Logical Matrices and Goldbach Problem

Abstract

The paper considers equivalent formulations of Goldbach conjecture in terms of sets of tautologies in sequences of logical matrices and single logical matrices. The significant part in this consideration belongs to concepts of tautology in a logical matrix, sums and products of logical matrices from sequence $K_{n+1}$ of Karpenko matrices. Thus the paper proposes an answer to A.S. Karpenko’s question about possible relations between sequences of logical matrices similar to $K_{n+1}$ and an open problem, known as binary Goldbach conjecture: every even natural number $n \geq 4$ may be represented as a sum of two prime numbers. The proposition that all finite-valued matrices in the sequence $M$ have tautologies iff the binary version of Goldbach conjecture ($G_2$) is true is proven. Using the properties of matrix product operation, it is proven that the infinite-valued matrix $M\otimes$ has tautologies iff $G_2$ is true. The paper also mentions that the set of tautologies of $M\otimes$ (id est the logical theory defined by $M\otimes$) is equal to the certain theory defined in terms of finite-valued Lukasiewicz logics $\L_n$ iff $G_2$ is true. These results were restated in terms of sequences of matrices and their products from a large class of logical matrices. Thus it was found out that $G_2$ has certain ogical aspects, as it is equivalent to existence of defined non-empty logical theories. DOI: 10.21146/2074-1472-2018-24-1-62-74