Logical consequence

In a valid argument, we can informally say that conclusion $B$ “follows” from a set of premises $A_1,...,A_n$.

Formally, we say that formula $B$ is a logical consequence of the set of formulae $\{A_1,...,A_n\}$, if the following implication holds for every interpretation $v$: $$ \text{If } v(A_i) = 1, \text{ for all } 1 \le i \le n, \text{ then } v(B) = 1. $$

Alternative terminologies include:

• $B$ is a logical consequence of $A_1,...,A_n$.
• $A_1,...,A_n\models B$.
• The argument $A_1,...,A_n\models B$ is valid.
• $B$ is semantically entailed (or implied) by $A_1,...,A_n$
• $B$ is a valid consequence of $A_1,...,A_n$