Lattice and metric completions of the classical logic metric space and a comparison

摘要

Based on the concept of truth degree for logical formulas, a pseudo-metric is constructed on the set of all classical propositional formulas, and a metric is naturally induced on the corresponding Lindenbaum algebra of Boolean type, which constitutes a metric space, called the classical logic metric space. We respectively study both lattice completions and metric completions of this space, and compare the two kinds of completions from the angle of lattice structure as well as metric structure. On one hand, it is proved that metric completions of the classical logic metric space are complete Boolean algebras, which also act as lattice completions of the Lindenbaum algebra. On the other hand, it is pointed out that the normal lattice completion of the Lindenbaum algebra constitutes a Boolean algebra as well, which is strictly smaller than metric completions of the classical logic metric space in the sense of order-embedding. Also, the normal lattice completion can be seen as a dense subspace of the metric completions in the sense of isometry.