Truth Factors

TL;DR

This paper refines the categorical equivalence between classification and conceptual structures by developing an algebraic, polar factorization approach to Galois connections, unifying key theories in knowledge representation.

Contribution

It introduces a polar factorization of Galois connections to unify classification and conceptual structures through an algebraic categorical framework.

Findings

Unified theorems for classification and conceptual structures

Developed polar factorization of Galois connections

Enhanced algebraic understanding of knowledge representation

Abstract

Truth refers to the satisfaction relation used to define the semantics of model-theoretic languages. The satisfaction relation for first order languages (truth classification), and the preservation of truth by first order interpretations (truth infomorphism), is a motivating example in the theory of Information Flow (IF) (Barwise and Seligman 1997). The abstract theory of satisfaction is the basis for the theory of institutions (Goguen and Burstall 1992). Factoring refers to categorical factorization systems. The concept lattice, which is the central structure studied by the theory of Formal Concept Analysis (FCA) (Ganter and Wille 1999), is constructed by a factorization. The study of classification structures (IF) and the study of conceptual structures (FCA) aim (at least is part) to provide a principled foundation for the logical theory of knowledge representation and organization. In an effort to unify these two areas, the paper "Distributed Conceptual Structures" (Kent 2002) abstracted the basic theorem of FCA in order to established three levels of categorical equivalence between classification structures and conceptual structures. In this paper we refine this approach by resolving the equivalence as the factorization of three isomorphic versions: relation, function and Galois connection. We develop the latter more algebraic version of the equivalence as the polar factorization of Galois connections. We advocate this abstract adjunctive representation of classification and conceptual structures.