By Existential Rule E4 $\mathscr B(t, t)\vdash (\exists x) \mathscr B(x, t)$. But how can we get back? How can we formalize $(\exists x) \mathscr B(x)\vdash \mathscr B(t)$? It is shown on page 74 and it is called “rule C (“C” for “choice”)".
But two questions:
- Designation. “… rule C deduction in a first-order theory K is defined in the following manner: $\Gamma \vdash_C \mathscr B$...” Wait a minute! What is the name of the theory? Usually $ \vdash_C$ means we have theory C, but here it looks like $ \vdash_C$ used to show that we use rule C? It is strange.
- “(d) there is a preceding wf $(\exists x) \mathscr E(x)$ such that $\mathscr D_i$ is $\mathscr E(d)$, where $d$ is a new individual constant (rule C).” A new individual constant? But in this case we have a new term and a new first-order theory. Why we cannot use already existing individual constant?