Locality in Residuated-Lattice Structures

TL;DR

This paper investigates the applicability of classical locality theorems in the context of residuated-lattice structures, revealing conditions under which these theorems hold or fail in non-classical many-valued logics.

Contribution

It analyzes how Hanf and Gaifman locality theorems extend to residuated lattices, introducing alternative approaches and syntactic tools for their validation in this setting.

Findings

Hanf's theorem fails with natural local neighborhoods but can be recovered for well-connected lattices.

Gaifman's theorem's main lemma can be recovered under certain syntactic conditions.

Order-interpreting connectives are crucial for encoding locality in residuated lattices.

Abstract

Many-valued models generalise the structures from classical model theory by defining truth values for a model with an arbitrary algebra. Just as algebraic varieties provide semantics for many non-classical propositional logics, models defined over algebras in a variety provide the semantics for the corresponding non-classical predicate logics. In particular, models defined over varieties of residuated lattices represent the model theory for first-order substructural logics. In this paper we study the extent to which the classical locality theorems from Hanf and Gaifman hold true in the residuated lattice setting. We demonstrate that the answer is sensitive both to how locality is understood in the generalised context and the behaviour of the truth-defining algebra. In the case of Hanf's theorem, we will show that the theorem fails for the natural understanding of local neighbourhoods, but is recoverable with an alternative understanding for well-connected residuated lattices. For Gaifman's theorem, rather than consider Gaifman normal forms directly we focus on the main lemma of the theorem from textbook proofs - that models which satisfy the same basic local sentences are elementarily equivalent. We prove that for a number of different understandings of locality, provided the algebra is well-behaved enough to express locality in its syntax, this main lemma can be recovered. In each case we will see the importance of an order-interpreting connective which creates a link between the modelling relation for models and formulas and the valuation function from formulas into the algebra. This link enables a syntactic encoding of back-and-forth systems providing the main technical ingredient to proofs of the main locality results.