The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2} \wedge \dots \wedge S_{n}\rightarrow S$ is satisfiable.

The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2}\wedge \dots \wedge S_{n}\rightarrow S$ is valid.

The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2}\wedge \dots \wedge S_{n}\neg S$ is consistent.

The sentence $S$ is a logical consequence of $S_{1},\dots,S_{n}$ if and only if $S_{1}\wedge S_{2}\wedge \dots \wedge S_{n}\wedge S$ is inconsistent.

$S_1, S_2,\ldots ,S_n$ can be considered as premises and S is conclusion. Argument $premises\implies conclusion$ is said to be valid(TRUR) if $S_1,S_2,S_3,\ldots,S_n \rightarrow S$ is valid. (False $\rightarrow $ or True $\rightarrow $ True.)