I'm reading part II of A. Kock's Synthetic Differential Geometry. It's my first introduction to SDG and I'm reading it for fun.
I'm confused by the use of the "forcing" symbol $ {\vdash} $. Fix a category $ \mathcal E $, which we assume to have enough structure to do everything we need to do. Let $ R\in \mathcal E $ (for example, $ R $ might be a ring object), and fix some $ X\in \mathcal E $. At some point the author claims that if we suppose that we know the meaning of $$ \vdash_Y \phi(b) $$ for every generalized element $ b\in_Y R $ (i.e., for every arrow $ b\colon Y\to R $), for every $ \alpha\colon Y\to X $ and for every "mathematical formula" (?) $ \phi $, then we can declare $$ \vdash_X \forall x\, \phi(x) $$ to mean that given any $ b\in_Y R $ and any $ \alpha\colon Y\to X $, then $ \vdash_Y \phi(b) $ holds.
My first couple of questions is: What is a "mathematical formula"? Why do we write $ \vdash_X\forall x\,\phi(x) $ with an $ X $ below the turnstile?
Another question. A couple of lines below the notion of stable formula is introduced. A formula (?) $ \phi $ is stable if "$ \vdash_X \phi $ and $ \alpha\colon Y\to X $ imply $ \vdash_Y \phi $". What's the meaning of $ {\vdash_X} $ or $ {\vdash_Y} $ here really? And how can I show that the formulas $ \forall x\, \phi(x) $ and some others are stable (Proposition 2.1)?