Definition of the Epsilon Operator in Leśniewski’s Ontology

Abstract

The epsilon operator, introduced in the predicate calculus by D. Hilbert, is indefinable in this system. The paper presents the definition of this operator in Leśniewski’s ontology. In Leśniewski’s ontology we can distinguish:
1) the epsilon operator occurring in the expression ε_aP(a), defined by D1 and read: an a such that P(a),
2) the epsilon operator restricted to objects, occurring in the expression ε_a′P(a), (i.e. in the expression ε_a(aεa P(a))), defined by D1’ and read: an object a such that P(a) (or read: an object having the property P).

The sense of this second operator corresponds to the sense accepted in the predicate calculus, since in this system the nominal variables are individual variables for which we can substitute the individual names of objects of a given domain.

By virtue of the definitions D1, D1’ the basic theorems concening these operators are proved (T1-T4, T1’-T4’).

According to the remarks given at the end of the paper the definitions D1, D1’ can be formulated also in the standard form accepted in Leśniewski’s ontology.