An essay on complex valued propositional logic

Abstract

In the decision making logic it is often necessary solving of logical equations for which, due to the features of disjunction and conjunction, no admissible solutions exist. An approach is suggested in which by introducing of Imaginary Logical Variables (ILV) the classical propositional logic is extended to a complex one. This provides a possibility to solve a large class of logical equations. The real and imaginary variables each satisfy the axioms of the Boolean algebra and of the lattice. It is shown that the Complex Logical Variables (CLV) observe the requirements of the Boolean algebra and the lattice axioms. Suitable definitions are found for these variables for the operations disjunction, conjunction, and negation. A series of results are obtained, included also the truth tables of the operations disjunction, conjunction, negation, implication, and equivalence for complex variables. Inference rules are deduced for them analogous to Modus Ponens and Modus Tollens in the classical propositional logic. Values of the complex variables are obtained, corresponding to TRUE (T) and FALSE (F) in the classic propositional logic. A conclusion may be made from the initial assumptions and the results attained, that the imaginary logical variable i introduced hereby is "truer" than the condition "T" of the classic propositional logic and ¬i--"falser" than the condition "F", respectively. Possibilities for further investigations of this class of complex logical structures are pointed out.