Full Idea
The standardly accepted formal theory of part-whole is classical extensional mereology, which is known in two logical guises, the Calculus of Individuals of Leonard and Goodman, and the Mereology of Lesniewski.
Gist of Idea
Two standard formalisations of part-whole theory are the Calculus of Individuals, and Mereology
Source
Peter Simons (Parts [1987], Intro)
Book Reference
Simons,Peter: 'Parts: a Study in Ontology' [OUP 1987], p.1
A Reaction
Simons catalogues several other modern attempts at axiomatisation in his chapter 2.