Question You come across four trolls playing bridge. They declare:
Troll 1: All trolls here see at least one knave.
Troll 2: I see at least one troll that sees only knaves.
Troll 3: Some trolls are scared of goats.
Troll 4: All trolls are scared of goats.
Are there any trolls that are not scared of goats? Recall, of course, that all trolls are either knights (who always tell the truth) or knaves (who always lie).
What I have done so far
I have represented all the four statements using propositional logic.
If troll $ p $ sees a knave lets say $ K(p) $
If troll $ p $ sees a troll lets say $ T(p) $
If a troll $ p $ is scared of goats lets say $ G(p) $
Thus our four statements can be written like so:
$ \forall p K(p) $
$ \exists p \exists q T(p) \land K(q) $
$ \exists p G(p) $
$ \forall p G(p) $
I see that perhaps to solve the problem, first I need to find out whether we are dealing with lying knaves or we are dealing with truth-telling knights. To do that we use statements 1 and 2. Once we know if they are knights or knaves we use that knowledge to find out whether any trolls are not scared of goats, that is, $\exists p \lnot G(p) $, from statements 3 and 4.
However I do not know how to deal with the first part or where to start from. Any help will be appreciated.