How to solve the liar problem?

Have you seen, that Mathematica is capable of many boolean computations using special boolean functions? Let's assume someone from the island makes a statement, then when the statement is true, whether or not he tells the statement is true, depends on whether or not he is a truth-teller. When we know, which kind he is, we know the correct statement through what he says. Therefore, let's define a function for this and check the truth-table

trueStatement[statement_,isTruthTeller_]:=!Xor[statement,isTruthTeller]    
BooleanTable[{a,b,trueStatement[a,b]},{a,b}]//TableForm
(*
True    True    True
True    False   False
False   True    False
False   False   True
*)

So, if A says a statement is true and A is a truth-teller, the statement is true for sure. On the other hand, if A is not a truth-teller, then the real statement is false.

Now we want to transform the two statements

A said: "B is a truth-teller." B said: "We two are different kinds of people."

without knowing whether a or b are liars or truth-tellers

eq = trueStatement[b, a] && trueStatement[a == ! b, b]

The statements read as: a says that b is a truth-teller and b says, that a is not of the same kind as b. Now we can simply do

SatisfiabilityInstances[eq, {a, b}]

(* {{False, False}} *)

Therefore, both, a and b are liars.

Cases[Tuples[{True, False}, 2], {a_, b_} /; 
  Equivalent[a, b] && Equivalent[b, Xor[a, b]]]

(*{{False, False}}*)

FindInstance[Equivalent[a, b] && Equivalent[b, Xor[a, b]], {a, b}, Booleans]

(*{{a -> False, b -> False}}*)