\}$ to have no models.

Necessary Condition
Suppose that $U \cup \left\{{\neg P}\right\}$ has no models.

There are the following possibilities:


 * $\neg P$ has no models, in which case it is itself a contradiction.

Hence from Tautology and Contradiction, $P$ is a tautology.

Hence $P$ is true under every model.

Hence whatever $U$ may be, every model of $U$ is also a model of $P$:
 * $U \models P$


 * $U$ has no models, in which case every model of $U$ is also a model of $P$ vacuously, and so:
 * $U \models P$


 * There exists at least one model of $U$, but each one does not model $\neg P$.

Let $\mathcal M$ be such a model.

We have that:
 * $\mathcal M \not \models \neg P$

so by definition of negation:
 * $\mathcal M \models P$

Thus $\mathcal M \models U$ implies that $\mathcal M \models P$.

Hence by definitinon:
 * $U \models P$

In all cases $U \models P$.

So $U \models P$ is a necessary condition for $U \cup \left\{{\neg P}\right\}$ to have no models.