The Ontological Vocabulary
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This Demonstration uses the ontological table to explain the meaning of certain one-place and two-place functors of Leśniewski's ontology, for which a better name might be the calculus of names [6, pp. 116].[more]
This shows the hierarchy of name-like expressions [5, pp. 18].
The ontological table is an extension of Euler's circles. Euler's circles do not take into account unshared names ("Socrates", "the Moon", …) or fictitious names ("Pegasus", "centaur", …). A black disk represents the only object named by an unshared name; a circle represents the many objects named by a shared name ("man", "philosopher", …); no disk or circle is used in the case of a fictitious name. The ontological table consists of two subtables, for one name (three diagrams) and for two names (16 diagrams).
Here is how to use the ontological table in the case of a statement involving a relation between two names and . The second part of the ontological table highlights those of the 16 diagrams for which the relation is true.
Example 1 (thumbnail): The functor of singular inclusion occurs in expressions of the type (to be read: is ). A proposition of this type is said to be true if and only if diagram 1 or diagram 3 (of the second table) illustrate the semantic status of the names and .
The nearest natural-language correlate of the may be found in languages that have no articles—for instance, the Latin est of Socrates est philosophus ("Socrates is a philosopher") or the Polish jest of Kraków jest miastem ("Krakow is a town"). In English there are is sentences—for instance, Elizabeth is queen [5, pp. 21].
So the following propositions are true: Elizabeth ϵ queen (diagram 3 of the second subtable) and Warsaw ϵ the capital of Poland (diagram 1). But Pegasus ϵ Pegasus is false, since "Pegasus" is an empty name.[less]
Contributed by: Izidor Hafner (March 2016)
Open content licensed under CC BY-NC-SA
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 )
 C. Lejewski, "On Leśniewsi's Ontology," in Leśniewski's Systems: Ontology and Mereology, The Hague: Martinus Nijhoff Publishers, 1984 pp. 123–148.
 E. J. Borowski and J. M. Borwein, Collins Dictionary of Mathematics, New York: HarperCollins, 1991, pp. 203.
 Wikipedia, "Euler Diagram." (Mar 29, 2016) en.wikipedia.org/wiki/Euler_diagram.
 Wikipedia, "Stanisław Leśniewski." (Mar 29, 2016) en.wikipedia.org/wiki/Stanis% C5 %82 aw_Le % C5 %9 Bniewski.
 D. P. Henry, Medieval Logic and Metaphysics: A Modern Introduction, London: Hutchinson, 1972.
 L. Borkowski, Elementy Logiki Formalnej (in Polish), Warsaw: Polish Scientific Publishers PWN, 1976.