Goedel Logics: On the Elimination of The Absoluteness Operator

TL;DR

This paper studies the eliminability of the Delta operator in Gödel logics, showing it can be eliminated at the propositional level under restricted semantics but not in general at the first-order level.

Contribution

It demonstrates conditions under which the Delta operator can be eliminated in propositional Gödel logic and analyzes the challenges in first-order logic.

Findings

Delta is eliminable at propositional level under restricted semantics.

Every formula with Delta is equivalent to a disjunction of chain formulas.

Delta-elimination fails in first-order logic but can be recovered with witnessed semantics.

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.