Details

Example 2 (first snapshot): When is the relation of strong inclusion true—that is, ? Diagrams 1, 3, 9, and 10 are highlighted. So the following statements are true: Socrates ⊏ philosopher, Socrates ⊏ Socrates, Philosopher ⊏ a human, and Philosopher ⊏ philosopher.

So the diagrams act as definitions of relations. Since there are three types of names, there are three types of identities. The statement is true if and only if the object is the same as . So London = the capital of Great Britain is true. The statement is true if and only diagrams 1 or 9 present the semantic status of the names. So the following two statements are true: Socrates ◻ Socrates, man ◻ man. The statement is true if and only if one of the diagrams 1, 9, or 16 is highlighted. So Pegasus ◯ Pegasus is true, but Pegasus ◻ Pegasus is false.

Example 3: Suppose the objects are the natural numbers . Let "N" mean "natural number", "odd" mean "odd number", "even" mean "even number", and "" mean "the solution of the equation ". Then the following propositions are true: , , , , , and (because "1/0" and "2/0" are both empty names). The following propositions are false: , , and (both names are shared names).

"Learning the ontological vocabulary, like learning any other vocabulary, is an informal affair. Any method is good as long as it gives the required results. The meaning of ontological constants might be best explained by translating them into English. The ambiguity of ordinary usage, however, renders this method less effective than we might like it to be, and compels us to have recourse to the other devices." ([1, pp. 126].)

Leśniewski formulated ontology as an axiomatic theory based on the functor of singular inclusion as the sole primitive term with a single axiom written in Peano–Russell notation as:

.

The following definitions that correspond to the algebra of logic are also interesting:

( is an object if and only if is )

( is an object that does not exist if and only if is and it is not true that is )

( is non- if and only if is and it is not true that is )

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, 1991, pp. 203.

[3] Wikipedia, "Euler Diagram." (Mar 29, 2016) en.wikipedia.org/wiki/Euler_diagram.

[4] Wikipedia, "Stanisław Leśniewski." (Mar 29, 2016) en.wikipedia.org/wiki/Stanis% C5 %82 aw_Le % C5 %9 Bniewski.

[5] D. P. Henry, Medieval Logic and Metaphysics: A Modern Introduction, London: Hutchinson, 1972.

[6] L. Borkowski, Elementy Logiki Formalnej (in Polish), Warsaw: Polish Scientific Publishers PWN, 1976.