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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.

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...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 $\phi$.

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