Curried propositional calculus and the matrix representation

Propositional calculus, which is an accomplished system for deductive reasoning, consists of binary operations. However representation of induction or abduction might need a calculation system based on unary operations. In the present paper, we propose a new calculation which consists of only unary operations, for propositional logic. The new calculation system, which is represented by 2 dimensional vectors and 2 × 2 matrices, can emulate truth tables of propositional logic. We call it curried propositional calculus. Moreover, we introduce diagrams which corresponds to logic gates for conventional logical connectives. Consequently, propositional logic which is one of the systems for deductive reasoning is connected to matrix algebra based on unary operations. This result gives a start point to reconstruct a new framework of mathematical logic to study the three types of reasoning, deduction, induction and abduction, comprehensively.