S-Logic with First and Second Imaginary States

Abstract

An imaginary logic, i-logic was introduced for solving of some unsolvable problems in the framework of the classical propositional logic. On the other hand similar unsolvable problems arise in the imaginary logic itself. Introduction of a second imaginary j-logic is suggested in the present work through which this insolvability in the i-logic is surmounted. For this purpose constraints are ushered in between the variables classical r-logic and those of the i-logic and j-logic. This gives rise to complex logics—s, s ₁ , s ₂ , respectively, in which functioning of the r, i, and j-logics is being interpreted. It is shown that all these logics are based on the algebraic structures Boolean algebra and lattice. A rule is proposed through which the contradictions may be avoided at the realization of associativity of disjunction and conjunction. A number of results are received for the behavior of conjunction and disjunction in the complex logics being concerned. On their base two truth tables are filled in and shown for the variables of indices r, i, j, s, s ₁ —for conjunction and disjunction separately. It is pointed out that the logical structures being proposed may be considered as a complex multiple-valued logic with 12 states in which three two valued logics with indices r, i, j are interpreted in an appropriate manner. The field of application of the logical structures being investigated are shown.