Simple question about of $\Vdash \varphi$

Question

Let $\mathbb{P}$ a poset. The following are equivalent.

$(1)$ $p\Vdash \varphi$.

$(2)$ $\forall r\leq p(r\Vdash \varphi)$.

$(3)$ $\{r: r\Vdash \varphi\}$ is dense below $p$.

I am confused when trying statements of the form $\Vdash \varphi$ where $p$ does not appear.

What does it mean $\Vdash \varphi$ ?

Can someone explain me please. Thanks

Answer 1

Usually when we write $\Vdash\varphi$ that just means that any $p\in\mathbb{P}$ forces $\varphi$ or equivalently $\{r;r\Vdash\varphi\}$ is dense in $\mathbb{P}$. If $\mathbb{P}$ has a maximum element $\mathbb{1}$ then we can also write $\mathbb{1}\Vdash \varphi$ for $\Vdash\varphi$.