# Smulliyan book suggests some sort of duality in logic.

Hi there

I'm not sure if this is the proper forum for this, if not, then my apologies.

I've been reading some Ray Smulliyan. For those who haven't heard of him, he writes puzzle books based on formal logic. The puzzles are all of the 'knight and knave'sort, where knights always tell the truth and knaves always lie.

In a lot of his puzzles, there seems to be some sort of duality going on between what you are told and what you can deduce.

For example. If you meet two blokes, A and B, and A tells you-

"If I'm a knight, then so is B."

then you can deduce that they are both knights. If on the other hand he had told you that they were both knights, then all you could deduce would be that if A's a knight, then so is B.

If A tells you that B is a knight, then you can tell that they're both the same (but not whether they're both knights or both knaves.)

If A tells you that B is the same as he, then you can deduce that B must be a knight (But know nothing about A).

So my question is: Is this a real duality? If so, does it have a name, and where can I find out more about it?

Thank you in advance

Stevie Hair

### Re: Smulliyan book suggests some sort of duality in logic.

Let X be a set. Given x in X and K a collection of subsets of X define the following product:

x.K = {A a subset of X| Either (x in A and A in K) or (x not in K and A not in K)}

It is easy to check that x.(x.K)=K.

That identity expresses the duality you refer to. (Apologies if you already knew this and were looking for something deeper).

The reason is that if X is your set of knights and knaves, and person x says that only the sets in K can be the set of knights, then if L is the collection of subsets which could actually be the set of knights, we have:

L=x.K

Similarly, if person x says that only the sets in L can be the set of knights, then the collection of subsets which could actually be the set of knights is:

x.L=x.(x.K)=K