More precisely, let $T$ be a theory of some first-order language. My question is then the following. What does it mean for two sentences $\sigma_1$ and $\sigma_2$ in the same language to be independent of each other in the theory $T$?
My "hypothesis" is that it means $\neg \, ( \, T \vdash \sigma_1 \leftrightarrow \sigma_2 \,)$. However, I could only come up with this "hypothesis" by looking at some particular examples; also I could not immediately find an online reference where this concept is defined, hence this question. Can somebody confirm this?
Note. I do know what it means for one formula $\sigma$ to be independent from a theory $T$. Namely it means that $\neg \, ( \, T \vdash \sigma\,) \; \wedge \; \neg \, ( T \vdash \neg \sigma)$.