I am trying to show that the following argument is valid.
- There is an email that is sent but it is not saved in the inbox.
- All emails are saved in the inbox or the inbox is full.
- If the inbox is full, then all emails are lost.
- Therefore, some email is lost.
Let the universe of discourse be the emails.
$P(x):\;x$ is sent.
$S(x):\;x$ is saved in the inbox.
$L(x):\;x$ is lost.
$Q:$ Inbox is full.
\begin{align} &1.\; \exists\;(P(x)\; \land \;\lnot\;S(x)) \\ &2.\; P(c)\;\land\;\lnot\;S(c)\;for\;some\;c \\ &3.\; \forall x\;S(x)\;\lor\;Q \\ &4. \; Q\;\rightarrow\;\forall x\;L(x) \\ &5. \; \lnot Q\;\lor\;\forall x\;L(x) \\ &6.\;\forall x\;S(x)\;\lor\;\forall\;L(x) \\ &7.\; S(c)\; \lor \; L(c) \\ &8. \; \lnot \;S(c) \\ &9.\; L(c) \\ &10. \; \exists x\; L(x) \\ \end{align}