A Coq-based Axiomatization of Tarski's Mereogeometry

TL;DR

This paper presents a formal Coq-based axiomatization of Tarski's mereogeometry, aligning it more closely with Lesniewski's original mereology and providing a clearer, more concise foundation for spatial reasoning.

Contribution

It introduces a type-theoretical representation of mereogeometry in Coq, reducing the axioms needed and aligning the theory with Lesniewski's original ideas.

Findings

More clear foundational basis for mereogeometry

Reduction from four to three axioms

Full compatibility with Lesniewski's mereology

Abstract

During the last decade, the domain of Qualitative Spatial Reasoning, has known a renewal of interest for mereogeometry, a theory that has been initiated by Tarski. Mereogeometry relies on mereology, the Lesniewski's theory of parts and wholes that is further extended with geometrical primitives and appropriate definitions. However, most approaches (i) depart from the original Lesniewski's mereology which does not assume usual sets as a basis, (ii) restrict the logical power of mereology to a mere theory of part-whole relations and (iii) require the introduction of a connection relation. Moreover, the seminal paper of Tarki shows up unclear foundations and we argue that mereogeometry as it is introduced by Tarski, can be more suited to extend the whole theory of Lesniewski. For that purpose, we investigate a type-theoretical representation of space more closely related with the original ideas of Lesniewski and expressed with the Coq language. We show that (i) it can be given a more clear foundation, (ii) it can be based on three axioms instead of four and (iii) it can serve as a basis for spatial reasoning with full compliance with Lesniewski's systems.