Can anyone explain what is the intuition behind the following definition?
Definition 4.25 Let $\Bbb P$ be a poset. Let $\phi(x_1,\ldots,x_n)$ be a formula, $p\in\Bbb P$, and let $\tau_1,\ldots,\tau_n$ be $\Bbb P$-names. We define $p\Vdash^*\phi(\tau_1,\ldots,\tau_n)$ by recursion on the complexity of $\phi$ as follows.
$p\Vdash^*\tau_1=\tau_2$ if and only if the following hold.
For all $\langle\pi_1,s_1\rangle\in\tau_1$, the set $$\{q: q\leq s_1\rightarrow\exists\langle\pi_2,s_2\rangle\in\tau_2(q\leq s_2\land q\Vdash^*\pi_1=\pi_2)\}$$ is dense below $p$.
For all $\langle\pi_2,s_2\rangle\in\tau_2$, the set $$\{q:q\leq s_2\rightarrow\exists\langle\pi_1,s_1\rangle\in\tau_1(q\leq s_1\land q\Vdash^*\pi_1=\pi_2)\}$$ is dense below $p$.
- $p\Vdash^*\tau_1\in\tau_2$ if and only if the set $$\{q:\exists\langle\pi,s\rangle\in\tau_2(q\leq s\land q\Vdash^*\tau_1=\pi)\}$$is dense below $p$.
- $p\Vdash^*\phi(\tau_1,\ldots,\tau_n)\land\psi(\tau_1,\ldots,\tau_n)$ if and only if $$p\Vdash^*\phi(\tau_1,\ldots,\tau_n)\text{ and }p\Vdash^*\psi(\tau_1,\ldots,\tau_n).$$
I know that the sign $p \Vdash \phi(x_1,...,x_n)$ somehow suppose to tell me that for any generic filter which contains $p$, $M[G] \models \phi(x_1,...,x_n)$. But, what is the connections to the definition here above?
Thank you