**Q: Russell's paradox is a model of the third valued proposition in four valued logic. So, what is a symbolic expression of the third valued proposition?**

Ａ：Russell's paradox "Set M which does not contain itself" is the third valued proposition which is neither true nor false.

If set M which does not contain itself contain itself, set M is not set M. If set M which does not contain itself does not contain itself, then set M is set M. So If set M contains M, set M is not element of set M, and then if set M does not contain M, set M is element of set M. This means that Set M is composed of two dual implications between contradictory propositions.

Let p= "M is not element of M".If not p, then p, and, if p, then not p. then p⇔not p.

Because this set M is contradictory to the true statement and the false statement, and has to be expressed by a dual implication between two contradictory statements, this set M is assigned the third truth value which is contradictory to "the true" or "the false". There might not be comprehensible proposition as an example of the proposition with the third truth value any further.

The third truth value is, at times, assigned to a possible proposition, and at times, assigned to a recurrent propositions, and at times, assigned to a moral proposition which has the political contents, and above all, assigned to a paradoxical propositions. And, such a phenomenon overflows in the world. The third truth value represents these paradoxical or reflexive or recurrent or possible propositions which contain itself anywhere or anything in it's propositions.

Let p a possible proposition, p depend on q which is something immature or future, but something q depend on a possible proposition p. Now, let {p=p(q)} p depend on q, then p=p(q) and q=q(p). So, p=p(q(p(q.......... . Let q=-p. Then, p=p(-p(p(-p(p(-p...... . This is the symbolic expression of a possible proposition p and its dual implication.

The statement of this symbolic expression are a possible proposition or a recurrent proposition or a future proposition or moral proposition, etc. So this type of proposition is natural events in our life.

Note: Denial of this third valued proposition

As these third valued propositions exist certainly, the denial of the third valued proposition becomes logical problem. So this four valued logic assumes this negative form of the third truth value to the fourth truth value. Then, this four valued logic is able to have the law of contradiction or excluded middles like 2-valued logic. This is the reason that this logic is said the simplest expansion of 2-valued logic.