The Architecture of Truth

TL;DR

This paper presents a formal framework for understanding the architecture of truth in institutions using indexed duality, satisfaction relations, and logical environments, emphasizing the preservation of truth structures through morphisms.

Contribution

It introduces a novel duality-based architecture of truth in institutions and formalizes the role of logical environments in enriching and preserving this architecture.

Findings

Defines truth architecture as an indexed duality with satisfaction relations.

Shows how logical environments axiomatize truth adjunctions.

Demonstrates preservation of truth architecture via morphisms.

Abstract

The theory of institutions is framed as an indexed/fibered duality, where the indexed aspect specifies the fibered aspect. Tarski represented truth in terms of a satisfaction relation. The theory of institutions encodes satisfaction as its core architecture in the indexed aspect. Logical environments enrich this truth architecture by axiomatizing the truth adjunction in the fibered aspect. The truth architecture is preserved by morphisms of logical environments. (Although not every institution is a logical environment, each institution has an associated logical environment defined via the intent of the structures of the institution, and each institution is represented by an indexed functor into the structure category of the classification logical environment $\mathtt{Cls}$.)