Leśniewski and Russell's Paradox: Some Problems

Abstract

Sobociński 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.

Acknowledgements

Work on this paper was supported by the Killam Graduate Scholarship for years 2006–2008 (at the University of Calgary). I owe gratitude to Richard Zach and Nicole Wyatt (both from the University of Calgary) for helpful comments.

Notes

The English translation is to be found in Srzednicki and Rickey 1984.

Stanisław Leśniewski (1886–1939) was a Polish logician with nominalistic inclinations who constructed his own logical systems as an alternative to the system of Principia Mathematica. His systems consist of three parts: (i) the propositional part named Protothetic, (ii) a theory of predication called Ontology, and (iii) a formalized theory of parthood, called Mereology. Almost all his papers have been translated to English and are available in two volumes as Surma et al. 1991. I will refer to Leśniewski's papers by the date of original publication, but all page references to Leśniewski's papers employ the pagination of the English edition.

Below is the list of Leśniewski's papers together with English titles and reference to their original place of publication. First, there are papers from the period 1911–1916:

Table

The 1916 paper was the last one written in the early period, and also the first that dealt with the issues discussed in details in the series of papers between 1927 and 1931, titled ‘On the Foundations of Mathematics I–XI’. The list of parts of this series together with their references is:

Table

While publishing this series, Leśniewski also published two papers which were more concerned with specific mathematical topics rather than with foundations of mathematics. ‘On Functions whose Fields with respect to these Functions are Abelian Group’ 1929c and ‘On Functions Whose Fields, w.r.t. These Functions are Groups’ 1929d simplify axiomatizations of two mathematical theories (Abelian group theory and group theory respectively).

In 1929 Leśniewski also published a paper concerned with a description of Protothetic – ‘Fundamentals of a New System of the Foundations of Mathematics, §1–§11’ 1929a.

In ‘On the Foundations of Ontology’ 1930b he presented an outline of his Ontology and in ‘On Definitions in the so-called Theory of Deduction’ 1931b he formalized his requirements put on definitions in his systems.

In 1938 he was preparing for publication in Collectanea Logica (a Polish journal which never came into being, mainly because of the Second World War) two papers: ‘Introductory Remarks to the Continuation of my Article: Fundamentals of a New System of the Foundations of Mathematics’ 1938a, where he elaborates on how he had formulated Protothetic and explains some of its basic principles, and ‘Fundamentals of a New System of the Foundations of Mathematics §12’ 1938b where he hints at another, so-called computative formulation of Protothetic. Those were never published, but the offprint copies survived in the Harvard College Library.

Another invaluable source is a set of notes made by Leśniewski's students. The majority of the text consists of formal symbols, theorems and proofs. The notes have been gathered and edited in Srzednicki 1988. There are two parts to that book: (a) foundations of mathematics, (b) Peano arithmetic and Whitehead's theory of events. The first part is divided into: ‘From the Foundations of Protothetic’, ‘Definitions and theses of Leśniewski's Ontology’, ‘Class Theory’. The second part consists of: ‘Primitive terms of arithmetic’, ‘Inductive Definitions’, ‘Whitehead's Theory of Events’.

Here Protothetic is reconstructed mainly on the following basis: Tarski's dissertation, the first eleven paragraphs of Leśniewski's Fundamentals of a New System of the Foundations of Mathematics Leśniewski 1929a (the twelfth paragraph being prepared for publication in 1938 survived as well) and some lecture notes taken by his students (edited by Srzednicki and Stachniak in their 1988). The basic collection of secondary literature on Protothetic is Srzednicki and Stachniak 1998. It contains reprints or English translations of: Simons 1993, Rickey 1976, Tarski 1923, Sobociński 1949a, Słupecki 1953, Sobociński 1961, Rickey 1973 (which is a continuation of Rickey 1972) and Le Blanc 1985. Tarski's research on functional completeness of Protothetic was continued in Sobociński 1949a, where other forms of various definitions are discussed. Sobociński's paper is also interesting for historical reasons. In the introduction of this paper he describes the fate of Collectanea Logica in 1939.

The game of finding the shortest possible axiomatization of Protothetic, already initiated by Leśniewski, was continued for quite a few years. Sobociński 1954b and Sobociński 1961 are good surveys of the result. Sobociński 1961 also gives an interesting metatheorem which states a sufficient condition for a set of formulas to be an axiomatization of Protothetic. Le Blanc 1985 provides another, shorter version of the axiomatization.

One of the best introductions to Protothetic so far is Słupecki 1953. The formulations that Słupecki focuses on are mainly conditional. His explanations of the rules of inference are somewhat handwavy, but he provides many examples of simple proofs which allow the reader to get an idea as to what rules he is using. He also discussed some interesting theorems that can be formulated in elementary Protothetic (e.g. the principle of bivalence, which normally is expressed in the metalanguage of classical propositional logic), which he also proves to be provably equivalent in elementary Protothetic (for one-place function symbols), at least, but a generalization of his results is to be found in Chikawa 1967.

Rickey 1972 and 1973 provide a really good insight into the nature of Leśniewski's terminological explanations, with special attention to how inscriptional description of the rules of Protothetic is to be understood.

Leśniewski's idiosyncratic wheel-and-spoke notation is interesting for at least two reasons: (i) symbols for propositional one- and two-place connectives encode their truth-tables with their own shape, (ii) geometric relations between different symbols correspond to various logical relations between the functions that they express. Recently it has been shown that there is a simple algorithm that extends this method to arbitrary first-order classical connectives (Urbaniak 2006b).

A monograph on the so-called ‘New Protothetic’ is also worthy of attention (López-Escobar and Miraglia 2002), but it focuses mainly on intuitionistic formulations. Another group of formulations of Protothetic, hinted at by Leśniewski, is called computative Protothetic, see Le Blanc 1991. An interesting research direction is many-valued Protothetics (see Scharle 1971 and Watanabe 1973). There is a connection between Protothetic and Henkin's theory of propositional types given in Henkin 1950 and 1963. Grzegorczyk 1964 shows that every one of Henkin's equality symbols is definable by means of relational abstraction and the equality symbol for the usual material equivalence (although still the set of axioms is denumerable). Meredith 1951 provides also a similar system, commented on in Arai and Tanaka 1966.

Actually, one can get the same result as if complex expressions could be used, introducing simple expressions ‘equivalent’ to the complex expression by means of definitions. Thus, the restriction of the substitution class to simple expressions is not an essential feature.

The order of a functor (connective) is defined by two conditions:

1. Every functor whose all arguments belong to the category of propositions, is a first-order functor.

2. Every functor whose at least one argument is of category n − 1 and no argument is of order n or higher, is an n-order functor.

The most important collection of secondary literature related to Ontology (and Mereology) is Srzednicki and Rickey 1984. It contains English translations of Kruszewski 1925 and Sobociński 1949b. Lejewski 1954b, Słupecki 1955, Lejewski 1958, Canty 1969, Iwanuś 1973, Sobociński 1954a, Clay 1966, Lejewski 1969, Clay 1970, Clay 1974a are also included.

The simplest subsystem of Ontology we get when we restrict ourselves only to name variables and we do not admit any quantification. A general account and metatheory of calculi of this kind can be found in Pietruszczak 1991. Quite independently, a similar subsystem of elementary Ontology called L ₁ was studied in Ishimoto 1977, 1982, 1997.

More general systems are generated by introducing quantifiers binding name variables only. Some authors call the subsystem of Ontology which contains only this sort of quantifiers ‘elementary Ontology’. Some other authors use ‘elementary Ontology’ to refer to a rather stronger system which admits also second-order variables and quantifiers binding them. Elementary Ontology is discussed especially in Iwanuś 1973 where it is also shown equivalent to the atomic classical Boolean algebra and the classical algebra of sets. Iwanuś also provides an interesting proof of decidability of Elementary Ontology.

Elementary Ontology is also discussed in the first part of one of the best surveys on Ontology (Słupecki 1955). Słupecki discusses also some aspects of non-elementary Ontology, which is obtained mainly by adding the rule of extensionality. He proves that definitions in the system satisfy the condition of translatability, i.e., for every φ which contains a defined term there is a provably equivalent expression φ′ which does not contain this term. Interestingly, definitions are nevertheless creative (but they can be made non-creative by introducing two axiom schemata). Słupecki also sketches the proof of the theorem that Ontology is consistent relative to Protothetic.

A specific type of functional completeness of Elementary Ontology with respect to ε has been proven in Urbaniak 2006a.

Lejewski 1958 is intended as a survey of Ontology. It contains a more elaborate intuitive explanations of the intended meaning of epsilon and some other connectives definable in Ontology. Some research has been done on replacing epsilon with another functor which would be more intuitive for English native speakers. Lejewski 1977 has shown that the functor of weak inclusion can serve the same purpose as epsilon.

Various equivalent axiomatizations of Ontology are discussed in Sobociński 1934.

Miéville 2004 is a recent text intended as an introduction to Ontology.

Canty 1969 contains an explanation of the relation between the epsilon, the distributive predication, and higher-order epsilon connectives, although the discussion is quite informal.

Urbaniak 2006c proves that in set-theoretic semantics the classical Ontology has at least ℵ₀ of non-standard models with a non-standard ε.

Kubiński 1969 argues that Ontology is absolutely non-categorical (i.e. that no set of formulas of Ontology defines a class of models up to isomorphism).

Munitz 1974 is an interesting book that covers some differences in expressing existence in various logical systems, Ontology included. Existential assumptions and commitments as expressed in Ontology were, it seems, first discussed in Lejewski 1954b. As contrasted with something intuitively false in a first order language:

it is provable in Ontology that:

that is, that for some a, a does not exist. This, first of all, indicates that in Ontology (in contrast with Quine's approach) the existential commitment is not expressed by quantification. An interesting commentary to Lejewski is Kearns 1969.

Prior 1965 argues that variables in Ontology represent class names rather than singular or empty or common terms. Sagal 1973 criticizes this paper.

Since the quantifiers in Leśniewski's systems do not (or at least, were not intended to) express ontological commitment, a perfectly legitimate question arises as to what those quantifiers were supposed to do and how we are to interpret variables that occur in the wffs of the system. For instance, there is some similarity between Leśniewski's account and that presented in Boolos 1985. There are of course some differences. For Leśniewski, quantification does not express existence (i.e. ∃_αφ, philosophically is not to be meant as stating that there exists an a such that φ). For Boolos individual quantification commits one to individuals and the higher-order quantification in the standard interpretation commits one to certain kinds of abstract entities, but monadic higher-order quantification interpreted as relating to plurals commits us to only whatever individuals intuitively have to exist to make the relevant sentence true. Also, Leśniewski assumes that the category of names contains singular names, empty names and plurals indiscerningly. Boolos treats plurals as a group in its own right. He has a syntactic distinction between two kinds of nominal variables. Some similarities and dissimilarities between Leśniewski and Boolos are informally discussed in Simons 1997.

The question of ontological commitment of Ontology is also raised in Simons 1995. Simons suggests a different criterion of ontological commitment and argues that Ontology has no ontological commitments. His claim that Ontology is valid if the domain of individuals is empty is certainly true. However, this does not seem to answer exactly the question whether it is free of any commitments. In his proof, the truth-conditions of higher-order statements tacitly do much of the heavy lifting, and the question whether their use does not involve any commitment has not been successfully answered. Simons attempted to provide some hints in 1985b, where he suggests his ‘combinatorial semantics’ in which name variables (are supposed to) range over ‘ways names can name’, but it is not a technically precise account and this approach has not yet been fully developed and assessed. It must be admitted, however, that he poses an interesting problem: how should one construct a formal semantics for Leśniewski's logic which performs the task a formal semantics normally performs, but does not assume explicitly or implicitly the existence of abstract objects?

Küng and Canty 1970 argue that the quantification in Ontology is not objectual (referential). The quantification in a given language is called referential (or objectual) with respect to a given semantics of this language if the truth in a model of quantified statements is dependent upon there being (or not) some objects in the domain of this model satisfying (or not) the formula resulting from the initial formula by deleting the quantifier. Or, in other words, in the objectual interpretation variables in a model are assigned objects in the domain of this model. The objectual reading is often opposed to the substitutional reading, where variables are rather associated with a substitution class – the class of expressions (which do not belong to the domain of the model), and the satisfaction of a formula in a model is defined via some results of substitution being true (or not). Technically speaking, Ontology as an axiomatic system can be given both an objectual and a substitutional interpretation. Presumably, the question of whether quantification in Ontology is objectual or referential boils down to something like: ‘which interpretation is in accordance with Leśniewski's views?’ Küng and Canty 1970 claim that the interpretation should be substitutional (although they admit that variables can be interpreted as referential over a realm of sets associated with a domain of objects). Kielkopf 1977 criticizes the view in its whole generality and introduces some other subtle interpretations of quantifiers which are neither simply substitutional nor obviously referential. Küng 1977 applies so-called prologue-functors (developed in Küng 1974) like ‘whatever extension the inscriptions equiform with the following item are taken to have … the following is asserted …’ to account for quantification in Leśniewski's systems. Whether this reading is nominalistic depends on how the notion of extension is understood, and providing a nominalistic explanation of what an extension is is not an easy matter.

A very interesting approach to semantics of Ontology has been taken by Rickey 1985. Quantified name calculi are interpreted in a first-order two-sorted models. One sort is the sort of individuals, another sort is the sort of names. A binary relation on the Cartesian product of those two sorts is added, which intuitively corresponds to the relation of naming or referring to.

Definite descriptions as expressed in Leśniewski's Ontology have also been studied. Hiż 1977 proved that the PM definition of description (⋆14.01) when formulated in the language of Ontology is in Ontology provably equivalent to the axiom of Ontology. Another discussion of Russell's theory of definite description from a Leśniewskian point of view is Lejewski 1960. Russell responded to it in 1960.

Creativity of definitions is an interesting subject in its own right. For example, Myhill 1953 constructs a version of arithmetic where some definitions are creative (he does not do it in Ontology, though).

It is argued that the system of free logic FL ^t is translatably equivalent to L4′ – a first-order modification of elementary Ontology in Lambert and Scharle 1967. The authors suggest that their translation ‘Provides for the first time … a way of interpreting at least the first order fragment of one version of Leśniewski's system called Ontology in more general parlance’. Simons 1981 correctly criticizes this claim, where he points out that L4′ (i) is not a subsystem of Ontology, because it has an additional axiom requiring that no term be general, and (ii) even if it were a subsystem, it would be a narrow part of Ontology and it is false that standard free logics provide a more general parlance to speak of Ontology. Interesting modifications of Ontology that allow to embed some free logics are developed in Simons 1985a.

Grzegorczyk 1955 compares Ontology to Boolean algebra and points out that Ontology formally resembles the theory of complete atomic Boolean algebras with constants and functions of an arbitrary high type. As Iwanuś 1973 (p. 202) points out this claim is only true to a certain extent. E.g. Boolean algebras are not provided with a rule of definitions, so the statement should be conditional: if we extend the theory of Boolean algebras with a good account of definitions (which would pretty much be a translation of Leśniewski's rules), then there is a correspondence. Also, the proof has been given only for elementary Ontology. Interestingly, it was Leśniewski's research that inspired Tarski 1935, which played an essential role in the development of Boolean algebras.

Stachniak 1981 develops a strengthening of Ontology, L _DF and a semantics for this calculus, where models are taken to be atomic Boolean algebras enriched by a specific set of functions and relations. Stachniak uses a fairly standard method of constructing models from constants to prove the completeness and compactness theorems. He also discusses the ultraproduct construction of models for Ontology. Definitions in L _DF are not creative.

Lebiediewa 1969a extends Ontology by introducing modalities (it is a paper based on her PhD dissertation Lebiediewa 1969b).

There is also an interesting modification of Ontology which is meant to deal with vague terms (Kubiński 1958). Kubiński 1959 developed also a modification of Ontology that is intended to deal with apparent contradictions of the form ‘a is b and a is not b’. Unfortunately, both papers are available in Polish only. Kubiński 1960 extends Ontology with means to express something like ‘a is more like b than like c’.

An ontological version of the axiom of choice is independent of Ontology. Kowalski 1977 studies the extension of Ontology obtained by adding this axiom, and Davis 1975 discusses various formulations of this axioms.

An interesting extension of Ontology is constructed when we add an axiom of infinity. When it is done, a theory that corresponds to Peano arithmetic is derivable. Gödel's theorem applies to this extension (Canty 1967).

Arguably, if semantics for Ontology is construed substitutionally, Ontology will turn out not to be a free logic, because the validity of its theorems would require that the substitution class(es) be non-empty. The claim that interpretation of quantifiers in Ontology has to be substitutional is a philosophical position which neither has a clear textual support nor is shared by all Leśniewskian scholars. Whatever the interpretation of quantifiers in Ontology one prefers, Ontology is a free logic at least in the following sense: theorems of Ontology hold even if nothing that can be in the language of Ontology called an object exists.

Historically speaking, Leśniewski used only the universal quantifier.

The original term is semantic category. Nowadays the term syntactic category is more common. The term which Leśniewski used when he formalized his theory of semantic categories will be preferred.

The only requirement is that those categories have to be previously introduced.

Although note that actually it doesn't even require that a be a set in order to be ⋆.

In this paper, for the sake of simplicity, only proof sketches are given. They are sufficient for reconstructing the full-blown proofs. Also, first three theorems in this paper gave been proven by Sobociński. Nevertheless, the proofs are interesting enough to at least go over them briefly.

In proofs all consequence-related expressions like ‘implies’, ‘follows’ etc. are taken to refer only to provability in Ontology.

I mean the discussion in Leśniewski 1927.

Indeed, given that ‘is’ is read as ε, if a does not name anything, it is false for any b that aεb. Moreover, if one uses name negation, it is also false that aε not − b.

A few remarks on secondary literature. Grzegorczyk 1955 argued that the models of Mereology and the models of complete Boolean algebra with zero deleted are identical. This claim has been criticized in Clay 1974a on the basis that the system discussed by Grzegorczyk was not Leśniewski's Mereology. Basically, Grzegorczyk's weakening seems to consist in restricting Mereology to a first-order theory with individual variables, instead of setting it in a framework of the full-blown Ontology. Clay 1974a also shows that to prove that complete Boolean algebra with zero deleted is Mereology, the definition of Boolean algebra with zero deleted has to include the condition that it does not have to include nonzero elements. Asenjo 1977 points out that the comparison works only if Mereology is taken to be atomistic.

A simplification of one of the axiomatizations of Mereology is to be found in Clay 1970, where it is shown that one of the axioms was redundant. Various axiomatizations of Mereology were discussed in Lejewski 1954a, 1955, 1962.

Sobociński 1954a is a decent introduction to Mereology. A variety of results pertaining to Mereology can be found in a PhD dissertation by Clay 1961.

Asenjo also discusses the historical role that Leśniewski's Mereology played in the development of non-classical set theories, such like that developed in Asenjo 1965.

Słupecki in his 1958 suggests that Mereology is formally nothing more than a particular elementary theory of partial ordering. However, he formulates it with Chwistek's simple theory of types as underlying logic, which as Clay 1974a points out, ‘fails to include Ontology or any other theory of the distributive class to use as counterpoint for the notion of collective class.’

Interestingly, Słupecki 1958 also attempts to reconstruct natural number arithmetic in Mereology. Clay 1965 is relevant to this issue: he proves that if the condition of weak discreteness is satisfied, the collective and distributive classes behave similarly in terms of equinumerosity.

Kubiński 1968 remarks that Kl_M(a) may be taken in the algebraic interpretation to the least upper bound of the a's.

The consistency of Mereology is proved relative to real numbers in Clay 1968. The domain is taken to be the set of all real numbers whose decimal expansions contain only zeros and ones with the exception of the number 0. The relation a is a part of b is taken to mean that every position where a has a one in its decimal expansion, b has it too. Another consistency proof is in to be found in Lejewski 1969. Another type of models (non-empty regular sets of a topological space) is discussed in Clay 1974b.

The interplay between the partition of a whole into parts and the attribution of properties to an object is the subject of Lorenz 1977.

Clay 1972 proves that if a is finite then Kl(a) is finite. In 1973 he shows that there is a simpler definition of class which does not require other defined terms.

At least two straightforward modifications of Mereology are available. First, it is possible to add an axiom stating that every object is either an atom or is constructed from those atoms which are its parts, or one can add an axiom stating that no atoms exist. Atomistic Mereology is studied in Sobociński 1971. Both atomistic and atomless Mereology are axiomatized in Clay 1975.

A weaker version of Mereology, but inspired by Leśniewski has been developed in Goodman and Leonard 1940. Richard Milton Martin in his 1988, 1992, also attempted to develop various applications of Mereology.

Note that this definition is just (3.2).

Of course some people reject this view, claiming that different mereological fusions are obtained from the same constituents depending on what is the structure of the whole. In response it can just be said that they do not have the Leśniewskian notion of parthood in mind.

This does not constitute any practical problem, since a person who'd believe that there are at most two different objects would be quite a rare specimen.

Of course, this notion is defined in Ontology, as applying to name variables or constants – the question to what extent mathematics can be founded on Ontology lies beyond the scope of this paper. The point here is only that mereological sets are indeed much different from the sets postulated in, say, ZF.

In the introduction to his first paper on mereology (1916) Leśniewski wrote: ‘The present work is the first link in an extended series of works, which I intend to publish in the near or distant future, desiring to contribute as much as possible to the justification of modern mathematics … The arrangement of definitions and truths, which I established in the present work dedicated to the most general problems of the theory of sets, has for me, in comparison to other previously known arrangements of definitions and truths (Zermelo, Russell, etc.) this advantage that it eliminates the ‘antinomies’ of the general theory of sets without narrowing the original domain of Cantor's term ‘set’ …and on the other hand, it does not lead to assertions which are in such startling conflict with intuitions of the ‘commonalty’…’

It is important to emphasize that the question I am concerned with is not whether there is a way of introducing into the language of Ontology a formal counterpart of the expression ‘the class of’ which would mimic correctly Leśniewski's understanding of this term (which was quite unusual), but rather how the expression ‘the class of’ in the nowadays most common, set-theoretic use can be imitated in the language of Leśniewski's systems. Thus, even if Leśniewski would not share the intuitions behind the examples above, it does not matter. The point is that the sense of ‘the class of’ in which the above examples come out true seems to be a legitimate and widespread understanding of the expression in question and mereology falls short of accounting for this plausibility.

Leśniewski would probably say that the only irreducible sense of the word ‘class’ is that formalized by mereology. There is an intriguing comment on Leśniewski's philosophical method made by Twardowski:

In general, those who follow Leśniewski, very arbitrarily demand an analysis where they find it convenient; however, whenever someone demands an analysis where it is inconvenient, they refer to intuition. And when the opponent in the discussion tries at some point to refer to intuition as well, they respond: ‘We cannot understand what you claim to be intuitively given’. (K. Twardowski's Diary, ms. 2407/3).

[‘W ogóle Ci, co postępują według wzoru Leśniewskiego, bardzo arbitralnie domagają się analizy tam, gdzie im to dogodne; gdy się jednak ktoś od nich domaga analizy tam, gdzie im to niedogodne, powołują się na intuicję. A gdy przeciwnik w dyskusji próbuje kiedyś również powołać się na intuicję, odpowiadają: ‘Nie rozumiemy tego, co według Ciebie ma być intuicyjnie dane.’’]

The quote comes from Kazimerz Twardowski's archive, located in the library of the Institute of Philosophy and Sociology of the Polish Academy of Sciences, Warsaw. The reference is to signatures in this collection.

In contexts like this one, by ‘false’ I mean ‘disprovable in Ontology or Mereology’ (the contexts determines which system is meant).

Wait a second! Is not our effort pointless? Is there really a need to introduce Kl? Is not Ontology a set theory already? Arguably, if one constructs a standard set-theoretic semantics for Ontology (name variables range over the power set of the domain of individuals etc.), something very much like the axiom of choice can be expressed in the language of Ontology by:

which in the set-theoretic interpretation can be read as stating that there is an f such that for every non-empty set b, f(b) is an element of b (see Davis 1975 for details).

Davis himself uses set-theoretic paraphrases throughout his paper, commenting briefly:

Set theoretic paraphrases will be used throughout this investigation since they are the most natural to use in discussions concerning the axiom of choice. (Davis 1975, p. 182).

This line of reasoning is familiar. It resembles the standard argument to the effect that the standard second-order logic is set theory.

The fact that set-theoretic paraphrases are considered natural in a discussion does not imply that the paraphrased language indeed says something about sets. At most, one can be inclined to say that if it is assigned the standard set-theoretic semantics then it is possible to formulate in it formulas that are true in a model iff a specific set-theoretic axiom (formulated in metalanguage) holds in the same model.

This, in a sense, indicates that at least some facts usually stated in set theory may be taken to be expressible in the language of Ontology. The problem is that this does not answer the initial problem. Even though formulas of Ontology may (in the standard set-theoretic semantics) have the same satisfaction conditions as certain set-theoretic claims, the language of Ontology itself still does not contain a systematic device which would emulate the behaviour of the usual set-theoretic jargon. I grant that there is a certain interesting translation along these lines:

Still the language of Ontology is far from the generality enjoyed by the language of set theory. Let us take a closer look at (6.2). Take the common-sense set-theoretic semantics. We start with a possibly empty set of urelements, then construct the power set to obtain the range of name variables, and similarly for all other semantic categories. In this setting (6.2) says only that for any non-empty set of urelements there is a function that selects an element of that set. This is not the same as saying that for any nonempty family of pairwise disjoint nonempty sets there is a function which assigns to each element of that family a unique element of the input set. Of course, in Ontology we can emulate ‘nonempty family of pairwise disjoint nonempty sets of urelements’ and similarly for any other ‘type’ or ‘order’ of the usual set-theoretic objects. But there is no single formula of Ontology which is true in a model if and only if the set-theoretic axiom of choice in its full generality is true in it. We have to introduce an axiom-of-choice-like formula for pretty much every semantic category separately. Now, since the language of Ontology has no upper limit on how complex semantic categories can get, this means that the best we could do is to suggest a choice schema under which all those particular choice axioms would fall. But this is quite different from being able to express the axiom by means of a single formula of set theory.

Formulae (7.2)–(7.6) were given and the whole approach was suggested in Lejewski 1985 and advertised in a different context in Henry 1972 (pp. 42–44). Henry, after introducing the above formulae has argued that

is not a theorem. He was also quite optimistic about his conclusions: ‘This gives one acceptable sense to the principle that there are classes of whatever sorts of objects one may specify. (Here ‘are’ has its higher-order sense.)’ Henry 1972 (p. 44). After giving (7.1) and denying it, he comments: ‘This is a contradiction which may be viewed as a version of Russell's paradox. It is of course avoided… In other words, intuitive clarity can be maintained throughout, this version of the paradox is avoided, and ad hoc evasive strategems of a formalist nature shown to be needless.’ (Henry 1972, p. 46). The rest of this section pursues some consequences and issues that (I think) arise from the consideration of these definitions, purporting to show that the issue is not as simple as Henry suggested.

What makes a connective a higher-order epsilon is an interesting question. The intuition is that f is an epsilon if the truth of f(δ, γ) requires some sort of uniqueness of the referent(s) of δ (presumably, up to coextensivity), and some sort of inclusion between the referent(s) of δ and the referent(s) of γ. This however is far from providing a formally correct and precise definition.

Consider e.g.