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

Question

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 $T$.

Answer 1

Short version: you're looking for (the closure of) $$\{\phi\rightarrow \psi: \psi\in T\}.$$

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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 $cl(S\cup\{\phi\})\supseteq T$.