Abstract

A logical space is a pair of a non-empty set A and a subset of . Since is identified with {0, 1} A and {0, 1} is a typical lattice, a pair of a non-empty set A and a subset of for a certain lattice is also called a -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.