Existential generalisation

An existential generalisation means that for every entity $c$ in the domain, we have a correct inference rule of the form $$ \boxed{ \begin{array}{c} \underline{P(c)} \\ \exists x P(x) \end{array} } $$

For example:

$$\begin{array}{lc}& \underline{\text{Marla is reading the lecture notes.}}\\ \therefore & \text{There is someone in the class reading the lecture notes.}\end{array}$$

By existential generalisation, given that $\varphi (x/t)$ is known to hold for a particular object $t$ of the domain, we may derive $(\exists x)\varphi (x)$:

$$\frac{\varphi (x/t)}{(\exists x)\varphi (x)}(\text{EG})$$

This can be seen as $\exists I$.