I am trying to derive: $-(\exists x) -Lx \vdash (\forall x) Lx$ As part of a broader understanding of relationships between universal affirmative; universal negative; particular affirmative; and particular negative.
But I am having trouble getting started with this one because of the negated Lx in the premise.
With a similar one, I was able to derive along the lines of:
$ 1 (1)\hspace{10 mm} -(\exists x) -Lx \hspace{10 mm}A \\ 2 (2)\hspace{10 mm} Lx \hspace{29 mm}A \\ 2 (2)\hspace{10 mm} (\exists x) Lx \hspace{21 mm}2,\exists I\\ $
And then introduce a contraditction between $-(\exists x)Lx \wedge (\exists x)Lx)$, but again, the issue with the negated Lx in the premise is causing me some difficulty.
Does anyone have any suggestions on how I might approach this? The logic system is restricted to primitive rules, so some of the obvious equivalencies are not allowed.
Your idea of deriving a contradiction with the premise is correct. To provoke a contradiction with $\neg \exists x \neg Lx$, you need to get $\exists x \neg Lx$. So just change your assumption on line 2 to $\neg Lx$. That's what you later want to apply the contradiction rule on.