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The following description of Leśniewski's system for the foundation of mathematics is given in [5, pp. xiv]. It consists of three parts: protothetic, ontology, and mereology. These can be described, in general terms, as follows.

Mereology is a theory of part-whole relations.

Ontology is a theory of the copula (linkage) "is". It comprises the theory of predicates, of classes, and of relations, including the theory of identity.

Protothetic is the logic of propositional forms with quantifiers binding propositional and functional variables.

Three definitions of the notion of class were discovered by Kuratowski and Tarski [5, pp. 327].

In [6], Lejewski studied atomistic and atomless mereology using two notions:

The expression ( is an atom) means that is an object that has no proper parts.

The expression means that is an atom of .

In this Demonstration, 1×1 squares could be considered as atoms. The set of all open disks in the plane is an example of an atomless mereology.

References

[1] C. Lejewski, "On Leśniewsi's Ontology," in Leśniewski's Systems: Ontology and Mereology, The Hague: Martinus Nijhoff Publishers, 1984 pp. 123–148.

[2] E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: HarperCollins Publishers, 1991 pp. 203.

[3] Wikipedia, "Euler Diagram." (Apr 4, 2016) en.wikipedia.org/wiki/Euler_diagram.

[4] Wikipedia, "Stanisław Leśniewski." (Apr 4, 2016) en.wikipedia.org/wiki/Stanis% C5 %82 aw_Le % C5 %9 Bniewski.

[5] S. J. Surma, J. J. T. Srzednicki, D. I. Barnett, and V. F. Rickey, eds., Stanisław Leśniewski Collected Works, Volume I, New York: Springer, 1991.

[6] C. Lejewski, "A Contribution to the Study of Extended Mereologies," Notre Dame Journal of Formal Logic, 14(1), 1973 pp. 55–67.