What's the right/typical way to describe a group or a monoid acting on a logic or, equivalently, the automorphisms/endomorphisms of the language of a logic that preserve the consequence relation?
Suppose I have a logic $S$ consisting of $L$ (for language), a set of well-formed formulas, and $\vdash_S$, a consequence relation. For the purposes of this question, I'm not peering inside the logic, it's just a set of sequences of symbols and a consequence relation.
Let's consider the case of classical propositional calculus.
I'll use the term "symmetry" to refer to an intuitive, preformal notion of symmetry.
Logics seem to have a lot of symmetry, in particular the following map $g$ is bijective on $L$ and preserves the consequence relation $\vdash_S$.
$$ g(\varphi) = \varphi \; \text{with every subexpression $a \land b$ replaced with $b \land a$} $$
This map is interesting because $\varphi \vdash_S g(\varphi)$ is true for all $\varphi$.
Another example of a symmetry is $f$, defined below.
$$ f(\varphi) = \varphi \; \text{with all occurrences of the variables $P$ and $Q$ swapped} $$
In this case, $\varphi \vdash_S f(\varphi)$ is not always true since $P \vdash_S Q$ is false.
It seems fairly clear that we can collect all the automorphisms of $L$ (with $\vdash_S$ being the structure that they have to respect) into a group.
It also seems clear that that we can collect each endomorphism $e$ of $L$ satisfying the rule below into a monoid. Let $\Phi_I$ be a family of well-formed formulas indexed by $I$.
$$ \Phi_I \vdash_S \varphi \implies \{ e(\Phi_i) : i \in I \} \vdash_S e(\varphi) $$
There are two semi-interesting monoids I can think of, the one that sends everything to $\top$ and the one that sends everything to $\bot$.
So far, though, I haven't been able to find anything relevant for groups or monoids acting on a logic or for endomoprhisms or automorphisms of a logic, which makes me think I'm describing what sort of thing I'm after in a nonstandard/weird way.