Definition:Logical 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 $P$.

Sources

• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.7$