Goedel Logics: On the Elimination of The Absoluteness Operator

Abstract

We investigate the eliminability of the absoluteness operator Delta in Goedel logics. While Delta is not definable from the standard connectives and disrupts important proof-theoretic properties, we show that it becomes eliminable at the propositional level under a restricted semantics in which all propositional atoms (except the truth constant 'True') are interpreted strictly below 1. Under this semantics, every formula containing Delta is equivalent to a disjunction of chain formulas, yielding a Delta-free normal form (standard and restricted semantics coincide w.r.t. valid formulas without Delta). We further analyze the situation in the first-order setting, where Delta-elimination fails in general due to recursion-theoretic and topological constraints, but can be recovered under witnessed semantics.