Theory of Completeness for Logical Spaces

Abstract

A logical space is a pair ${(A, {\mathcal{B}})}$ of a non-empty set A and a subset ${{\mathcal{B}}}$ of ${{\mathcal{P}} A}$ . Since ${{\mathcal{P}} A}$ is identified with {0, 1}^A and {0, 1} is a typical lattice, a pair ${(A, {\mathcal{F}})}$ of a non-empty set A and a subset ${{\mathcal{F}}}$ of ${{\mathbb{B}}^A}$ for a certain lattice ${{\mathbb{B}}}$ is also called a ${{\mathbb{B}}}$ -valued functional logical space. A deduction system on A is a pair (R, D) of a subset D of A and a relation R between A* and A. In terms of these simplest concepts, a general framework for studying the logical completeness is constructed.