Logical Consequence with Union

Theorem

Let $U$ be a set of propositional formulas.

Let $P$ be a propositional formula.

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

Then:

$U \models P$

iff:

$U \cup P \models P$

Proof

First we note that, from the Law of Identity:

$P \models P$

Suppose $U \models P$.

So we have:

• $U \models P$
• $P \models P$

So by definition of model:

$U \cup P \models P$

It also follows that if $U \cup P \models P$, then:

• $U \models P$
• $P \models P$

Thus we have shown that $U \models P$ iff $U \cup P \models P$.

$\blacksquare$