Thinking about the seemingly isomorphic nature of theories (I take a theory here as a deductively closed set of sentences) and filters has lead me to ask myself the following question:
Can the deductive closure of a consistent set of axioms be characterized (up to formulae equivalence) as the smallest set of sentences $T$ such that if $p$,$q\in T$ then $p\wedge q\in T$ and if $p\in T$ and $r$ is a sentence then $p\vee r \in T$?
I have only studied propositional and predicate first order logic and hence would like to place the question is both these settings.
...Yes! At least if the logic is compact, as with propositional or first-order logic.
Suppose that $\phi$ is a consequence of the set $S$ of axioms. By compactness, $\phi$ is a consequence of some conjunction $\psi$ of elements of $S$. Let $T$ be the closure of $S$ under conjunction and disjunction (as you specify). Then $\psi\in T$, and so $\phi\lor \psi\in T$. But $\phi\lor\psi$ is equivalent to $\phi$.