[Paper Review] The Axiomatic Foundation of Space in GFO

Ringo Baumann, Heinrich Herre|arXiv (Cornell University)|Sep 23, 2011

Semantic Web and Ontologies50 references3 citations

TL;DR

This paper presents an axiomatic foundation for space within the General Formal Ontology (GFO) using first-order logic, formalizing phenomenal space through four primitive relations: spatial region, spatial part, spatial boundary, and spatial coincidence. The theory, called $β$-Theory ($\mathcal{BT}$), is inspired by Franz Brentano’s philosophy and establishes a rigorous, logically consistent framework for spatial entities, including the construction of an absolute Brentano space as an ideal limit entity.

ABSTRACT

Space and time are basic categories of any top-level ontology. They are fundamental assumptions for the mode of existence of those individuals which are said to be in space and time. In the present paper the ontology of space in the General Formal Ontology (GFO) is expounded. This ontology is represented as a theory BT (Brentano Theory), which is specified by a set of axioms formalized in first-order logic. This theory uses four primitive relations: SReg(x) (x is space region), spart(x, y) (x is spatial part of y), sb(x, y) (x is spatial boundary of y), and scoinc(x, y) (x and y spatially coincide). This ontology is inspired by ideas of Franz Brentano. The investigation and exploration of Franz Brentano's ideas on space and time began about twenty years ago by work of R.M. Chisholm, B. Smith and A. Varzi. The present paper takes up this line of research and makes a further step in establishing an ontology of space which is based on rigorous logical methods and on principles of the new philosophical approach of integrative realism.

Motivation & Objective

• To formalize the ontology of space in the General Formal Ontology (GFO) using rigorous axiomatic methods.
• To develop a logical theory ($\mathcal{BT}$) grounded in first-order logic that captures the intuitive structure of spatial entities based on Brentano’s ideas.
• To distinguish between phenomenal space (subject-dependent) and extension space (subject-independent), aligning with integrative realism.
• To provide a formal basis for spatial reasoning in top-level ontologies, supporting applications in AI, knowledge representation, and cognitive modeling.
• To explore the construction of an absolute Brentano space as a theoretical limit of increasing space regions, serving as a container for all spatial entities.

Proposed method

• Formalizing space using four primitive relations: $SReg(x)$ (x is a space region), $spart(x,y)$ (x is a spatial part of y), $sb(x,y)$ (x is a spatial boundary of y), and $scoinc(x,y)$ (x and y spatially coincide).
• Constructing the theory $\mathcal{BT}$ as a first-order axiomatic system to capture mereo-topological structure of space.
• Introducing the concept of absolute Brentano space ($\mathcal{ABS}$) via a limit construction of an infinite chain of space regions, where each region is a proper part of the next.
• Using axioms to ensure closure under mereological sums and to enforce spatial continuity and dimensionality (0D to 3D entities).
• Distinguishing pure space entities from material objects by introducing the notion of occupation between material boundaries and space regions.
• Introducing additional unary predicates like $ball(x)$ and $cube(x)$ to model standard spatial forms, with axioms capturing intuitive properties such as connectedness of mereological differences.

Experimental results

Research questions

• RQ1 How can space be formally axiomatized in a top-level ontology using first-order logic while preserving intuitive spatial structure?
• RQ2 What is the logical status of the absolute Brentano space as a theoretical limit entity, and how is it constructed from increasing chains of space regions?
• RQ3 How do the primitive relations $SReg$, $spart$, $sb$, and $scoinc$ capture the mereo-topological structure of space and its boundaries?
• RQ4 What is the relationship between phenomenal space (subject-dependent) and extension space (subject-independent), and how is this captured in the theory?
• RQ5 How can standard spatial forms like balls and cubes be formally introduced and characterized within the mereotopological framework?

Key findings

• The theory $\mathcal{BT}$ is proposed as a consistent, first-order axiomatic system for the category of phenomenal space, grounded in Brentano’s philosophy and integrative realism.
• The absolute Brentano space ($\mathcal{ABS}$) is constructed as a theoretical limit of an infinite chain of space regions, where each is a proper part of the next, and $\mathcal{ABS}$ contains all space regions as proper parts.
• The theory supports a four-partite spatial universe with 0D (points), 1D (lines), 2D (surfaces), and 3D (regions), with a subtle distinction between lower-dimensional entities and boundaries.
• The theory is undecidable if consistent, but sub-theories $\mathcal{BT}(0)$, $\mathcal{BT}(1)$, and $\mathcal{BT}(2)$ are believed to be decidable, with some complete characterizations possible (e.g., a line with two endpoints).
• The mereological sum of all balls within a topoid is not necessarily unique, and two topoids with identical balls may differ in boundary structure, indicating non-trivial mereological complexity.
• The introduction of standard forms like $ball(x)$ and $cube(x)$ with axioms (e.g., connected difference between two balls) allows for formal modeling of intuitive spatial morphology within the mereotopological framework.