(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Richard is either a knight or a knave. Knights always tell the truth, and only the truth; knaves always tell falsehoods, and only falsehoods. Someone asks, "Are you a knight?" He replies, "If I am a knight, then I'll eat my hat."

a) Must Richard eat his hat?

b) Set this up as a problem in propositional logic. Introduce the following propositions: p = "Richard is a knight" and q = "Richard will eat his hat." Translate what we are given into propositional logic i.e. re-write the premises in terms of these propositions.

c) Prove that your answer from part (a) follows from the premises you wrote in (b)

3. The attempt at a solution

I didn't have an off-the-cuff answer, so I skipped to part (b). We have

p: Richard is a knight

q: Richard will eat his hat

r: Richard's response is true

If Richard is a knight, then his response is true. So we have:

[tex] p \Rightarrow r [/tex]

and if he is a knave, his reponse is false

[tex] \neg p \Rightarrow \neg r [/tex]

But Richard's response amounts to "If p, then q", or [itex] r = (p \Rightarrow q) [/itex]

So we have:

[tex] p \Rightarrow (p \Rightarrow q) [/tex]

From this we can conclude that if Richard is a knight, he is telling the truth when he says that if he is a knight, he will eat his hat. Therefore, since is IS in fact a knight, he WILL eat his hat. What if Richard is a knave?

[tex] \neg p \Rightarrow \neg (p \Rightarrow q) [/tex]

We can show that

[tex] \neg (p \Rightarrow q) = p \wedge \neg q [/tex]

So it seems that if Richard is a knave, his statement is false. Which means that the negation of his statement is true. The negation of his statement is:

Richard is a knight and Richard will not eat his hat

This doesn't really make sense to me. It seems paradoxical since Richard is a knave in this case. Furthermore, the recursive nature of p --> (p --> q) is bothering me. Have I approached this problem in the right way?

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# Homework Help: Propositional Logic problem

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