Logical Consequence with Union

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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 $U \cup P \models P$.

$\blacksquare$