Four valued logic----Hidden paradox

Four valued logic Q&A 1999.3

17:Dual Implication will be hidden in the 3rd valued proposition except the pure paradox

18:The opposite propositions will be decomposed easily to dual Implication.

19:The extension of the correspondence theory of 2-valued logic

20:Truth table of negation (corrected at 1999.03)

　

Q&A17：Dual Implication will be hidden in the 3rd valued proposition except the pure paradox.

The 3^rd valued proposition p has the dual implication (p<=>-p) which has self-negative proposition as the antecedent or consequent. But, this dual implication (p<=>-p) in this 3^rd valued proposition hide oneself ordinarily. So we can not know this dual implication easily.

Now, most typical case is that dual implication hides itself under the opposite propositions which appears often in the political opinions or in the discriminatory opinions. If this dual implication hides itself under the political opposite propositions or another, though we can observe these opposition many times, we know already something tips of the dual implication.(The paradox expose this dual implication. So, paradox is the model of the 3^rd valued proposition.) It is easy to find this dual implications in our ordinal opinions, but it is necessary to know the basic method which decompose the opposite opinions to the 3^rd valued proposition.

This method is simple, because this dual implication does not exist in our unconscious world or under our hidden presupposition, or it is not the hidden apprehension in our ethics. We must check two points of dispute to find this dual implication in a opposition. One point is to decompose this opposition to two contents "self-denying affirmation" and "lack desiring realization", 2nd point is to decompose formally a proposition to the paradoxical implication (p<=>-p).

Q&A18: The opposite propositions will be decomposed easily to dual Implication.

The 3^rd valued proposition p has the dual implication (p<=>-p) which has self-negative proposition as the antecedent or consequent. So We must find two indicators for this dual implication to assign truth value for a ordinal proposition. One indicator is that this proposition has paradoxical contents "self-denying affirmation" and "the lack desiring realization", 2nd indicator is to find the paradoxical notation (p<=>-p). (Q&A17)

This "self-denying affirmation (p=>-p)" is the negative expectation of itself, but it is the affirmation for the time being. So this affirmation is timid affirmation. While, "the lack desiring realization (-p=>-p)" is the real recognition of the lack for realization, but the lack is the hard desire truly. So this negation is the true desire. In this another aspect, this 3^rd valued proposition is dual implication which is firm implication and timid implication.

So it is not strange to misunderstand this dual implication to opposite true or false proposition. Or, it is not strange to misunderstand the conjunction of this dual implication to the conjunction of the opposite propositions. This misunderstanding develop to the recognition of infringement of "the law of the contradiction". One misunderstanding gives birth to the new misunderstanding.I think that this misunderstanding gives birth to the intermediate truth value which is one truth value of many-valued logic without "the law of the contradiction".

　

Q&A19:The extension of the correspondence theory of 2-valued logic

If the proposition p is the 3^rd valued proposition, the proposition p has the dual implication (p<=>-p) which has self-negative proposition as the antecedent or consequent. This notation is p|T3=>(p<=>-p).

In the 3^rd valued proposition, antecedent is inseparable to the consequent which is the negation of antecedent. So the correspondence of the 3^rd valued proposition is equal to dual implication in the 3rd valued proposition. The dual implication does not show that it is not inevitable to battle against opposite opinions, but show that it is necessary to find the dual implication in the opposite opinions or another type of opinions. That is to say, the 3^rd valued proposition is the discovery of dual implication or dual correspondence in the opposite opinions or another. But it is difficult to find pure dual implication, because antecedent is not inseparable from consequent in the 3^rd valued proposition.

This inseparability of antecedent and consequent in 3^rd valued proposition is basically different from the divisibility of true proposition or false proposition. We need the extended-theory of correspondence which include the 2-valued theory of correspondence for 4-valued logic.

Next correspondences are basic of "truth", "(antecedent)proposition","consequent".

truth	proposition	correspondence	consequent
T1	A	signify	True fact
	a statement is true if it says of what is that it is.( Aristotle definition)
T2	A	signify	Wrong, False fact or Lie
	a statement is true if it says of what is not that it is. (Aristotle definition)
T3	A(inseparable)	<=>, dual implying	(inseparable-A) Possible, Future, Moral idea
	A =>-A=>A=>-A=>A=>.......
	A = A(-A(A(-A(A.....
	a statement is true idea if it has a self-denying statement as a antecedent or consequent of the conditional.
T4	(A<=>-A)inseparable	<=>, dual implying	(inseparable-(A<=>-A)) Anxious idea, nothingness.
	(A<=>-A)=>-(A<=>-A) =>-((A<=>-A)=>-(A<=>-A))=>-(-((A=>-A)=>.......
	A = - A(-A(-A(-A(-A....
	A statement is true anxious if it is the self-negative proposition to keep denying the reality recognition unlimitedly by which oneself is found (corrected at 1999.08)

Q&A20： Truth table of negation (corrected at 1999.03)

In our natural language, the negative sentence "neither true nor false" is useful often to signify the different truth value from "the true" or "the false". This 4-valued logic define this negation as the normal logical negation-.

Do you think this negation- is different from 2-valued negation ? I think that it is not different.

The exclusion by which Russell excluded his paradox from true or false proposition, is different from the 2-valued negation, because we can define Russell's paradox contradict to 2-valued propositions. While the negation of 2-valued logic is the connective of true or false proposition which contradict each other, or is the normal application of "the law of contradiction". But Russell's exclusion is the extended application of "the law of contradiction" in 2-valued logic. Because his exclusion is equal to original negation, this extended negation is not different from original negation of 2-valued logic.

This 4-valued negation is defined by next truth table of negation.(corrected at 1999.03)

p　 -p -p -p -p

T1　 T2 T3 T4

T2 T1 T3 T4

T3 T1 T2 T4

T4 T1 T2 T3　