This paper is centered around logical distinguishability. It gives a non-trivial topological basis in which to relate logical expressions. Using this basis we build Propositional Logical Topologies (ProLT’s). Notions common to topology theory, such as open and closed sets, are given meaning in ProLT’s each which correspond to logical expressions. A measure of distance or distance metric is build which allows a measure of separation of logical expressions, this degree of separation i.e. Hausdorff tells how different one logical expression is from another. A further measure of distance, the truth distance, is used to compare relative truths of ProLT’s corresponding to the positive valuations their respective isomorphic logical expressions.
As illustrated in the picture the intersection of ProLT’s give rise to the notion of (topo)logical Inference resulting in various logical syllogysms. This work gives a basis for the work done in homology which aims to both define the notion of a ‘Logical Homology’ as well as give a way to build up higher dimensional logical simplicial complexes. These complexes are similar to the ideas of ‘mental points’ as seen in Khrennikov’s work, though the construction is different.