Tense logics based on posets

TL;DR

This paper investigates tense logics based on partially ordered sets (posets), exploring how dynamic truth values evolve over time and proposing new approaches to integrate tense operators into poset-based logical systems.

Contribution

It extends prior work by developing methods to incorporate tense operators into poset-based logics, enabling the modeling of temporal changes in truth values.

Findings

Proposed multiple approaches to tense logics on posets

Enhanced understanding of dynamic truth values over time

Inspired further research in temporal poset-based logics

Abstract

Not all logical systems can be captured using algebras. We see this in classical logic (formalized by Boolean algebras) and many-valued logics (like Lukasiewicz logic with MV-algebras). Even quantum mechanics, initially formalized with orthomodular lattices, benefits from a simpler approach using just partially ordered sets (posets). This paper explores how logical connectives are introduced in poset-based logics. Building on prior work by the authors, we delve deeper into "dynamic" logics where truth values can change over time. We consider time sets with a preference relation and propositions whose truth depends on time. Tense operators, introduced by J.Burgess and extended for various logics, become a valuable tool. This paper proposes several approaches to this topic, aiming to inspire a further stream of research.