Four valued logic Q&A 1997.8
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.
Q：Which truth value have to be assigned to the assignment proposition of truth value to a certain proposition?
Ａ：The judgment proposition of truth values is either true or false proposition.
Four valued logic is able to define a proposition which assigns the truth value to a certain proposition.
The assignment proposition of one truth value from four truth values is a proposition by which it is judged whether the truth value is suitable for the proposition. So it's assignment is either true or false. As 2-valued logic has the excluded middle which any statements is either true or false, the assignment of truth values is a proposition in the 2-valued logic.If the proposition will be a future proposition, or a possible proposition, or a moral proposition, or paradoxical proposition, the third truth value can be assigned to the proposition. Moreover, this four value logic applies the fourth truth value to a self-negative proposition. Therefore, four valued logic applies any of four kinds of truth values to any proposition without fail.
Determinism that decides the truth value of all propositions to the true or the false, nevertheless, even voluntarily distinguish the proposition which is not decide either. Therefore, 2-valued logic allows a proposition to which the truth value can not be provided. Because 2-valued logic is logic only concerning the true or false proposition, it exclude the proposition which "true or false" is not assigned, to out of it's logic. The distinction between 2-valued logic and 4- valued logic appears here.
Proposition p=q(t) which define the truth value to a proposition q in 4-valued logic
truth value ofｑ ｐ=q(Ti)
true T1 p=q(T1)=T1 or p=q(T2,T3,T4)= T2
false T2 p=q(T2)= T1 or p=q(T1,T3,T4)= T2
third T3 p=q(T3)= T1 or p=q(T1,T2,T4)=T2
fourth T4 p=q(T4)= T1 or p=q(T1,T2,T3)=T2
Proposition p=q(t) which define the truth value to a proposition q in 2-valued logic
truth value of ｑ ｐ=q(Ti)
true T1 ｐ=q(T1)=T1 orｐ=q(T2)=T2
false T2 ｐ=q(T2)= T1 orｐ=q(T1)=T2
? ? ｐ=q(?)=?
Q： How to confirm the third valued proposition? Or, what is the identity of the third valued proposition very different from the identity of the true or false proposition?
Ａ：The identity of the third valued proposition cannot be verified as the existence of the meaning just like the true or false proposition. But, we can say that the third valued proposition is the same if it can be verified that dual implication in itself is the same.
The definition of the identity of third valued proposition is equal that dual implication in a proposition is the same. This is different from the identity of the true or false proposition. Because the third valued proposition is a possible proposition, ethics proposition, and an reflexive proposition, it does not consist of an 100% external object, or it is not defined independently from speaker itself . Therefore, the verification of third valued proposition might be missed to objectivity compared with the verification of the true or false proposition.
Q： Is it necessary to position the third truth value between "true" and "false"?
Ａ： It is not necessary that the third truth value have to position between "true" and "false". It is located outside from "true" and "false". Many valued logic of J.Lukaisiewicz locates the 3rd value and the 4th value between the true or the false. Therefore, his many valued logic has not the excluded middle and the law of contradiction. This should be called a big fault for the logic which everyone uses anytime and anywhere. As the contradiction law is an most familiar rule for the logic, I think that the logic without it's law is not easy to accept to people. However, if J.Lukasiewicz can locate the third truth value outside of the true and the false, I think that his many valued logic was welcomed accurately as the extension of 2-valued logic based on a principle of two truth values from Greek era.
1997.8
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