Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a specific reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Leśniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated.

Leśniewski and Russell's Paradox: Some Problems

The main aim of this paper is to show that a natural extension of the predicate calculus with identity (hereafter LI) enriched with some logico-linguistically proper devices leads us in quite a natural way to the axiom of Lesniewski's Ontology, and that Ontology in turn implies the theses of the LI thus enriched. In addition the degrees of extensionality of the enriched LI will be discussed. In showing this, the Russellian notion of description appropriately modified will play an essential role. Indeed it is evident that, in its semantic intention, the axiom of Ontology is quite similar to the Russellian theory of description.1 Hiz [1977] shows in fact that the Russellian definition of description is inferentially equivalent to the axiom of Ontology in a sense to be specified2, namely for semantic categories higher than that of names. The present paper will show the logical fact that LI suitably enriched is inferentially equivalent to Ontology for the lowest semantic category as well. As a consequence Lesniewski's Ontology is not and should not be regarded as a system which is rather unique and specific, being isolated from the usually accepted logical framework. It is a logically natural extension of Frege-Russellian logical system with identity and description. It is something which could well have been proven before, but as far as I know it has not yet been proven. This is certainly because of the ontological prejudice which has dominated the main stream of logical analysis since the very time when it was created by its founders3.

Ontology as a Natural Extension of Predicate Calculus with Identity Equipped with Description

We propose an intuitive understanding of the statement: ‘an axiom (or: an axiomatic basis) determines the meaning of the only specific constant occurring in it.’ We introduce some basic semantics for functors of the category s/n,n of Lesniewski’s Ontology. Using these results we prove that the popular claim that the axioms of Ontology determine the meaning of the primitive constants is false.

/pdf/some-non-standard-interpretations-of-the-axiomatic-basis-of-40bnv7t0ag.pdf

Some non-standard interpretations of the axiomatic basis of Leśniewski’s Ontology

After describing basic intuitions underlying the system, I briefly point out how it differs from the classical logic. (The main difference is that there is no distinction between predicates and singular terms: there is only one class of name variables, and names can refer to any number of objects.) Then, I describe its language and its original axiomatization. I give some examples of definitions in Ontology to highlight the flexibility of its language. Next, I discuss Leśniewski’s argument against universals formulated in the language of Ontology, the relation between Ontology and Russell’s description theory and Leśniewski’s attempt to apply Ontology to a paradox concerned with four-dimensional objects. Secondary literature pertaining to Ontology is surveyed in the last section.

Leśniewski’s Ontology

Abstract This article is an attempt to recreate the intuitions which accompanied Leśniewski when he was creating his calculus of names called Ontology. Although every reconstruction is to some extent an interpretation, and as such may be defective, still, there are reasons justifying such reconstruction. The most important justification is the fact that both Leśniewski and his commentators stressed that ontology originated from reflections about ordinary language, in which sentences such as A is B appear in one of the meanings associated with them in Ontology, and that the users of the Polish language use such sentences accordingly and properly identify them. Assumed it is so, let us try, based on Leśniewski’s guidelines as well as comments and elaborations on Ontology (Leśniewski 1992: 364-382, 608-609; Kotarbiński 1929: 227-229; Rickey 1977: 414-229; Simons 1992: 244; Lejewski 1960: 14-29), to evaluate the accuracy of this approach, referring also to certain knowledge of the Polish language. To make it clear, this article is not about Ontology as a formal theory of language. It is solely an attempt to assess whether some syntactical constructs of the Polish language and this language’s properties are significant conditions of a proper understanding of Ontology, and whether Ontology is, in fact, in a relationship with the ethnic language of its author.

/pdf/on-the-formal-approach-to-describing-natural-language-notes-2w1oib67g1.pdf

H. Hiż

Papers

Descriptions in Russell's theory and in ontology