# Weakest theory such that adding a sentence gives another specified theory.

Let $$T$$ be a theory and let $$Mod(T)$$ denote the set of models of $$T$$. For a set of models $$M$$, let $$Th(M)$$ denote the theory which is the set of all sentences true in all the models in $$M$$. Let a theory $$T$$ be reffered to as closed if $$Th(Mod(T))=T$$ and let $$cl(T)=Th(Mod(T))$$ denote the closure of $$T$$.

Consider some specific closed theory $$T$$ and some sentence $$\phi$$ in $$T$$. Let $$S=\{S_i:i\in I\}$$ be the set of all closed theories, indexed by index set $$I$$, such that $$\forall i\in I, cl(S_i\cup\{\phi\})=T$$. Must it be the case that $$T=cl(W\cup \{\phi\})$$ where $$W=\cap_{i\in I}S_i$$?

Put less rigorously, I want to consider all theories such that adding the sentence $$\phi$$ gives back the original theory once you close it. I want to intersect these theories to get a new theory $$W$$, and then determine whether adding $$\phi$$ to $$W$$ results in getting back $$T$$.

The motivation for this is that I want to find the 'weakest' or 'smallest' theory that one can add some specific sentence to in order to retrieve some particular theory $$T$$. It seems to me the easiest way to get such a 'weakest' theory would be to intersect all theories for which this is possible. However, I am not sure that the resulting intersection would in fact have the original desired property, that property being that adding $$\phi$$ retrieves the original theory $$T$$.

Short version: you're looking for (the closure of) $$\{\phi\rightarrow \psi: \psi\in T\}.$$
Note that for any theories $$S,T$$ we have $$cl(S\cup\{\phi\})\supseteq T\iff S\models\phi\rightarrow\psi$$ for each $$\psi\in T$$. If we further assume that $$S$$ is closed, this is equivalent to $$\phi\rightarrow\psi$$ being in $$S$$ for each $$\psi\in T$$. And this gives a description of the desired "smallest (closed) theory:" set $$T_\phi:=cl(\{\phi\rightarrow\psi: \psi\in T\}).$$
Then $$T_\phi$$ is deductively closed, $$cl(T_\phi\cup\{\phi\})=T$$, and $$T_\phi\subseteq S$$ whenever $$S$$ is closed and $$cl(S\cup\{\phi\})\supseteq T$$.