  The meaning or interpretation of the quantifiers is a more subtle question. It seems very likely that Leśniewski thought of the quantifiers as a notational necessity when a domain could be infinite, since infinite disjunctions and conjunctions were not possible. In Protothetic, as its computational variant makes clear, there is strictly no need for quantifiers, since each semantic category, no matter how high in the hierarchy, has only finitely many possible (extensional) values. But in Ontology, where there is no logical requirement that the domain of individuals be finite, the quantifiers are indispensable. One thing for sure is that they cannot be given an existentially committed reading, where ‘for some’ means ‘there exists’. The reason is that the laws of quantification and the definability in Ontology of a necessarily empty term ‘\(\Lambda\)’, for which it is true that no \(\Lambda\) exists, entail the truth of the quantified proposition ‘for some \(a\), no \(a\) exists’ (cf. T.127 of “Definitions and Theses of Leśniewski’s Ontology” in S. Lesniewski’s Lecture Notes in Logic). This would be contradictory if ‘for some’ meant ‘there exists’. Leśniewski always used the expression ‘particular quantifier’ rather than ‘existential quantifier’. So the question is how the quantifiers are to be understood if not in the standard way.