On the use of fuzzy logic and learning automata optimization to resolve the Liar and related paradoxes

We show that logical paradoxes based on self-reference (of which the Liar is the best known example) are equivalent to the non-existence of solutions to a numerical system of equations, the so-called truth-value equations. Furthermore, we show that in many cases a self-referential system which does not posses a crisp (Boolean) solution can be solved by expanding the solution set to include fuzzy solutions. Then we formulate the computation of these fuzzy solutions as an optimization problem and, by numerical experiments, we demonstrate that teams of Learning Automata (of a type intermediate between finite action and continuous action automata) can be succesfully used to solve the optimization problem. In this manner, the combination of fuzzy logic and learning automata resolves a wide class of paradoxes.