# Making deductions: Are there any trolls scared of goats

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:

1. $$\forall p K(p)$$

2. $$\exists p \exists q T(p) \land K(q)$$

3. $$\exists p G(p)$$

4. $$\forall p G(p)$$

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.

• Your translation of statement 2 into propositional logic is not correct. Must you use propositional logic to solve this? – player3236 Nov 11 '20 at 5:06
• @player3236 What other options do I have – GilbertS Nov 11 '20 at 5:07
• @player3236 Why is my translation false? – GilbertS Nov 11 '20 at 5:08
• It currently reads: There exists some troll that sees a knave. – player3236 Nov 11 '20 at 5:11

I’d not bother with the formal notation.

If Troll 1 is a knave, the other three must all be knights in order for his statement to be false. However, Troll 2’s statement would then be false, so Troll 1 must be a knight. That means that there must be at least two knaves amongst the other three trolls in order to make has statement true.

The only person who could possibly see only knaves is Troll 1, since he’s a knight. This means that Troll 2’s statement is true if and only if Trolls 1, 2, and 3 are all knaves, in which case Troll 2’s statement cannot be true. Thus, Troll 2’s statement must be false, and every troll sees at least one knight. Of course Trolls 2, 3, and 4 see Troll 1, so what that really tells us is that Troll 1 sees a knight: one of Trolls 2, 3, and 4 is a knight, the other two are knaves, and Troll 2 is one of the knaves. We conclude that one of Trolls 3 and 4 is a knight.

Note that if Troll 4 is telling the truth, so is Troll 3, so if Troll 4 is a knight, so is Troll 3. This is impossible, so Troll 4 must be a knave, and we conclude that not all trolls are scared of goats.

• I thought all trolls are either knights or knaves – GilbertS Nov 11 '20 at 5:55
• I think I misunderstand that last statement. So we can have some knights and the other knaves – GilbertS Nov 11 '20 at 5:58
• @GilbertS: Yes, what is meant is that each troll is either a knight or a knave, not that all four trolls are the same type. – Brian M. Scott Nov 11 '20 at 6:00
• Is there a way to answer this question using truth tables. Because in the book, deductions are proved using truth tables, not brute force logic. I was making the propositions so I can put them in a truth table and check for that line where they have true values so I can deduce the conclusions that arise. But I dont know what propositions I should consider – GilbertS Nov 11 '20 at 6:04
• @GilbertS: You could make something similar to a truth table. It would have $2^4=16$ rows, one for each possible assignment of knight or knave to Trolls 1, 2, 3, and 4. Besides a column for each troll, it would have a column for each of the four statements: in each row you’d mark a column true if its statement is true for that particular combination of knights and knaves and false otherwise. If you do this correctly, you’ll find that the row making Trolls 1 and 3 knights and Trolls 2 and 4 knaves is the only one that makes all four statements true. – Brian M. Scott Nov 11 '20 at 6:13

If Troll 2's statement is true, then the troll referred to in his statement sees Troll 2, so that Troll 2 is a knave. This is impossible, since a knave wouldn't make a true statement, so Troll 2 is a knave. Suppose that no troll is afraid of goats. Then Trolls 3 and 4 are also knaves, and Troll 1 sees only knaves, so that Troll 2's statement was true after all. We have seen that this is impossible, so some trolls fear goats, and Troll 3 is a knight. If Troll 1 is a knave, then his statement is true, since everyone see Troll 1 or Troll 2. This is absurd, so Troll 1 is a knight. Now Troll 2 must see a knave and that can only be Troll 4.

Not all trolls are afraid of goats.

• But the question states that all trolls are either knights (who always tell the truth) or knaves (who always lie) – GilbertS Nov 11 '20 at 5:57
• Ohh. That doesnt meant that either all are knights or all are knaves. Ok – GilbertS Nov 11 '20 at 5:58