Logical Consequence

This is all a bit basic, but worth saying because it interacts with foundational questions (for example, "what are the relata of logical consequence?" or "what is more basic, semantics or inference?"). Suppose that $A=\{a_1,a_2,a_3\}$ is an alphabet. Suppose that ${\sigma }_1$ and ${\sigma }_2$ are non-empty strings from $A$. That is, they are finite sequences of the form

${\sigma }_i:[0,n]\to A$.

Consider a claim of the form:

${\sigma }_2$ is a logical consequence of ${\sigma }_1$.

I think this claim is seriously underspecified, because logical consequence can only make sense given a class of interpretations of the strings.

Suppose that $A$ above is the alphabet of an interpreted language $L$. Suppose that this also determines a special class $Sent(L)$ of strings, and a class $\mathrm{\Sigma }(L)$ of interpretations for $L$ in such that way that, for any $I\in \mathrm{\Sigma }(L)$, and any $\varphi \in Sent(L)$,

$I\models \varphi$

is defined, meaning,

the $L$-string $\varphi$ is true in $I$.

Then consider the claim,

${\sigma }_2$ is a logical consequence in $L$ of ${\sigma }_1$.

On the Bolzano-Tarski definition, this means,

${\sigma }_1,{\sigma }_2\in Sent(L)$ and, for any $L$-interpretation $I\in \mathrm{\Sigma }(L)$,

if $I\models {\sigma }_1$ then $I\models {\sigma }_2$.