Distribution Laws in Weak Positional Logics

Abstract

A formal language is positional if it involves a positional connecitve, i.e. a connective of realization to relate formulas to points of a kind, like points of realization or points of relativization. The connective in focus in this paper is the connective “R”, first introduced by Jerzy Łoś. Formulas “R_αφ” involve a singular name α and a formula φ to the effect that ϕ is satisfied (true) relative to the position designated by α In weak positional calculi no nested occurences of the connective “R” are allowed. The distribution problem in weak positional logics is actually the problem of distributivity of the connective “R” over classical connectives, viz. the problem of relation between the occurences of classical connectives inside and outside the scope of the positional connective“R”.