Definition:Semantic Consequence

Logical Formula
Let $$P$$ and $$Q$$ be logical formulas.

Then:
 * $$P$$ is a logical consequence of $$Q$$, or $$P$$ is logically implied by $$Q$$

iff:
 * every model of $$Q$$ is a model of $$P$$

or alternatively, iff
 * $$P$$ is true in every model for $$Q$$.

We write:
 * $$Q \models P$$

and we can say:
 * $$P$$ follows from $$Q$$

Set of Logical Formulas
Let $$U$$ be a set of logical formulas.

Let $$P$$ be a logical formula.

Then:
 * $$P$$ is a logical consequence of $$U$$, or $$P$$ is logically implied by $$U$$

iff:
 * every model of $$U$$ is a model of $$P$$

or alternatively, iff
 * $$P$$ is true in every model for $$U$$.

We write:
 * $$U \models P$$

and we can say:
 * $$P$$ follows from $$U$$

Alternative terms
An alternative term to logical consequence is semantic consequence or semantic entailment.

Thus, $$U \models P$$ means $$U$$ semantically entails $$P$$.