The way of truth

THE WAY OF TRUTH

JERZY PERZANOWSKI

July 1992

Abstract: In the framework of qualitative ontology the standard Parmenidean Identity Principles are scrutinized. A being is defined as a subject of qualities. Five conjugate notions (a nonbeing, the being, the nonbeing, Being and Nonbeing) are also defined.
By application of classical logic the following contribution to the controversy concerning being and nonbeing between Plato and Parmenides is made: Beings are. Nonbeings aren’t. The being is. The nonbeing is. Being is. And, Nonbeing is.
Finally, two basic abstracting operations: collection and unification, used frequently in ontological definitions, are discussed in a way leading to axiomatization of collections and ideas. Several theorems concerning the universe with ideas are given,including the claim that the universe is infinite.

Content

1. Introduction
2. Beings, the Being and Being
3. Ontological Connection
4. Towards a Theory of Ontological Connection
Language
Logical Axioms
The Basic Ontological Definitions
Objectiva
Collections
Ideas
5. Some Classical Ontological Questions
6. A Linguistic Intermezzo
Some Etymology
Comparative Grammar: Greek, Latin, English, German, Polish
Instead of a Conclusion
7. An Outline of a Primitive Theory of Being – PTB
The Theory
Results
Models
Comments
An Alternative Approach
8. Towards an Extended Theory of Being – ETB
Collections and Ideas
Empty Ideas
Connections of Abstract Concepts
Identity
Qualities of Abstractors
ETB
Supports and Forms
Infinity
Final Remarks
9. Parmenidean Statements Reconsidered and Classical Questions
Answered
10.Summary

Acknowledgments
References

THE WAY OF TRUTH

JERZY PERZANOWSKI

1. Introduction

1.1 The Parmenidean “way of truth” concerns what there is and what there is not:
estin te kai os ouk esti me einai [1]
It concerns the basic ontological items: beings and nonbeings, as well as (the) being and (the) nonbeing.
As we have learned from Parmenides, Zeno and Plato [2] the way of Parmenides is the way of difficult truth, the way of metaphysical paradox.

1.2 Quite often the principal truth of Parmenides is formulated as the ontological principle of identity: being is and nonbeing isn’t. Usually this principle is considered tautologous [3] or even trivial.
I disagree. Triviality presupposes clarity. The principle,hovever, is neither clear nor evident. Also it is not obvious.
Is it true?

1.3 Both “being” and “is” are immediate derivatives of the verb “to be”. The verb itself has several variants.
Can all these derivatives and variants be presented in a uniform way? Is, for example, “Being is” a more adequate expression of the thought of Parmenides than “Whatever is, is”?
Next, to which items does the Parmenidean statement refer: to particular beings – like me, you, a ship, this pencil; or to their totality – the being; or to their unity – Being? Should Parmenides’ statement be understood as “the being is and the nonbeing isn’t”, or rather as “a being is and a nonbeing isn’t”,i.e., “any being is and no nonbeing is” or “beings are and nonbeings aren’t”?

1.4 The problem was pointed out and discussed by Plato in Sophist as the crux of his refutation of the sophistic claim that nothing is a false.
Parmenides’ spokesman, the Eleatic Stranger, is arguing there for Plato’s conclusion that “nonbeing has an assured existence and a nature of its own”, recalling at the same time the warning of Parmenides: “For never shall this thought prevail, that non-beings are, but keep your mind from this path of inquiry”. [4]

1.5 The answer to Plato’s problem clearly depends on an explication of the four notions involved: being, nonbeing, is and isn’t.
From a metalogical point of viev it is also determined by the related logics: the logic of our reasoning and an appropriate logic of being.

1.6 Hereafter, the ontological notions are explained according to the qualitative approach to the notion of being: a being is a subject of some qualities; the being is the totality of all beings; Being is the unity of all beings.
These, quite ancient but yet obscure, formulas are crucial for traditional ontology and they therefore deserve clarification.
Such a clarification needs an appropriate theory of qualities, as well as a suitable theory of ontological connection connecting qualities with subjects. It is the latter, above all, which will be outlined in the present study.

1.7 Clarification comes, inter alia, through formalization.
Formalization needs logic. In what follows I rely exclusively on classical logic. To be more exact, standard classical logic is used as the logic of reasoning, whereas a suitable applied version of classical logic will serve as our logic of being.

1.8 In what follows a very general theory of ontological connection is provided.
In spite of its generality this theory enables us, as you will see, to reconsider the classical ontological claims of Parmenides and to refute an anti-ontological claim that the notion of being is syncategorematic.
Also certain ontological theorems will be proved, including: Being is and Nonbeing is (sic!). A being is, wheras a nonbeing isn’t. Also: whatever is, is – which is shown to be equivalent to Whatever isn’t, isn’t.

1.9 The paper is organized as follows: I start with general remarks concerning ontology and different approaches to the notion of being. Next, several classical questions of traditional ontology are discussed. After making our problems clear, I will introduce a formalism enabling us to study them in their full generality. Finally, the results of the paper are discussed in a manner introducing perpectives for a subsequent theory of qualities.

2. Beings, the Being and Being

2.1 Ontology is the discipline of being. It is the theory of what there is, why and how.
As the tradition makes clear, the verb “to be” used here is ambiguous. It refers either to the domain of existing objects , depending therefore on an appropriate theory of existence, or to what really exists (in metaphysical sense) – to logos which is behind existing items and behind the facts. The latter realm emerges when emphasis is placed on the second part of the above definition,i.e., on the questions: why and how?
The answer we are looking for is of the form: there is x because x is possible and in addition for existing objects x satisfies certain specific conditions of existence. Possibility is frequently explained as a matter of consistency or coherence, whereas existence conditions are specified in terms of stability, homeostasis, actualization, etc.

2.2 Ontology is distinguished by its extremal generality and by the richness and fertility of its basic notions.
The most basic ontological notions are notions of a particular being and the being. Both notions can be approached in at least three ways: possibilistically, connectionally (or qualitatively), and through what we shall call verb-type-ontologies.

2.3 In the verb-type-approach both notions – of a being and the being – are obtained from the verb “to be” by transformations and nominalizations.
The theory behind these latter is quite complicated, much more than is the grammar of the verb “to be” itself, and this is complicated enough. In most Indo-European languages we must distinguish at least 11 variants of the verb “to be”, which leads to a rather rich variety of verb-type-ontologies.
A being here is defined as any item which is in a sense specified according to an appropriate variant of the verb “to be”.
The verb “to be” has its basic form in the context “S is P”, where it denotes a binary relation. The most general verb-type-ontology is therefore the general theory of relations. By specification of variants of the verb “to be” we obtain variants of the verb-type-ontology, for example the ontology of things and properties and the set-theoretic ontology.

2.4 In the possibilistic approach a being is defined as any possible object, hence the ontological universe is understood as the space of all possibilities. Its ontology is therefore the general theory of possibility.

2.5 The qualitative or connectional approach deals with the most traditional concept of a being, defined as any item having some quality (or as a subject of qualities).
Here at least three topics need further elaboration: the ontological connection itself and the items connected – qualities and subjects. In consequence, there are four variants of this type of ontology: the qualitative one – stressing qualities, the subjective one – putting emphasis on subjects (individuals), the connectional one – stressing the formal side of the ontological connection, and the eclectic one – which tries to develop all three factors in tandem with each other.
In the present essay I am going to consider a few fundamental topics of connectional ontology.

2.6 The above three approaches differ in their ideas of particular beings. They do however use the same devices to define the being as the totality of all beings and Being as the unity or the idea of all beings.
These are quite abstract concepts. To formalize them we need two abstractors; one – to collect objects satisfying a suitable condition and another one – to unify objects belonging to some family.

3. Ontological Connection

3.1 Consider a typical qualitative statement: My sweater is red. Here three components can immediately be differentiated: the subject “my sweater”, its qualitative phrase “red”, and the connector “is”.
According to the qualitative approach, the sweater is indeed a being, for it is the subject of some quality, or it is connected with a quality – in our case with redness.

3.2 Hereafter we put emphasis on the ontological connection. Usually such a connection is made by means of a suitable connector, in most cases by the verb “to be”.
The verb-type and the connectional approaches will meet each other exactly at this point, because the connector “is” is a derivative of the verb “to be”, which is basic for verb-type-ontologies.

3.3 Now, let’s ask: Is an explicit connector, like “is”, necessary for a connection to be made?
Certainly not. In English we say “John is at home” but in Russian we can say this in a much simpler way “Ivan doma”, without any explicit connector.
Therefore, looking for a general view on ontological connection, we should abstract from particular connectors and simply write “yx” to express the situation that an item x is ontologically connected with an item y, with further idea in mind, that x is a subject of the quality y.

3.4 What is connected in a given ontological connection?
Consider “yx”. Here the right-hand argument is the connection’s subject.
The idea of a subject is rather clear. A subject is any object characterized, determined, formed, framed, modified or described by its connected companion. Simply, a subject is the subject for something or of something.
What the left-hand argument is, however, is messy and ambiguous.
Notice the relative and formal character of both connected items. In particular, an item that is subject in one connection can be a feature or someting similar in another.

3.5 To be free from misleading associations let’s fix our terminology in a more neutral, purely formal way. The right-hand argument, its subject, will be called the objectum, whereas the left-hand argument will be called the connection’s objectivum.

3.6 In different connections the place of the objectivum is taken by: qualities, determiners, characters, traits, modifiers, properties, attributes, features, concepts, etc,; in the place of the objectum there can occur things, individuals, psychic subjects, situations, facts, events, processes, etc.
As you can see, both families are very broad and heterogenous.

3.7 The ontological connection in general is also not uniform. Two basic intuitions concerning it should at least be distinguished: yx means either that x has y, that y describes or characterizes x; or that y determines, forms or frames x.
The first intuition has dominated logico-linguistic investigation of connection and prevails in tought- and language- ontologies, whereas the second plays the key-role in scientific investigation of the world and prevails in what might be called
being ontologies in the proper sense. [5]
Notice the grammatical difference between typical objectiva of the first and second sorts. The former is expressed in an adjective-like form like red, the latter in a noun-like form, like spin of an electron or platonity of Plato.

3.8 The above remarks are very sketchy. They mark only the distinctions which have to be made clear. In particular, the notion of a quality needs more detailed discussion, for which cf. [25] and [26].

4. Towards a Theory of Ontological Connection

4.1 I am going to outline a theory of ontological connection suitable to study the controversy between Parmenides and Plato concerning being and nonbeing.
We start by introducing a language appropriate to study both beings and the being defined along the lines of the qualitative approach. To this end we need a language with a binary connective for ontological connection and suitable abstract operators to define the being and Being.
Next, we use classical logic to deduce ontological theorems from these definitions and some further theorems via additional axioms.

4.2 Which version of classical logic should we use?
Ontology is the most general discipline; it deals with any object at all. At the beginning of the ontological investigation objects should not be differentiated without essential reasons. Each of them has to be taken at the same level with every other.
It is important, therefore, not to start with any classification of beings imposed by the language we use, even with an elementary distinction between names and sentences or individual expressions and predicates. Natural classifications should arise as a result of research.

4.3 To express the most basic ontological notions we need quantifiers and abstract operators. We need therefore a suitable version of the classical quantifier logic.
But not a predicate one. The differentiation between names and predicates, individuals and complexes, etc., which is usual for predicate calculi is clearly not ontologically innocent. Predicate languages are too connected with language and thought ontologies (in particular with common ontologies of things and properties) to be accepted as the starting point for general ontology.
Thus we need a more neutral version of classical quantifier logic.

4.4 Hence in general ontology and in being-ontology we shall employ languages with only one basic category of expression. These ontologies are thus analogous in their linguistic machinery to calculi of names, to propositional logic and to algebraic logic.
Because we need quantifiers and abstraction operators we will work within a version of classical propositional logic with quantifiers and abstractors, similar to the protothetics of Leœniewski, cf. [15].

Language

4.5 The alphabet. We use only one sort of designators: x, y, z,… denoting objects, because we like to speak of all objects without prior differentiation.
Eight designators are fixed, for the sake of the definitions to follow: B, , N, , U, , D, .
The alphabet includes also: the concatenation symbol “*”, usually omitted, to express the ontological connection, two abstract operators {_ :___} and [_ :___] as well as the standard functors of classical logic: negation ¬, conjunction , disjunction , implication , equivalence , major and minor quantifiers ” and $, and identity =.

4.6 Expressions, like objects are either simple or complex.
Atomic expressions are built up, at the first stage, by concatenation of designators and, on the next stages, by application of the two operators of abstraction.
To be more exact:
(i) Any designator is an atomic expression,
(ii)If A and C are atomic expressions, then (A)(C) is also an atomic expression.
We accept the convention of cancelling brackets containing designators as well as brackets joining to the left. For example, instead of (x)(y) we write simply xy; ((x)(y))(z) is equivalent to xyz and (x)((y)(z)) to x(yz).

4.7 Notice that complex connections, like xyz or x(yz), are not available in elementary predicate languages, because the iteration of predicates is there syntactically forbidden.
This is, in fact, the major advantage of propositional (or algebraic) languages in the realm of general ontology where, for essential reasons, we need iterated connections.

4.8 Expressions are produced in the usual way, cf. Mendelson [17], from atomic expressions by means of the classical conectives: ¬, , , , , $, “, =.
Also the notion of a bound and a free occurrence of a designator (variable) and the notion of a substitution of a designator by atomic expressions are standard.
A(x/C) denotes the result of replacing all free occurrences of x in A by an atomic expression C, provided that C is free for x in A, i.e., each variable occurring freely in C remains free after substitution.

4.9 Concept expressions. Concepts are objects made by abstraction or ideation in several ways, including comparison, classification, joining together, unification, variation, etc.
To form concepts we use operators, usually one for each way. In our case, it is convenient to work with at least two such operators:
{__:___} – to form collections, and
[__:___] – to form ideas.
Indeed, in qualitative ontology we are interested in distinguishing extensional and intensional aspects of totalities.

4.10 We extend the formation rules of the language by the following:
(iii) If A is an expression, then both {x: A} and [x: A] are atomic expressions.
Both abstractors bind designators which are their first argument. Designators therefore can be bound either by means of quantifiers or by means of abstractors.
Finally, to obtain all expressions of the extended language we repeat formation rules suitable number of times.
Notice that the syntax of our language is more Leœniewskian than Hilbertian.

Logical Axioms

4.11 We accept the usual axioms of classical quantification logic with identity (cf. again Mendelson [17]) adapted to our propositional language.
This is acceptable because the ontological universe is clearly non-empty. Indeed, you are reading this essay. Hence at least you, me, the essay and the reading are objects in the universe of all objects, i.e., in the ontological universe.

4.12 In what follows two families of specific axioms of our theory will be introduced: in the first step – the basic ontological definitions and in the next step – axioms for abstractors.

The Basic Ontological Definitions

4.13 Let me start with an informal clarification of abstractors.
The inscription A(x) is used to mark the fact that the designator x occurs freely in the formula A. A(x) is connected in a usual way with its extension, i.e., the family of all A’s containing all objects satisfying it.
Now, [x: A(x)] is understood as the idea of A, i.e., as the general concept of A’s. It is therefore a natural objectivum:
[x: A(x)]y means that y falls under the concept A or, using the terminology of Plato, that y participates in the idea of A. Conversely, y[x:A(x)] means that y is an objectivum, i.e., a determiner, a property or an idea of the concept A.
As regards collectors, {x:A(x)} is understood as the collection of all objects satisfying A. This is also a typical objectivum. {x:A(x)}y means that y belongs to {x:A(x)}, which in the set-theoretical notation can be expressed by y{x:A(x)}. The reverse, objectum position of collectors will be discussed later.
Both concepts clearly conjugate. Quite often they are identified, as in the standard Zermelo-Fraenkel set-theory ZF; sometimes they are implicitly [6] or explicitly differentiated.
The above clues will be used in the future as the starting-point for the axiomatization of our two abstractors.

4.14 Observe that in the present framework we can distinguish four aspects of a given object: two “extensional” and two “intensional”.

Q(x):={y:yx}
The qualification of x, in which all qualities (objectiva) of x are collected.

E(x):={y:xy}
The extension of x, collecting all the objecta of x.

F(x):=[y:yx]
The form of x, which is the unity, togetherness, of all its qualities (objectiva).

S(x):=[y:xy]
The support of x, which is the unity of all the objecta of x.

The above concepts are clearly interconnected, in a way which will become clear once we have introduced suitable axioms for abstractors.

4.15 Fix now a given connection yx. This is an open formula with both x and y occurring freely in it.
Binding one of these designators and using negation in all possible ways we obtain eight elementary formulas expressing basic ontological situations. Four of them are positive: $y yx, “y yx, $x yx, and “x yx; four are negative: ¬$y yx, ¬”y yx, ¬$x yx and ¬”x yx.
In consequence, we get [7] exactly eight basic ontological qualities, eight connected totalities and eight ideas.

Objectiva

4.16 Notice that the first formula “$y yx” says that the object x is qualified, i.e., that it has a quality or, more formally, an objectivum y. This, according to the qualitative approach, means that x is a being. On the other hand, its negation “¬$y yx” says that x is a nonbeing or a naked object, i.e., an item without qualities.
Formally, let us put:
Bx:= $y yx
x:=¬$y yx
defining thus two really basic ontological notions: x is a being and x is a nonbeing.

4.17 The remaining six notions are defined as follows:

Nx:=$y xy
x is nonempty iff x is not a flying objectivum,i.e., it is possessed by some objectum

x:=¬$y xy
x is empty iff x is an unrealized objectivum,i.e. without objectum

Ux:=”y xy
x is universal iff it is a tautologous objectivum,i.e., such that any object is its objectum

x:=¬”y xy
x is non-universal iff it is an objectivum which some object does not have

Dx:=”y yx
x is defective iff any object is its objectivum, i.e., its qualification is full

x:=¬”y yx
x is non-defective iff it is an objectum possessing only some, if any, objectiva

We have introduced eight basic ontological categories. Four of them are positive: B, N, U, D. Four are negative: , , , . Four are general: , , U, D; four are particular (existential): B, N, and .

Collections

4.18 Collecting suitable objects together, we obtain eight basic ontological collections:

B:={x: Bx}
The being, which is the collection of all beings

:={x: x}
The nonbeing, which is the collection of all nonbeings

N:={x: Nx}
The nonemptiness, which is the collection of all nonempty objects

:= {x: x}
The emptiness, which is the collection of all empty objects

U:={x: Ux}
The universality, which is the collection of all universal objects

:={x: x}
The nonuniversality, which is the collection of all non-universal objects

D:={x: Dx}
The defectiveness, which is the collection of all defective objects

:={x: x}
The nondefectiveness, which is the collection of all nondefective objects

Ideas

4.19 Analogously, unifying suitable objects into one, we obtain the following eight basic ontological ideas:

B:=[x: Bx]
Being, which is the unity of all beings

:=[x: x]
Nonbeing, which is the unity of all nonbeings

N:=[x: Nx]
Nonemptiness, which is the unity of all nonempty objects

:=[x:x]
Emptiness, which is the unity of all empty objects

U:=[x:Ux]
Universality, which is the unity of all universal objects

:=[x: x]
Nonuniversality, which is the unity of all nonuniversal objects

D:=[x: Dx]
Defectiveness, which is the unity of all defective objects

:=[x: x]
Nondefectiveness, which is the unity of all nondefective objects

4.20 As in the case of qualities, so also ontological collections and ontological ideas can be categorized into positive and negative, and also into general and particular.
A collection or an idea is respectively positive, negative,
general or particular, if its definitional quality is such.

4.21 Notice that by definitional replacement the above concepts can be made more explicit: B={x: $y yx}, B=[x: $y yx], ={x: ¬$y yx}, =[x: ¬$y yx], etc.

4.22 Observe the self-referential, quasi paradoxical character
of our collections and ideas.
Being, i.e., the idea of beings, falls under itself, if Being is a being (we will see that it indeed is). Similarly, the being, i.e., the collection of all beings, belongs to itself if the being is a being (which indeed holds: cf.§7.12).
Analogous claims can be made concerning our other concepts as well.

All of this is nothing surprising. Similar connections are well-known for the general (Cantorian) notion of set. They, however, are notorious for their power to generate inconsistencies. We must therefore control them very carefully, for we would of course like to produce consistent ontology.

4.23 Definitions are implicit axioms.
The above twenty four definitions are rudimentary axioms of the theory of ontological connection.
Later on, in the chapter 8, our list of axioms will be extended by suitable axioms for abstractors. But even the present list enables us to answer several traditional questions of ontology, including some raised by Parmenides, Plato, Aristotle and Kant.

5. Some Classical Ontological Questions

5.1 Let’s start with the famous problem of Parmenides answered by him in the positive:

Q1. Is there being? I.e.: Is being in being?

Notice the ambiguity of these questions caused by the ambiguity of the name “being”. Having distinguished three variants of the corresponding notion, we can either ask:

Q1a. Is there a being? I.e.: Is a being in being? Or

Qlb. Is there the being? I.e.: Is the being in being? Or finally

Q1c. Is there Being? I.e.: Is Being in being?

5.2 Observe that the claim There is x, which purely ontological meaning is: x is in being, is expressed in our formalism by the formula Bx: x is a being.
Therefore Q1 in each of its three versions can be paraphrased into:

Q1’a. For a being x, does Bx hold?

Q1’b. Does BB hold?

Q1’c. Does BB hold?

5.3 Observe first that Q1’a can trivially be answered in the positive.
Indeed, it in turn can be paraphrased into: Given Bx, does Bx hold? I.e., Does Bx  Bx hold?
The last implication is a case of a well-known theorem of classical logic. Therefore, Q1’a has the emphatic answer: YES.

5.4 The question Q1 in its remaining versions (including Q1a) is, however, somewhat more tricky.
Notice, that putting “B” or “B” instead of “x” in the
formula “Bx” and applying next the definition of qualification Q we immediately obtain

(1) BB  $y yB  Q(B) _  and BB  $y yB  Q(B)_ 
In plain words: BB (or BB), i.e., the being (Being) is in being iff it is a being iff it has some gualities,i.e., its qualification is nonempty, or it has a content.

The above explication made clear the self-referential character of both concepts involved: B – the being and B – Being in itself.

5.5 The story now is clear: Both general concepts of being are self-referential iff they are from an essential point of view “nonempty”, i.e., if they have content.
Indeed, by (1), Parmenides’ question can be reformulated as follows

Q2a. Has the being content or not? In other words: Is the being
qualified or not?
Q2b. Has Being content or not? I.e.: Is Being qualified or not?

Both answers occur in philosophy. The answer YES is given by Parmenides and his followers, the answer NO by Aristotle. [8]

5.6 The question of Parmenides is not only of historical importance. Using the above reformulation we see that it is indeed crucial for qualitative ontology.
For if Being (the being) has a content which is not trivial, then ontology itself is not tautologous.
Parmenides’ position means, therefore, that the notion of a being as well as the notions of the being and Being are categorematic. Hence ontology, which is the most general discipline at all, is still essential.
On the other hand, the Aristotelian position in its extreme form means that the notion of the being (Being) is syncategorematic, hence ontology is a purely formal, “empty” discipline.
What is going to be shown in this paper seems to lead Aristotelians in qualitative ontology into troubles, provided that they respect classical logic.

5.7 The universality question is as follows: Is “to be a being” universal or not? In other words: Is any object a being?
In formal terms:

Q3a Does “x Bx hold? I.e., Is B universal: UB?

Observe that

(2) “x Bx  “x$y yx  “x Q(x)_
Every object is a being iff every object has some quality iff every object has a content.

5.8 In the future refinement of the present formalism, after clarifying the meaning of the connections Bx and Bx, a similar explication will also be made regarding the universality question concerning the being and Being:

Q3b Do “x Bx and “x Bx hold? In other words, Are B and B universal: UB?, respectively: UB?

Both Q3a and the first part of Q3b will be answered in the positive, whereas the second part of Q3b will be answered in the negative.

5.9 We may ask a similar question regarding qualities: Are there universal, or tautologous, qualities?
More formally:

Q4 Is there an object y such that “x yx? In other words: Is the extension of the idea of universality empty or not?

5.10 By classical logic and under the proviso that the ontological universe is non-empty, which we took for granted at §4.11, we have:

(3) The answer YES for Q3 implies the answer YES for Q4.

5.11 Finally, let me ask a rather naive question:

Q5 Is the being (or Being) an object, or not?

Observe that

(4) The answer NO for Q1 but YES for Q3 implies the answer NO for Q5.

5.12 The conclusion of (4) is somewhat paradoxical. Some philosophers enjoying, like Twardowski and Quine, the assumption of (4), might interpret it as a refutation of the categorematicity of the notion of being.
However, they usually equate the notion of a being with the most general notion of all – of an item or object.
Let me stress that in qualitative ontology beings are qualified objects, i.e., objects enjoying some qualities.

5.13 In what follows you will see that a suitable version of classical logic enables us to answer all five questions in the positive.

6. A Linguistic Intermezzo

6.1 Since their origin in ancient Greek philosophy ontological expressions like those studied in the present paper have been notorious because of their difficulty and obscurity.
As was previously mentioned in footnote 1, even the translation of small fragments of Parmenides’ poem causes serious problems, like Patin’s problem of “the subject of estin” at B2 (cf.Bodnár [1]).

6.2 I do not like to go into details as to the complex philological and historico-philosophical discussion concerning this matter. The content of the Parmenidean poem is, in its main, not a problem of philology and history of philosophy, but a problem of ontology. In discussions of it, both philological and historical analyses are auxiliary, not decisive.
The problem needs rather a discussion and development of an appropriate theory (or theories) of being.

6.3 A brief linguistic discussion can still, however, be rewarding.
It is interesting to know, inter alia, whether we can easily express in natural languages the meaning of expressions like BB and other formulas used in the previos chapter.

6.4 Indeed, it is an interesting philosophical exercise to collect and compare ways in which the Parmenidean formula BB, with its conjugate forms, can be expressed in several languages.
First of all, a discussion of such matters can illuminate the problem itself. On the other hand, it can throw light on a metaphilosophical question: to what extent is philosophy, which – like any human activity – is produced by human beings born, educated and thinking in particular languages, determined by them?

Some etymology

6.5 To begin, let me report, after Brückner [4], Heidegger [9] and The Shorter Oxford English Dictionary [l8], what is known about the etymology of the verb “to be”.
The verb itself and its derivatives – including “is”, “beings”, “the being”, etc. – came from at least three different roots. The first two are common to all Indo-European languages, the third occurs only in some of them.

6.6 The first Indo-European radical is Snskrit bhu, bheu, bhawati – which means “to grow” or “to emerge”. It is present in Greek: phuo, phumai; in Latin: fui, fuo, futurus; in English: be, been, become; in German: bin, bist; in Polish: byæ, bawiæ, przebyæ, przybyæ.
The second stem is the Aryan es-, or Sanskrit as-, asus, asmi, esmi, esi, esti – which means “to live”, “to be and stand in itself”. In Greek: es-, estin, eimi, einai; in Latin: es-, ens, est, esum, esse; in English: is; in German: ist, Seiende, Sein; in Polish: jest, mieszkaæ.
The third stem is wes-. In Sanskrit was-, wasami which means “to remain”, “to dwell”, “to sojourn”. In English: was; in German: was, war, es west, wesen, gewesen; in Polish: bytowaæ, bywaæ, przebywaæ.
In English and in several other languages also a fourth stem ar- can be distinguished, present in the plural of “is”: are. Its origin is unknown.

6.7 The word “being” is equivalent to Greek “eon”(“on”) or “einai”, to Latin “ens” or “esse”, to German “das Seiende” or “das Sein”, to Polish “byt”, to French “l’tre”, to Italian “essere”, to Spanish or Portuguese “ser”.
All of these are obtained by production outlined above.

6.8 The story, however, is more complicated. The notion of being is also related to the conjugate family of synonyms belonging to the above second and third etymological line, including the Latin or Italian verb “sta, stare”, German “stehen”, English “to stay”, Polish “staæ”, “pozostaæ”, Spanish or Portuguese “estar”.
Observe that the notion of “a state”, meaning conditions or manner of being, is derived from this fifth family in a way similar to the generation of the notion of being from the first.
As a matter of fact, in ontology beings and states are closely related. It seems that, at least from an etymological point of view, they can be treated as parallel bases for ontology.

6.9 Observe, however, a certain remarkable improportionality.
The usual ontological conceptual network is built by means of derivatives of the verb “to be”. It would be interesting to know a parallel net of concepts derived from the verb “to stay”.
Parallel to a being is, of course, a state of affairs or a situation. Which item is, however, the collection of all states? Is it the being? Or, is it the ontological space?
Is a complex the most general notion of the ontology of states?
Next, identify the unity, or the idea, of all states. Is it the form, like essence, logos or nous in the case of beings?

6.10 To conclude, the concept of being is a common abstraction mixing different sources, which results in a remarkable wealth of variants and makes quite a lot of applications possible.

Comparative Grammar

6.11 I hope now that we are sufficiently awake to concentrate on the question: which variant of the verb “to be” is used, in different languages, to express the statements of the Parmenidean sort.
I will test five languages in this regard: Greek, Latin, English, German and Polish.

6.12 We are particularly interested in ways used to differentiate particular beings, their collection and their idea.
That means that we should like to distinguish in a careful way between six notions involved: a being, the being, Being, a nonbeing, the nonbeing and Nonbeing.
The chief mark of distinction is that the four general notions have only singular form – no plurals for the being, Being, etc.; whereas both particular notions have also plurals: beings and nonbeings.

Greek

6.12 The standard Greek ontological vocabulary is drawn from the verb eimi (esti), which belongs to the line generated by the second Indo-European stem es-.
Three forms of it are particularly important: the infinitive einai, the feminine form ousia and the neutral form eon (on).

6.13 A particular being, like you, me or an earthquake I am experiencing just now, is to eon (to on). In the plural: ta eonta. Notice that to eon (to on) can also denote the plurality of all beings, i.e., the being.
The infinitive einai is also ambiguous. As a matter of fact, it can be used to denote all three ontological objects under scrutinization: a particular being, all beings, the idea or essence of beings.
Ousia is usually reserved for the third use denoting, for example, in Plato’s writings the being itself or its unity, its nature. In Aristotle’s works on the other hand it denotes substance.

6.14 The grandfathers of ontology had at their disposal a quite sophisticated, but ambiguous, vocabulary.
Therefore from the very beginning they tried to delineate both the linguistic and the theoretical distinction between the three notions involved.
To this end, Parmenides was playing with the idea that there is an intimate connection between apprehension and being: to gar auto noein estin te kai einai – thinking and being is the same (cf. Diels [7], Parmenides B5). This leads to the idea that the unity of being, i.e. Being, is Nous, i.e., Reason or the Mind.
On the other hand, Heraclitus seems to be the first to deal with the idea of logos as the intrinsic togetherness, unity, of beings: tou logou d’eontos sonou (Kirk and Raven [13]: “..although the Logos is common [to essents-J.P.]..”, Manheim in [9]: “..though the logos is this togetherness in the essent..”).
Notice that it was exactly in this way Plotinus was using his notion hen (One) to enforce the notion of Being.

6.15 In fact, Greek philosophers sharply distinguished three notions of being. They used to eon (ta eonta) for the first, to eon or to einai for the second, and to einai, to ousia, nous or logos for the third, most general, notion.

6.16 To produce Parmenidean sentences we need two types of negation: nominal – which is used to produce negative notions and sentential – which is used to produce negative statements.
The grammar of Greek is standard. Names are negated by means of the negation particle me: to me eon – a nonbeing, ta me eonta – nonbeings, me einai – the nonbeing, to me einai – Nonbeing. The negation of a simple affirmative sentences is achieved by means of the adverb ouk.
Thus the basic Parmenidean statements are as follows:
to eon esti – a being is
to me eon esti – a nonbeing is
to me eon ouk esti – a nonbeing isn’t
ta eonta eisi – beings are
ta me eonti ouk eisi – nonbeings aren’t
esti gar einai – the being is
einai ouk esti – the being isn’t
me einai esti – the nonbeing is
me einai ouk esti – the nonbeing isn’t
to einai esti – Being is
to einai ouk esti – Being isn’t
to me einai esti – Nonbeing is
to me einai ouk esti – Nonbeing isn’t

6.17 Returning to the problem of the metaphysical subject in Parmenides’ poem. In most cases Parmenides is using the ambiguous einai. In the crucial point, which was emphasized twice by Plato in Sophist (cf.ft.4), he used, however, the plural form eonta: ou gar mepote touto damni einai me eonta (for never shall this thought prevail, that nonbeings are..”).
Taking into account the distinctive mark of plurality, which has been pointed in §6.12, we can conclude that Parmenides was considering beings at the beginning of his lesson, approaching next more abstract levels of them.
Therefore claims which can and have to be attributed to him are as follows: Beings are. Nonbeings aren’t. Also, Being is.
Whether Parmenides refuted the claim Nonbeing is remains unclear.

Latin

6.18 Philosophical Latin was created over the centuries after the influence of Greek distinctions. [9]
Its basic vocabulary is drawn from the verb sum (fui, esse), in which the first two Indo-European stems are mixed: es- is present in sum and esse, bhu- is present in fui and futurus.
In its developed form the vocabulary contains: ens and entia denoting respectively a being and beings. The name ens is also used to denote the being.
In order to avoid misuse, medieval scholars sometimes used the Greek determinate article to to mark a more general use of the word: to ens equals the being.
For the most general notion, i.e., for Being the word Esse or Essentia is usually reserved.

6.19 Latin syntax, too, is regular.
First, notice that both the nominal and the sentential negation is the same: non. Hence non ens means either a nonbeing or the nonbeing, non esse – Nonbeing, etc.
The basic Parmenidean statements are as follows:
A being is – Ens est, or Ens existet
Beings are – Entia sunt, or Entia existent
The being is – (to) Ens est, or (to) Ens existet
Being is – Esse est
A nonbeing is – Non ens est, or Non ens existet
………….
Nonbeing isn’t – Non esse non est, or Non esse non existent

6.20 The above duplication of forms: est – existet is remarkable. It seems that the first predicate “est”, coming from the same stem as ens – esse, is the natural one.
The second predicate “existet” is, however, more technical. It comes from existere, which is drawn not from the verb sum, but from its ontological counterpart (cf.§6.8) sto – stare: existere=ex+sistere=ex+stare, which means, literally: extract or essence of staying or standing in itself.
This invention of Latin philosophers was spread among most European philosophical languages. In this way two ontological lines converged, connecting beings with states. In particular, the question of being comes close to the problem of existence (esse=existentia, etc.), in a way which sometimes seems however rather to obscure than to clarify the problems in hand.

English

6.21 Not only Being is self-referential. Also a text can be such. Now writing in English, hence using its fundamental be-constructions I would like to discuss them.
As a matter of fact, the previous text makes quite clear what I am going to summarize here.

6.22 The English vocabulary as to being contains two families of terms.
One is used throughout this paper. It comes from the verb “to be” and contains the name “being” obtained from the verb via gerund-nominalization.
In this essay, having in mind the importance of singular – plural symmetry; and using both available articles, I shall exploit the following sequence of terms listed in the order of increasing generality: a being, beings, the being, Being.
The second family is adopted from Latin: an entity/an essent – entities/essents, the entity/the essent, Essence.

6.23 The level of mixing different stems in the forms of the verb “to be” is also worth observing. Each of the four stems is, in fact, present: bheu in to be, es in is, was – in the past form was, and ar in the plural are.
The last one is really engaging, because it suggests a lost verb used to speak about the plurality of beings.

6.24 English syntax, like that of Latin, is quite regular.
There are three basic negations: not, which is both nominal and sentential, no – which is sentential, and non – which is nominal.
Negative concepts follow immediately: a nonbeing or a not-being, etc.

6.25 As regards Parmenidean sentences, I will, for the sake of further discussion carefully list sentences from the first group, concerning beings, both in the positive and in the negative form:

1 There is a being. There is not a being.
2 A being is. A being is not.
3 A being exists. A being does not exist.
4 A being is in being. A being is not in being.
5 A being is in existence. A being is not in existence.
6 Beings are. Beings aren’t.
7 There are beings. There is no being.

Sentences concerning the being and Being are constructed after similar manner.

6.26 Positive Statements. The first positive statement, using an abstract predicate-phrase “there is”, is the most neutral.
The next two are standard: 2 is still neutral, whereas 3 – imitating a technical Latin phrase expresses existential presupposition.
Notice that 1 and 2 preserve the connection of the state of being with being itself, for both “being” and “is” are derivatives of the verb “to be”, coming, however, from two different stems. Taking this into account and recalling §6.6 we can risk the suggestion that the original meaning of both sentences is “what emerges and stands in itself”.
Sentences 4 and 5 are more technical, they develop the second statement. Their form, however, is quite complicated introducing problems with additional use of “is” and “in”, as well as – in 5 – delicate questions of existence.
In conclusion, the two first forms seems be the best way to express in English the first Parmenidean statement.

German

6.27 The general structure of the relevant German phrases is quite similar to the Latin and English ones.
The basic ontological vocabulary comes from the verb “sein”. It includes: ein Seiendes for a being, Seiende for beings, das Seiende or alle Seiende for the being, and das Sein for Being itself.
To build negative forms we operate with the negation nicht, both nominal and sentential.
The corresponding negative concepts are thus: ein Nicht-Seiendes, Nicht-Seiende, das Nicht-Seiende and das Nicht-Sein.
All relevant forms, with exception of existieren accommodated from Latin come from the stem es-.

6.28 The paradigmatic sentences are as follows:

1a Es gibt ein Seiendes. There is a being.
1b Ein Seiendes ist. A being is.
1c Ein Seiendes existiert. A being exists.
2a Es gibt Seiende. There are beings.
2b Seiende sind. Beings are.
2c Seiende existieren. Beings exist.
3a Es gibt das Seiende (alle Seiende). There is the being.
3b Das Seiende ist. The being is.
3c Das Seiende existiert. The being exists.
4a Es gibt das Sein. There is Being.
4b Das Sein ist. Being is.
4c Das Sein existiert. Being exists.

Similarly for the remaining forms, including negations. For example: Ein Nicht-Seiendes ist nicht – A nonbeing isn’t, etc.

6.29 Four comments follow:
First, German has a quite distinctive structure of ontological notions, differentiating what indeed should be differentiated.
Second, “es gibt” is the most neutral affirmator, like “there is” or “there are” in English.
Third, the verb “existieren” in German is transparently not neutral, because it presupposes at least existence in time.
Fourth, the regularity and symmetry of German phrases is amazing. For each case you have at least three expressions increasing in ontological commitment.

Polish

6.30 Finally, let me pass to my mother tongue.
In the main this is similar to the four languages which were studied previously. It, however, shows also some interesting dissimilarities.

6.31 The basic Polish ontological vocabulary is drawn from the verb “byæ” which, in its different form, refers to all three of the relevant Indo-European stems: byæ to bheu, jest to es, bywa – bytowanie both to bheu and was.

6.32 The standard line of the basic ontological notions is as follows: byt – both for a being and the being, byty – for beings, Byt or Istota for Being.
It is interesting to observe that in traditional Polish we can find a different line: jestestwo Рjestestwa -Jestestwo. Moreover, in old Polish Being was named by means of BytnoϾ, made from Byt by means of the abstractor -oϾ (i.e. -ness).
To sum up, I propose: poszczególny byt for a being, byty for beings, byt for the being, and Byt or Istota for Being.

6.33 Negative terms are built up by means of the negation nie which in general is both nominal and sentential.
Hence we have: niebyt for nonbeing, etc.

6.34 The most peculiar is, perhaps, the form of the negative Parmenidean statements.
Let me illustrate this by analysing statements concerning Being:

1 Byt bytuje.
2 Byt jest. – Being is.
Bytu nie ma. – Being isn’t.
3 Byt istnieje. – Being exists.
Byt nie istnieje. – Being does not exist.

Positive statements occur in three different forms (1-3). The last: Byt istnieje in Polish is as specific as in German; istnienie in its natural meaning presupposes reality of some kind.
The second sentence Byt jest is neutral, like Being is; whereas the first Byt bytuje is purely ontological.
Bytowanie is the Gerund form of bytowaæ, like bycie is the Gerund of byæ. Therefore, the meaning of the Polish equivalent of BB: Byt bytuje is: Being is in its characteristic state of being.
The negative statement Bytu nie ma is, as was said before, rather peculiar: ma comes from mieæ (to have) taken in the following meaning: it is possible to find (between something which is in your disposal). Therefore, nie ma has a purely logical meaning: it is impossible to find.

6.35 Both pecularities of ontological Polish effects in its comparative flexibility to speak, at least in the realm of ontology, in a neutral and abstract way.

Instead of Conclusion

6.36 I think that the above comparison says nothing for or against the philosophical flexibility of particular natural languages.
It is rather an argument for not limiting ourselves to this or that language when doing philosophy. It also suggests the superiority of suitable formal languages in discussion of formal philosophical questions. Usually such languages are more neutral and more adaptable to the problems under discussion than natural languages are.
Sometimes natural languages are not so flexible as we need

7. An Outline of a Primitive Theory of Being – PTB

7.1 What we need is to prove ontological theorems concerning both being and nonbeing. To this end we must rely on some formalism.
Is the formalism introduced in chapter 5 relevant? And is it sufficiently flexible and useful?
To find out we should use it, try to answer our questions, judging theory by judging its fruits.

The Theory

7.2 The theory PTB, which I am going to develop in the present chapter, is primitive in this sense, that its theorems are deduced from the first eight definitions (given in the sections 4.16 and 4.17) alone, without referring to further definitions concerning abstractors and to additional, specific axioms.
Therefore, its primitives are these nonlogical symbols which occur in the definitions of §§4.16 and 4.17, i.e., the binary connective * and the eight constants: B, , N, , U, , D and . The remaining l6 ontological constants, which denote the abstract concepts, are distinguished not in the present framework, but in its subsequent extension ETB.

7.3 The following list of axioms of PTB is provided to avoid any misunderstanding:

LOGICAL AXIOMS

(A0) The standard axioms and rules of classical elementary logic, including the rule of extensionality

Specific Axioms

(A1) Bx  $y yx
(A2) x  ¬$y yx
(A3) Nx  $y xy
(A4) x  ¬$y xy
(A5) Ux  “y xy
(A6) x  ¬”y xy
(A7) Dx  “y yx
(A8) x  ¬”y yx

ONTOLOGICAL META-PRINCIPLES

(M1) Everything we can speak about is, prima facie, an object.
An ontological universe contains, without any differentiation , all objects which can be treated consistently.

(M2) The objectiva B, , N, , U, , D and are ontological objects.

(M3) Variables range over objects in the ontological universe.

7.4 The ontological universe of PTB is any ontological universe satisfying the above axioms. I.e., it is an ontological universe with a natural network of basic ontological notions, characterized by (A1) – (A8).
Does such a universe exist at all? In other words, can the basic network of ontological concepts be used both in accordance with classical logic and in a consistent way?
We are going to prove that this indeed can be done; not in all universes, however, but only in some.

7.5 Metaprinciples are introduced to clarify presuppositions of the PTB-proofs.
(M1) expresses the extreme generality of ontology which, by definition, concerns everything, i.e., each and every object. In ontology, in particular, we are trying to differentiate consistent (possible) objects from inconsistent (imposible) ones. In the main, ontology deals with consistent objects. Inconsistent objects are also treated, under the proviso, however, that they can be elaborated in a consistent way.
(M2) expresses our presumption that the basic ontological notions can indeed be investigated, whereas (M3) guarantees unrestricted use of the classical calculus of quantifiers.

7.6 Specific axioms (A1) – (A8) were commented upon already in §§4.16 and 4.17. It would be useful, I think, to comment upon them once again, before use.
We should distinguish two ways of understanding the specific axioms: purely formal and ontological.
From an applicative, ontological point of view axioms are formulas of qualitative ontology. I.e., the connection yx expresses an ontological connection between the objectivum (quality) y and the objectum (subject) x. Hence, for example, the formula $y yx says: x is qualified by y, or x is the subject of the quality y.
On the other hand, from a purely formal, relational point of view the connection yx says only that y is related to x. Now, the formula yx says that something is related to x, i.e., that x is complex with respect to a given relational network.

7.7 For the sake of further discussion let me introduce the standard relational terminology.
Consider a relational frame ordered by the relation R. We say that:
x is simple iff nothing is related to it: ¬$y yx
x is complex iff it is not simple, i.e., something is related to it: $y yx
x is co-simple iff it is related to nothing, i.e., ¬$y xy
x is co-complex iff it is not co-simple, i.e., it is related to something: $y xy
x is a root iff it is related to anything: “y xy
x is a co-root iff everything is related to it: “y yx
If x is the only root (or the only co-root), it is called the smallest (or the biggest) one.

7.8 Now, it is easy to grasp the meaning of axioms under scrutinization.
(A1) says, ontologically, that B is a quality characteristic for items possessing qualities, i.e., a quality characteristic for beings. On the other hand, from a relational point of view, (A1) says that B indicates complexes, i.e., is related to complexes only.
Therefore, from a purely formal point of view, beings are qualitative complexes.
Interpreted in ontological terms, (A2) says that is the characteristic quality of nonbeings, i.e., items without qualities; whereas, in relational terms, indicates (is related to) simples. From a purely formal point of view, nonbeings therefore are qualitative simples.
Notice the paradoxical character of a quality characteristic for nonbeings, or a relational indicator of simples. Cf. th.7 below.
(A3) and (A4) characterize respectively N and as the qualities of nonemptiness or emptiness. In formal terms, N and serve respectively as an indicator of objects which are not co-simple or as an indicator of co-simples.
In (A5) and (A6), U and are presented as characteristic respectively for universal (non-universal) qualities. In relational terms, they are, respectively, indicators of roots (not-roots).
Finally, (A7) and (A8) say something similar for objecta. D and are, respectively, characteristic for defective and non-defective objecta, i.e., items possessing all (not all) qualities. In relational terms: D indicates co-roots, whereas do the same for objects which are not co-roots.

7.9 The special axioms of PTB, on first reading, are interesting from the point of view of qualitative ontology; second reading is interesting from the point of the verb-type, or relational ontology however. They are among the bridge-formulas and concepts which connect two of the three basic types of ontology – qualitative and relational ones.
Results

7.10 By classical logic we have the following interconnections between our eight notions:

(5) i) Dx  Bx: Anything defective is a being
x  x: Nonbeings, if there are any, are not defective
ii) Ux  Nx: Universals are nonempty
x  x: Empty qualities are not universal

On the other hand:

(6) Bx, Dx, Ux and Nx are in contradiction with respectively: x, x, x and x.

Therefore, in general, the above eight formulas form two logical squares:

Dx x Ux x

Bx x Nx x

In PTB, however, the first square can remarkably be reduced.

7.11 Observe first that:

(7) Anything is a being: “x Bx

Proof. Take an arbitrary object x. By (M2), both Bx and x are meaningful and legitimate. By the law of the excluded middle, either x is a being or it is not: Bx¬Bx. By (A1) and (A2),¬Bx  x. Hence, either x is a being or it is a nonbeing: Bxx.
To finish the proof it suffices, by the disjunctive syllogism law, to refute x. To this end suppose that x holds. Applying again (A2) we obtain: ¬$y yx. On the other hand, by the existential generalization law, which due to (M2) and (M3) can be used here, implies $y yx, which contradicts the previous claim.
Therefore ¬x, hence Bx. In conclusion, “x Bx, for x was taken arbitrarily.

7.12 A number of corollaries follow:

(8) The quality of being is universal: UB, hence nonempty: NB. On
the other hand, the quality of nonbeing is empty: , hence not-universal: .

Proof. Apply (A5) to (7), next use (6).

(9) The extension of being is universal: E(B)=OB, where OB is the family of all ontological objects.

(10) Everything is complex: “x$y yx.

Proof. By (7) and (A1).

(11) i) Everything has a content: “x Q(x) _ 
ii)The intersection of all contents is non empty.

Proof. Ad(i): By (7) and definition of Q.
Ad(ii): Indeed, B is a common objectivum of everything.

(12) i) Both the quality of being B and the quality of nonbeing are beings: BB, B.
ii) Moreover, the remaining six ontological qualities are also beings: BN, B, BU, B, BD, B.
iii)If appropriate ontological concepts: B (the being), B (Being), (the nonbeing), (Nonbeing) etc. satisfy (M1) then they are beings: BB, BB, B, B,etc.

Proof. Ad (i) and (ii): Apply (7), (M2) and (M3).
Ad(iii): Similarly, because (M1) is the general formal proviso for (M2) and (M3).

Notice that 12(i) is particularly striking in the case of the quality of being B. Indeed, the formula BB makes clear the self-referential character of the notion of being.
Observe also that for the conclusion of 12(iii) we need an appropriate investigation of the concepts involved. This will be done in the extended theory ETB.

Finally, notice the negative reformulation of (7) and (10):

(13) There are no nonbeings, i.e., nonbeings aren’t: ¬$x x.

(14) There are no simples: ¬$x”y¬yx.

7.13 The above proofs clearly depend on use of some characteristic laws of classical logic. This I will comment on later. I would now like to analyse the role of the axiom (A2).
First of all, it expresses, in a straightforward way, the idea of nonbeings in qualitative ontology, as items without qualities. As it was noted previously and made clear in the proof of (7), this idea is paradoxical.
This is even more transparent when we are using the relational explication: indicates simples, i.e., items to which nothing is related, hence in any given relational network they cannot be indicated. Characteristic for simples, if there are any, is that they cannot be indicated.
This does not mean, however, that simples cannot be collected, but only that no collection of simples is based on or generated by an indicative quality of them.
An important lesson follows: collection does not imply or presuppose the existence of any characteristic quality for the collected items.

7.14 By the way, with the above remarks in mind, it is easy to see the level of logical consequence in traditional philosophy and teology which, indeed, is remarkable.
Recall traditional discussions of simples and God. Quite often, God was claimed to be simple either with respect to analyses of the world, or with respect to essential or qualitative analyses, or with respect to both (being in this case absolutely simple).
The first option was taken, among others, by Leibniz; the second and the third by several gnostic or mystical thinkers, including Meister Eckhart.
To be consistent anyone should either consider God (and any natural simple being as well) to be complex from the point of view of qualitative analysis, or to claim that God (and any qualitative or absolute simple) cannot be qualified and indicated, that It is (they are) out of any essential characterization.
It is nice to recognize, that these moves were indeed taken, the former – inter alia by Leibniz, the latter – among others in negative theology.

7.15 Return to deduction. The second objectivum which, in fact, cannot be instantiated is the indicator of defectiveness D. Indeed

(15) For any x, x is defective iff x is contradictory.

Proof. For the right-hand implication assume that x is defective: Dx. By (A7), “y yx. By (M2) and (M3), D and are ontological objects, both in the range of y. Therefore, by general particularization, Dx and x hold. Applying (A7) and (A8), we obtain x  ¬Dx. Thus Dx¬Dx, hence x is contradictory.
For the reverse implication assume that x is the subject of contradictory characterization: A(x)¬A(x). By Duns Scotus’law, A(x)¬A(x)  “z zx. Hence “z zx, i.e. Dx, as required.

Observe now

(16) Each object is not defective: “x x.

Proof. Assume, for the sake of contradiction, that x is defective: Dx. Hence x is contradictory. For example, Dx¬Dx. This, however, contradicts the well-known classical principle of non-contradiction: ¬(Dx¬Dx).
Hence, for any x, x, as claimed.

7.16 Again, a number of corollaries follow.
The first square of §7.10 collapses, for

(17) “x Bx  “x x and “x x  “x Dx.

Proof. By (7) and (16), the formulas from the first equivalence are true, whereas these from the second equivalence are false.

(18) The quality of nondefectiveness is universal: U, hence nonempty: N. On the other hand, the quality of defectiveness is empty D, hence non universal: D.

(19) There is no qualitative co-root: ¬$x”y yx.

Proof. By (16) and (A8).

(20) Nothing has the full content: “x Q(x)_OB.

Confronting this with (11) we obtain

(21) Everything has an intermediate content: “x _Q(x)_OB.

(22) Each of the eight ontological objects under investigation is nondefective: B, , N, , U, , D, .

7.17 Iterations of the eight ontological objects under investigation were several times before shown to be important. Now, I am going to study them systematically.
There are 64, i.e. 88, such iterations which are divided into two basic families.
The first one, with 32 formulas, concerns the size of whatever is the relevant content, This is so because

(23) For any x:
Bx iff Q(x)_, x iff Q(x)=
x iff Q(x)_OB, Dx iff Q(x)=OB

Previously, in (7) and (16), the formulas occurring on the left side of the equivalences from the first column were proved. In consequence, in (12) and (22), the following sixteen iterations were included among the theorems of PTB: BB, B, BN, B, BU, B; BD, B; B, , N, , U, , D, .
The remaining sixteen iterations, which start with or D, are not provable in PTB, because they contradict PTB-theorems and in the next subchapter PTB will be shown to be consistent.

7.18 To characterize the remaining 32 iterations take one, say , and, using axioms calculate:
 ¬$x x by (A4)
 ¬$x¬”y xy by (A0) and (A5)
 “x”y xy by (A0).
Hence N is equivalent to the standard quantifier formula saying that everything is connected with everyting.

7.19 Taking into account that for a given two-variable connection “xy” there are exactly 12 combinations of negation with both general and existential quantifiers binding variables x and y, and repeating the above calculation for the remaining 31 formulas we obtain

(24) i) Each of the conditions , UU, UD, is equivalent to “x”y xy, i.e. to the thesis that everything is ontologically connected to everything else, or: Any object is both objectum and objectivum to each and every other object.
ii) Each of the conditions N, U, D, N is equivalent to ¬”x”y xy: Some objects are not ontologically connected.
iii) Each of the conditions NN, , , NB is equivalent to $x$y xy: Some objects are ontologically connected.
iv) Each of the conditions N, U, U, B is equivalent
to ¬$x$y xy: Nothing is ontologically connected.
v) Each of the conditions NU, is equivalent to
$x”y xy: Something is an objectivum (a quality) of everything.
vi) Each of the conditions U, U is equivalent to
¬$x”y xy: Nothing is an objectivum of everything.
vii) Each of the conditions UN, is equivalent to
“x$y xy: Any object is an objectivum (quality) for something.
viii) Each of the conditions N, N is equivalent to
¬”x$y xy: Some object cannot be an objectivum.
ix) Each of the conditions UB, is equivalent to
“y$x xy: Any object is an objectum for something, or: Everything in the qualitative universe is complex.
x) Each of the conditions B, N is equivalent to
¬”y$x xy:Something cannot be an objectum.
xi) Each of the conditions ND, is equivalent to
$y”x xy: Some object is the universal objectum.
xii) Each of the conditions D, U is equivalent to
¬$y”x xy: Nothing is the objectum for everything.

The above twelve quantifier formulas describe all general situations occurring in graphs of binary relations: everything is connected, something is not connected, something is connected, nothing is connected, etc. The second group of ontological iterations contain therefore expressions having a quite clear relational meaning.
Which of them hold in PTB?

7.20 As a matter of fact, PTB with respect to these general possibilities is complete: exactly half of suitable formulas are its theorems.
They are listed, with the previous numbering, below.

(25) The following formulas are theorems of PTB:
(ii) N, U, D, N and ¬”x”y xy
(iii) NN, , , NB and $x$y xy
(v) NU, and $x”y xy
(viii) N, N and ¬”x$y xy
(ix) UB, and “y$x xy
(xii) D, U and ¬$y”x xy.

Proof. Notice that, by (25), the formulas collected in the successive rows of the above theorem are known to be equivalent. Therefore, to check that they indeed are theorems of PTB, it suffices to check that at least one of them is such.
Ad (ii). By (18), N is a theorem of PTB.
Ad (iii). By (8), NB is a theorem of PTB.
Ad (v). Again by (8), UB is a theorem of PTB, a fortiori NU is such.
Ad (viii). By (13), , hence by (A0), (M2) and (M3),
$x x. Applying (A4) we obtain that N is a theorem of PTB.
Ad (ix). By (8), UB is a theorem of PTB.
Ad (xii). By (18), U is a theorem of PTB.

7.21 In conclusion: Exactly half of all sixty-four possible iterations happen to be theorems of PTB. The remaining thirty-two are their negations. Hence they cannot be theorems of PTB, provided that the theory is consistent.
And this, indeed, is the case.

Models

7.22 To prove consistency we need models. A model of a given theory is any interpretation of its language, which respects logical entailment and satisfies the axioms of the theory.
The language of PTB is that part of a language introduced in chapter 4 which suffices to express the axioms of the theory. Because abstractors are not used, they can be here left out of account.
The syntax of the language of PTB is therefore standard.

7.23 Which models model PTB?
PTB is a theory of the ontological connection and the eight notions involved: B,, N, , U, , D and .
Its natural models are therefore relational frames with eight distinguished elements, endowed with corresponding interpretations: F = .
Here b,b,n,n,u,u,d,d belong to the universe O; whereas R is a binary relation on it.
Let FOR denote the family of all formulas in the language of PTB.
An interpretation is any mapping f: FOR — O such that
(i) f(B)=b, f()=b, f(N)=n, f()=n, f(U)=u, f()=u, f(D)=d and f()=d;
(ii) f(AC)=f(A)Rf(C);
(iii)f respects classical connectives and quantifiers.

We say that a model is a PTB model if all axioms of PTB are valid in it.

7.24 It is easy to see that quite a lot of relational models are not PTB models.

By (24), PTB models must:
i) Interpret all specific symbols of PTB, hence they must include indicator of simples b;
ii) They therefore must contain only complexes;
iii)Also, they must have roots and co-simples, but not co-roots;
iv) They cannot be discrete (for something must be connected);
v) They cannot be full (for something should not be connected).

The crucial point is to verify (A2). There are no simples, hence b should indicate them in vain. Therefore b must be co-simple.

7.25 It is easy to see that the six-element model given below is a PTB model:

D=l

l

N
l

l

l
U

l
B=

The model verifies, in fact, a stronger theory than PTB, for, in addition, it verifies

(A9) “x(Bx  x), i.e., B and as well as and D are equivalent.

7.26 In conclusion

(26) PTB is consistent.

Therefore, by remarks in §7.21, PTB really answers all primitive questions which can be expressed in its language.

Comments

7.27 Some comments are in order. Are the above results convincing? Are they made in an ontologically innocent way? What are their presuppositions?

7.28 Two main questions can be raised. The first concerns the language of PTB, the second draws attention to the logic of the above proofs.

7.29 It is well-known that standard, i.e. classical-like logics enjoy a certain freedom in considering all syntactically correct formulas as meaningful despite ease of introduction of self-contradictory formulas.
The proofs under investigation depend on the correctness of all formulas involved, inter alia: Bx, x, BB, NB, etc. They are, for sure, made in a correct way and have a rather clear meaning. However, they presuppose that B can be both objectivum and objectum, that both B and can be objectiva of any object, etc.
These linguistic presuppositions are, however, implicit in the metaprinciples (M1) – (M3).

7.30 On the other hand, our language might be thought to be too poor.
We are using only one primitive ontological connective (and eight parameters). Someone might prefer to use more connectives: one – to define “to be being” and “to be nonbeing”, another one – for connection between an object and its qualities or ideas, etc.
In such a case, more freedom is available, for the conceptual framework is broader. As a matter of fact, this option was developed by F.J.Pelletier in his recent book [20]. For its outline see the next subchapter.

7.31 In particular, someone can prefer to use in “Being is” distinct symbols for “Being”, say B, and for “is”, say E. I.e., to write “EB” instead of “BB”.
This, however, suggests the basic, primitive distinction between “being” and “is”, despite the results of our previous
linguistic discussion. Remember, that both “is” and “being” are derivatives of the verb “to be”. And I like to preserve this connection.
The proper proportion of formal distinctions is perhaps the most subtle factor of any formalization. We can make, for sure, as many verbal or notational distinctions as we wish. However, each distinction needs further elaboration. The more notions we have, the more work is needed for their elaboration.
I prefer a reasonably economic framework. Taking into account the extreme generality of the pure ontological investigation and the status of the present investigation, which is the starting point for further development, I chose the most economical ontological framework of all: with only one primitive connection, plus parameters and, of course, logical functors. The rest is derived.

7.32 Turn now to the logical side of the arguments under scrutinization. I.e., reconsider the proofs of (7) and (16).
Both proofs depend heavily on use of a few characteristic classical laws: the law of the excluded middle, which in use seems to have a rather strong ontological meaning, the law of existential generalization, the double negation law, Duns Scotus’law, the law of disjunctive syllogism.
Notice that the formal proviso to existential generalization: nonemptiness of the ontological universe, is indeed acceptable (cf.§4.11). But what about an ontological proviso, if any.
Observe that a suitable assumption is implicit in our metaaxioms (M2) and (M3).

7.33 The law of the excluded middle has a deep ontological tenor in itself. Indeed, read A(x)¬A(x): For each (property) A and each (individual) x, either A(x) is the case or it isn’t.
In our case, its tenor is even stronger. Bxx: For each x, either x is a being or it is a nonbeing. Can such a formula be ontologically innocent?
It seems that it cannot. Recall the well-known arguments of £ukasiewicz [16] against an ontological justification of the law of the excluded middle and the law of noncontradiction as well as Scholz’s [35] arguments for the claim that they are, in fact, ontological statements.

7.34 Therefore, the laws of classical logic on which my proofs depend in the present ontological context should be treated not as purely formal, neutral logical principles, but rather as specific, ontological axioms of the theory.

7.35 Someone can argue that this is sufficient reason to limit the use of classical logic in ontology. But which logic should be used instead?
I am not going so far in my self-criticism. Any revision is revision of something. Before revising onto-logic, i.e. logic of
being, we must therefore develop it in a standard, classical way. And this is my present task.

7.36 Finally, a metaphilosophical remark. Some pecularities of the present formalism are, in a sense, illuminating.
It is half-predicative, half-algebraic. Our formulas, like BB, resemble rather the formulas of combinatory logic than those of the standard, predicate calculus.
This option has been chosen for essential reasons, discussed in chaper 4. In practice it indeed show some usefulness.
It seems therefore that the algebraic framework is more appropriate for general ontological investigation; whereas the standard, predicate formalism is more adequate to deal with language ontologies and perhaps also with the metaphysics of the common-sense world.

An Alternative Approach

7.37 Let us compare the framework of the present essay with that outlined by Pelletier at the end of his impressive analysis of Plato’s “Sophist”, cf.[20].
Pelletier adopts as primitives two binary predicates
DK(_,_) and X(_,_) introduced to formalize two types of participation. He also uses two parameters “b” and “n” for suitable forms of Being and not-Being.
Pelletier’s principles are as follows:

(P1) “y[$xDK(x,y)  X(y,b)]
(P2) “y[¬$xDK(x,y) X(y,n)]

7.38 It is easy to find equivalents of Pelletier’s formulas among ours:

By:= X(y,b)
y:= X(y,n), and
xy := DK(x.y).

Thus (P1) and (P2) correspond respectively to our definitions of a being and a nonbeing.
The axiom behind this reduction is following one:

(PP) DK(x,y)  X(y,x)

7.39 Substituting “b” and “n” for “y” respectively in (P1) and (P2) we obtain the following four formulas:

(P3) $xDK(x,b)  X(b,b)
$xDK(x,n)  X(n,b)
¬$xDK(x,b)  X(b,n)
¬$xDK(x,n)  X(n,n)
We immediately have

(27) X(b,b)  ¬X(b,n)
X(n,b)  ¬X(n,n)

7.40 By (27), the following three pairs of formulae constitute all maximally consistent subcollections of X-formulas, which are involved in (P3):

I := {X(b,b), X(n,b)}
II := {X(b,b), X(n,n)}
III:= {X(n,b), X(b,n)}

In my notation, I={BB,B}, II={BB,} and III={B,B}.

I corresponds to the family asserted in PTB, II constitutes Pelletier’s axiomatic choice of Parmenidean statements, whereas a rather counterintuitive family III constitutes a logical challenge to Parmenideans in the framework of Pelletier.

7.41Let me stress, that the family I occurs, mutatis mutandis, both in PTB and in the more liberal approach of Pelletier; whereas II and III are lost in PTB.
Restrictiveness is reasonable not only for economic reasons. Sometimes in a restrictive framework we can prove something which is lost in more liberal one.
In Pelletier’s approach, I is one of the three options which can be chosen axiomatically; in the more restrictive PTB, formulas belonging to it are provable.

8. Towards an Extended Theory of Being – ETB

8.1 In the Extended Theory of Being, in short ETB, which is an extension of PTB, we deal in addition with two abstract concepts of being: the being B, which is the collection of all beings and Being B, which is the unity or idea of all beings.
Both notions are abstract concepts arrived at by means of suitable abstractors: B:={x: Bx} and B:=[x: Bx].
ETB therefore relies on an appropriate theory of both abstractors, which is rather a delicate topic.

Collections and Ideas

8.2 Hereafter, the full language introduced in chapter 4 will be used, with all its pecularities commented upon there.
Recall that the inscription A(x) is used to mark that the variable x occurs freely in the formula A. A(x/C) denotes the result of correct substitution of a variable x by the formula C in the formula A.
A(x) is connected in the usual way with its extension, i.e., the family of all objects, say A’s, realizing it. Clearly, the exact definition of realization must presuppose either a given theory (in which A is provable), or a class of models satisfying it.

8.3 The collector {_:__} is understood as an operator collecting families of differentiated and determined objects. In particular, {x: A(x)} is the collection of all objects realizing A.
It is a typical objectivum, studied in set-theory.
{x: A(x)}y means that y belongs to {x: A(x)} (what, in set-theoretical notation, is expressed by y{x: A(x)} ).

8.4 Ideas or unities are built from multiplicities by their unification, if possible. Unvoid unification clearly presupposes compatibility, i.e., that items realizing a given condition can be taken into one.
Formally, ideas are formed by means of unifiers [_:__]. Thus, [x: A(x)] denotes the idea of A’s or the unity of all A’s. It is also a typical objectivum: [x: A(x)]y means, in Plato’s terms, that y participates in the idea of A’s.

8.5 Both abstractors need a careful axiomatic elaboration. This can be done in at least two ways.
In the standard option our attention is directed to two appropriate relations: the well-known relation of membership , characteristic for collectors; and the not so elaborated relation of participation, say «, characteristic for unifiers. Here both abstractors are implicit.
I prefer a more economic option: to work with one connection plus two abstractors. They can occur either as objectivum, on the left side of connection, or as objectum, on the right side, or even on both.

8.6 As a matter of fact, we need to characterize four situations:
{x: A(x)}y – y belongs to the collection of A’s
y{x: A(x)} – y qualifies or characterizes or contains the collection of A’s
[x: A(x)]y – y participates in the idea of A’s
y[x: A(x)] – y qualifies or determines or forms the idea of A’s

8.7 Usually people are dealing only with the objectivum position, working with:

Comprehension Principle: y{x: A(x)}  A(x/y), or

Abstraction Axiom: [x: A(x)]y  A(x/y).

The Comprehension Principle was common in intuitive set-theory and is still used in some of its refinements, whereas the Abstraction Axiom is one of the basic principles in formal theories of properties. Both of them are notorious for their capacity to produce paradoxes.
In what follows I am trying to approach the problem from its roots.

8.8 Consider first the case of abstractors as qualities.
Two fundamental, partly opposed intuitions concerning the interrelations between realization, collection and unification are leading.

8.8.1 According to the first, unification presupposes collection and, in turn, collection presupposes realization. Suitable implications are accordingly here accepted as axioms:

From unities to collections:
(UC) [x: A(x)]y  {x: A(x)}y
If y participates in the idea of A’s then y belongs to the collection of A’s (is one of the A’s) .
From collections to realized cases:
(CR) {x: A(x)}y  A(y)
If y is among the A’s, then y realizes A.

The question arises: Do the reverse implications hold as well?

8.8.2 In the second option exactly the reverse implications:
(RC) A(y)  {x: A(x)}y

(CU) {x: A(x)}y  [x: A(x)]y

are chosen as basic axioms.
I.e., it is claimed that if y realizes A, then y belongs to the collection of all A’s and, in turn, if y belongs to this collection, then y falls under the general concept of A’s, i.e., it participates in the idea of A’s.
Here it seems to be taken for granted that everything can be collected and that each collection can be unified. The first assumption seems to be reasonable, provided that collecting is considered to be a purely mental, a priori activity. The second assumption, however, is much more doubtful, because unification requires objective compatibility and that there is a corresponding form.

8.9 Combinations of both options are, of course, possible and are indeed often practised.
Notice that the Comprehension Principle is equivalent to the conjunction of (CR) and (RC), whereas the Abstraction Axiom easily follows from the conjunction of all four axioms.

8.10 If we choose, as I do, the first option, but not everything from the second, then the question of restrictions for appropriate equivalences occurs naturally.
By our previous discussion they should be suitable compatibility conditions. Looking for such conditions we see that in the present framework there are, in fact, only two natural candidates for such a condition. And both will be used.

8.10.1 For a given formula A, its downward compatibility, DC(A), means that some object z is the common objectivum of (or, is related to) all objects realizing A:
DC(A):= $z”y(A(y)  zy)
Thus, for a down-compatible formula A, examples of A’s enjoy a common quality, which is the reason for their compatibility.

8.10.2 Similarly for upward-compatibility:
UC(A):= $z”y(A(y)  yz)
Items realizing an up-compatible condition A are instantiated by (or, are related to) the same object, which is another sort of quarantee of A’s compatibility.

8.11 Using the above conditions, it is easy to distinguish different kinds of tranformation unifying many into one.
They are expressed in the following three axioms, accepted for further use:

Comprehension Axiom
(C) {x: A(x)}y  A(y)
Everything described by a linguistically correct condition can be comprehended, i.e., collected mentally.

Comprehension, however, does not presuppose compatibility. Hence not all mentally achieved collections are coherent.
Coherence appears in stages:

Proper Collection Axiom
(PC) [x: A(x)}y  {x: A(x)}yDC(A)
Proper collections are multiplicities produced by down-compatible conditions.

Unification Axiom
(U) [x: A(x)]y  [x: A(x)}yUC(A)
Unities are proper collections achieved by up-compatible conditions.

8.12 Immediately from these definitions we obtain

(27) i) [x: A(x)}y  A(y)DC(A)
ii) [x: A(x)]y  A(y)DC(A)UC(A)

Hence, y participates in the idea of A’s iff it realizes A, which can be unified, i.e., is both down- and up-compatible.

Observe that by the axiom (U), unities of incompatible collections are not excluded. Such ideas, however, must be empty.

(28) If the collection of A’s is incompatible, i.e. ¬DC(A) or ¬UC(A), or if A is not realizable, then no y participates in the idea of A’s: ¬$y [x: A(x)]y.

Similarly for incompatible proper collections, which also must be empty.
Now consistent collections are defined as proper collections with nonempty, i.e. instantiated, ideas.

8.13 Observe that (U) entails the Abstraction Axiom for compatible formulas:

(29) DC(A)UC(A)  ([x: A(x)]y  A(y)).

8.14 It is instructive to see the meaning of the above axioms in the set-theoretical realm (cf.[8] and [17]).
Down-compatibility on A is like Zermelo’s restriction, introduced by him for reasons similar to ours. To see this, recall Zermelo’s Separation Axiom:
For every formula A and every set a there exists a set whose elements are exactly those of a realizing the formula A:
$x”y[yx  A(y)ya] (x is not allowed to occur in A).
But in set-theoretical terms DC(A) is equivalent to:
$x”y(A(y)  yx), which, to some extent, resembles Zermelo’s condition.

On the other hand, up-compatibility means that the family of all A’s is centred, which is an important condition in the theory of filters:
UC(A)  $x”y(A(y)  xy) {y: A(y)}_

Both conditions of compatibility have therefore a well-known and recognized meaning in the realm of sets. Proper collections are sets in a modified Zermelo’s sense:
y[x: A(x)} iff A(y)x”y(A(y)  yx);
whereas unities, or consistent collections are proper collections which generate proper filters:
y[x: A(x)] iff A(y)($x”y(A(y)yx))({y:A(y)}_).

Empty Ideas

8.15 It is reasonable to accept the thesis that there are unities for all collections, i.e., for all conditions. But in some cases they are nonempty, i.e. something really participates in them; whereas in some cases they are empty, because nothing participates in them.
As a matter of fact, there are quite a lot of cases leading to empty ideas. I am going to characterize some of them. First of all

(30) Inconsistent condititions lead to empty collections and, in turn, empty collections end with empty ideas.

8.16 On the other hand conditions tolerating inconsistencies also lead to empty ideas.

To be more exact, a formula A is said to tolerate inconsistencies iff there are y and y, both realizing A, such that y is a quality opposite to y: $y$y(A(y)A(y)z(yz¬yz)).

Now, applying the principle of noncontradiction it is easy to see that

(31) If A tolerates inconsistencies, then the idea of A’s is empty: ¬$y [x:A(x)]y.

Proof. Take some A tolerating inconsistencies. Assume UC(A), i.e., $z”y(A(y)yz).
By assumption, for some opposite y and y: A(y) and A(y). Hence, by UC(A), yz and yz, i.e.,yz and ¬yz, which contradicts the principle of noncontradiction. Thus ¬UC(A), therefore, by (U), the idea of A’s must be empty.

In conclusion

(32) Very general conditions lead to empty ideas.

Indeed, what is truly general is common for everything, hence also for inconsistencies. Hence a truly general condition tolerates inconsistencies, Therefore, by (31), its idea is empty.

8.17 Further, notice that:

(33) If A tolerates simples or co-simples, then it produces an empty idea.

Proof. By definition, A tolerates simples (co-simples) iff it is realized by some simple (co-simple) object: $y(A(y)¬$x xy) or $y(A(y)¬$x yx).
In the first case, ¬DC(A); whereas in the second, ¬UC(A). Therefore, by (U), no y participates in [x : A(x)].

8.18 Finally, let me pass to qualitative particulars and universals.
Particulars are qualified, but not qualifying objects. I.e., x is a particular iff it is qualified but not qualifying: Q(x)_, or $z zx, but x itself is not a quality: ¬$z xz.
Universals are qualifying objects, i.e., x is a universal iff for some z, x is its quality: xz.
Void objects, i.e., objects both simple and co-simple, are neither particulars nor universals.
Particulars are unvoid co-simples. The rest are universals.

Observe that

(34) Only universals can be unified in a fruitful way.

This is because, by (33),

(35) If A tolerates (i.e. is realized by) void objects or particulars then its idea is empty.
Connections of Abstract Concepts

8.19 Abstract concepts are given in order of increasing abstraction.
We start with particular cases of A: A(y), which are then mentally collected into {x: A(x)}. This collection sometimes can be turned into the proper collection [x:A(x)} which, in turn, sometimes can be fruitfully unified into the idea [x:A(x)].
It is rather reasonable to expect that each of the above concepts falls under its more general companions. I.e., that not only the cases of A’s, but also their collections and ideas participate in the idea of A’s.

8.20 To be more exact, let a denote one of the A – abstractions: {x:A(x)}, [x:A(x)}, [x:A(x)]. Ask: when does a participate in [x:A(x)], i.e., [x:A(x)]a ?
By the axiom (U), [x:A(x)]a is equivalent to the conjunction of the three conditions:
(i) A(a). In particular:
(*) A({x:A(x)}), or A([x:A(x)}), or A([x:A(x)]),
which are typical fixed-point conditions;
(ii) DC(A),
(iii) UC(A).
By UC(A), for some z:
(**) If A(y) then yz.
By (*) from (i), the following holds: {x:A(x)}z, or
[x:A(x)}z, or [x:A(x)]z. Applying suitable axioms and (**) we finally obtain that zz.
Hence we proved

(36) If at least one of A-abstractions participates in the idea of A then
(i) The condition A enjoys a suitable fixed-point property,
(ii) The ontological universe is not irreflexive:
$z zz.

8.21 Therefore

(37) In a standard set-theoretical universe no abstractor participate in its conjugate idea.

This is so, because the standard set-theoretical realm is regular, hence ¬$z zz. In the set-theoretical universe the connection zz means zz, hence regularity contradicts (36.ii).

8.22 In which way should our axiomatics be completed?
Quite a lot of candidates can, in fact, be considered. Most of them, however, are either artificial or produce paradoxes.
In the present essay, I am limiting myself to rather safe conditions, introduced in order to guarantee logical extensionality: the same effect for equivalent formulas.

Identity

8.23 First consider the problem of identity, which is usually characterized by coextensiveness.
Let A and C be any conditions expressible in the language of qualitative ontology. They are either formulas with one free variable or they are abstractors.

In the case of abstractors we accept the convention that they are realized by objects instantiating them. I.e., for a given abstractor a, a(x):= ax.

Conditions can be compared. In particular, two conditions are said to be coextensive, AC iff they are logically equivalent, i.e., they are realized by the same objects:
AC:= “x(A(x)  C(x)).

Observe that by the convention and by the axiom (C) coextensivity concerning collections is reduced to coextensivity of appropriate formulas. For example:
{x:A(x)}C(x)  “y({x:A(x)}(y)C(y))  “y({x:A(x)}yC(y))
 “y(A(y)C(y))  AC
Similarly, but with additional requirements involved, in the case of the remaining two abstractors.

8.24 Following Leibniz’s clue we accept the principle of the identity of indiscernibles, with unavoidable relativization to language:

(L) x=y iff “A(A(x)  A(y))
Two objects are identical, if they cannot be distinguished (by conditions expressible in the language).

8.25 We can try to weaken this principle further. Are all conditions necessary for identity? Can we limit ourselves to some basic conditions?

In the framework of qualitative ontology we can try limitation only to conditions delineating objects’ qualities and subjects. This leads to the idea of coequity:
xy iff “z(zxzy)z(xzyz)
Two objects are coequal iff they have the same qualities and they qualify the same objects. I.e.

(38) xy iff Q(x)=Q(y)E(x)=E(y).

Clearly, in our language equity implies coequity:

(39) x=y  xy.

8.26 What are the reasonable conditions to reverse this implication? As a matter of fact, in the real ontological universe even a weaker condition should suffice, because the original principle of Leibniz says:

(LP) x=y iff Q(x)=Q(y)

Objects with the same qualities are identical, which is the principle accepted in qualitative ontology.

8.27 Observe that all requirements outlined above exclude quite a lot of relational frames from the realm of qualitative ontologies. For example:

l

F x l l -x

l

is not ontological, because x and -x are different but indistinguishable in the F – network.
This, in fact, is an ontological reason for troubles with ontic negation. To solve it, we must refer to structures
big and complex enough to differentiate items which must be distinguished.

8.28 We accept the usual axiom of extensionality for collections

(EC) {x:A(x)}={x:C(x)} iff A  C

8.29 What about ideas? First of all, observe that

(40) Ideas generated by coextensive conditions have the same extension: AC  E([x:A(x)])=E([x:C(x)]).

Indeed, it is easy to see that the conditions of participation in ideas defined by equivalent formulas are equivalent.

However, the above implication is not reversible (cf.the next subchapter). Hence the coextensivity of two conditions is logically stronger than the coextensivity of the corresponding ideas.

Therefore, coextensivity might be used to characterize identity of ideas, as in the following axiom of extensionality for ideas:

(IE) [x:A(x)]=[x:C(x)] iff AC

Immediately

(41) [x:A(x)]=[x: C(x)] iff {x:A(x)}={x:C(x)}
Identity of ideas is equivalent to identity of collections.

8.29 However, some doubts concerning (IE) remain.
Surely, coextensionality regulates the problem of equality of extensions. Is it sufficient for equality of ideas, which seems to depend on some essential or intensional factors?
Observe that coextensive ideas can be different. For example, Being B and Nonbeing are coextensive (cf. (49) below) but, by (IE), they are different, for conditions defining them are not equivalent.

Qualities of Abstractors

8.30 Consider now the problem of characterization of the determiners or qualities of collections or ideas.
A minimal requirement seems to be preservation of provable equivalences, expressed in the following axiom of preservation:

(CP) C{x:A(x)}  “E((EA)CE)

This axiom immediately entails

(42) If C{x:A(x)}, then CA

(43) If EA, then Q({x:E(x)}=Q{x:A(x)}

Therefore (CP) entails the weak form of extensionality:

(44) If EA, then {x:E(x)}{x:A(x)}

8.31 In the next subchapter, it will be pointed out that, in general, the qualities of conjugate collections and ideas are different.

(45) Neither C{x:A(x)} implies C[x:A(x)] nor, reversely, does
C[x:A(x)] imply C{x:A(x)}.

Therefore the analogous axiom of preservatiom for ideas is not acceptable.

An important and difficult question as to what are the acceptable axioms characterizing qualities of ideas is left open for further study.

ETB

8.32 The Extended Theory of Being, in short: ETB, is an extension of PTB obtained by: i) use of the full language, as defined in chapter 4, and ii) application of the list of definitions from chapter 4 plus the above axioms concerning abstractors.
Thus in ETB axioms (C), (PC), (U), (LP), (CE), (IE) and, with some reserve, (CP) are applied to ontological concepts determined by the eight ontological qualities: B,, N, , U, , D and .

8.33 First of all, observe that by (7) each condition is down-compatible, hence in each linguistically detactable case collections and proper collections are coextensive, hence, by (CE), equal.

(46) For any formula A, DC(A). Therefore
“y([x:A(x)}y  A(y)  {x:A(x)}y), i.e.,[x:A(x)}{x:A(x)},
hence [x:A(x)}={x:A(x)}.

Indeed, by (7), “y(A(y)By). Therefore $x”y(A(y)xy), i.e. DC(A). Next, use axioms (C) and (CP) and logic.

8.34 In consequence, all appropriate ontological collections defined in §4.18: B, , etc. are proper and co-extensive with suitable qualities:

(47) For any y: ByBy, yy, NyNy, yy, UyUy, yy,
DyDy and yy.

Therefore, by suitable theorems proved in PTB

(48) i) B and are universal collections: B=OB=,
ii) and D are empty: ==D; whereas
iii)N, , U, are intermediate, i.e., neither universal nor empty: _N_OB, etc.

Proof. Cases i) and ii) immediately follow from (7) and (16); whereas for iii) it suffices to remind oneself that NB, , UB and .

8.35 Return to the paradoxical quality . By the axioms of PTB it produces a legitimate collection :={x: x}, which is empty and equal to {x:¬$y yx}.
As was pointed out in §§7.8 and 7.13 simples cannot occur in PTB models. This means that in universes with qualitative simples the axiom (A2) must be cancelled.
In such cases delineation of simples, for example by the collection {x:¬$y yx}, cannot be treated as objective, i.e. determined by quality which really is in the universe, but at best as purely subjective, a priori, delineation.

8.36 As regards ideas:

(49) Four of them: B, , D and are, for general reasons, empty.

Proof. By (7) and (16), qualities B and are universal, hence tolerate inconsistencies. Therefore, by (31), they must be empty.
On the other hand, qualities and D are empty. Hence, by (30), their conjugate ideas are also empty.

(50) The idea of nonuniversality is empty as well: E()=.

Proof. Notice first that its conjugate quality is known to be non-empty. For example: . Therefore, by (46), we should check only whether is up-compatible: UC()?
It is not. Assume UC(), i.e., $z”y(yyz). Fix this z. Therefore, inter alia, z. By the example mentioned previously and by detachment, z. This, however, is in contradiction with the known PTB description of , which must be co-simple.

On the other hand

(51) The idea of universality is not empty: E(U)_. In particular, qualities of being and nondefectiveness participate in it: UB, U.

Proof. For the first claim, it suffices to check up-compatibility of the quality U.
By (A5), “y(Uyyx). Therefore, $x”y(Uyyx), i.e. UC(U), as required.
As to the second claim, it suffices to remind ourselves that, by (7) and (16), both UB and U. Applying now (29) we obtain that UB and U.

Finally

(52) Both the idea of nonemptiness and the idea of emptiness are empty: E(N)==E().

Proof. As usual, it suffices to disprove an alleged up-compatibility of scrutinized ideas.
Assume first UC(N), i.e., $x”y(Nyyx). Fix this x. We know that both UB and , hence NU and N. Therefore Ux and x, i.e.,
x is contradictory. This, however, itself contradicts the principle of noncontradiction.
Assume now UC(), i.e., $x”y(yyx). Fix x. We know that nonbeing is empty: . Therefore x, which, by (7), is impossible.

8.37 To resume: From the eight basic ontological ideas, only one idea is nonempty and instantiated. It is the idea of universality: U.
It clearly delineates the subrealm of general or universal beings, as items participating in it.

8.38 Now, it is easy to see reasons against that ontological extensionalism in which the investigation of abstract notions is reduced to investigation only of their extensions.
Indeed, the ideas of being and nonbeing, i.e., Being and Nonbeing or B and , by (49), are co-extensional. But they are transparently different, and differentiated even by (IE), bacause they are not coextensive. I.e., E(B)=E(), but ¬(B).
Extensionalism is simply too strong, and therefore must produce paradoxes, like the paradox to the effect that Being and Nonbeing are the same.

8.39 We now also see reasons for the reservations expressed in §§8.28 and 8.32.
In fact, (40) is not reversible, because as was pointed out just above, B= but ¬(B).
On the other hand, (45) is indeed correct. As we know, NB but B. Hence, CB does not imply CB, and also CB does not imply CB.

Supports and Forms

8.40 Finally, turn to the four general notions introduced in §4.14, i.e., for arbitrary x the form, qualification, support and extension:
F(x):= [y:yx]
Q(x):= {y:yx}
S(x):= [y:xy]
E(x):= {y:xy}

First observe that by the two axioms of extensionality, (CE) and (IE), we have the following particular case of (41):

(53) F(x)=F(y) iff Q(x)=Q(y), and S(x)=S(y) iff E(x)=E(y):
Identity of forms is equivalent to identity of qualifications and identity of supports is equivalent to identity of extensions.

Whereas the first equivalence is rather difficult to apply because of lack at the present time satisfactory knowledge of qualities characteristic to ideas and other objects studied here, the second one generates some interesting results.

8.41 To see this, notice that the supports of our eight ontological qualities are their corresponding ideas: S(B)=B, S()=, S(N)=N, S()=, S(U)=U, S()=, S(D)=D and S()=.
Thus the logical relations, including identity and difference, between those ideas are, by (53), reduced to relations between appropriate extensions. But they, at least in part, are known:

(54) i) E(B)=OB=E()
ii) E()==E(D)
iii) The remaining four extensions: E(N), E(), E(U) and E() are neither empty nor full, and each of them is different from the remainig seven.

Proof. Cases i) and ii) immediately follow by (7) and (16).
Ad iii): Consider first E(N). It differs: from E(B)=E(), because E(B)-E(N); from E()=E(D), because BE(N)-E(); from E(), for BE(N)-E(); from E(U), because UE(N)-E(U); and from E(), because E(N)-E().
Next, take E(). It is neither empty, as E(), nor full, for E(), hence it differs from four extensions considered in i) and ii). E()_E(N), by the example provided previously. E()_E(U), because E()-E(U); and E()_E(), because NE()-E().
Finally, consider E(U). It suffices to note that E(U) differs from E(), even more – they are in fact disjoint.

8.42 Applying now (53) we obtain

(55) i) B = and = D,
ii) B, , N, , U, differ one with another.

Observe that the above situation is in full accordance with the model of §7.25.

8.43 Taking into account the results of §8.36, we obtain

(56) The following seven ideas are coextensive: B   N    D  , wheras the idea of universality U differs in
extension from each of them.

8.44 By the two previous claims we have that

(57) In ETB coextensivity of ideas does not imply their identity.

Indeed, by (IE), identity of ideas is equivalent not to their coextensivity, but to coextensivity of conjugate collections.

8.45 Consider, for example, Being and Nonbeing.
By (53) and (54), they are different: B_, because the extension of their conjugate qualities differ one with another: E(B)_E().
But, by (49), both ideas are co-extensive, i.e. they have common extension: E(B)=E().
In conclusion: ETB is not an extensional theory. Some objects, like B and are provably different in it, with, however, common extension.

8.46 Observe that Being and Nonbeing have the same support: S(B)=S(), provided the standard set-theoretical axiom of extensionality.
Indeed, by (53), S(B)=S() iff E(B)=E(). But, under the above proviso, E(B)==E(), which entails the conclusion.

Query: Characterize the common support of Being and Nonbeing.

8.47 Last, but not least, consider forms.
The form of x, F(x), is the unity of all its qualities or, in other terms, traits or determiners: [y: yx]. Is it real, or coherent, unity? What is the participation in a form?
As a matter of fact, in general we know very little about forms. In ETB, however, we can answer both questions in the positive.

(58) For each x: i) Its form, F(x), is coherent; and
ii) Participation in the idea of x is equivalent to qualification or determination of x:
F(x)y  yx.

Proof. For a form F(x) its corresponding condition is Fx(y):= yx. By (7), it is, like any condition, down-compatible: DC(Fx).
It is also up-compatible. This is so because UC(Fx):= $u”y(Fx(y)  yu), which is equivalent to $u”y(yx  yu). But this formula is a theorem of classical logic. Hence UC(Fx).
In consequence, F(x) is coherent, hence nonempty.
Applying now (29) we obtain: F(x)y  [z:zx]y  yx.

8.48 Notice that each universal quality is formal (in the ontological sense). I.e.

(59) Universal qualities participate in all forms: “y”x(Uy  F(x)y)

Infinity

8.49 As a matter of fact, there are quite a lot of forms, a fortiori quite a lot of ideas. Namely, infinitely many.
To see this first recall (53): F(x)=F(y) iff Q(x)=Q(y), which combined with Leibniz’s Principle (LP) gives:

(60) F(x)=F(y) iff x=y.

I.e., the mapping F: x  F(x) is one-to-one.

On the other hand, by (58), each form is a coherent, i.e. nonempty, idea.

Consider now the family of all objects OB, i.e. the full ontological universe, and its subcollection FM containing all forms. Clearly

(61) FM is a proper subcollection of OB: FM_OB.

Proof. Take, for example, the idea B. By (49), B is empty. Therefore it is an object which does not belong to FM.

In conclusion, there is one-to-one mapping F of the ontological universe OB onto its proper subcollection FM,
F: OB  FM.
Therefore

(62) The ontological universe OB is infinite in the Bolzano- Dedekind sense.

But FM itself is equinumerous with OB. Hence also

(63) The family of all forms FM is infinite.

8.50 The above infinity results are rather unexpected in purely ontological research (cf. Russell [34]). Therefore its proof deserves scrutinization.
First of all, the results are not immediate. In a sense, the main body of ETB is used in the proof.
Indeed, in the proof that the mapping F is one-to-one all three axioms of extensionality: (CE), (E) and (LP) were used. On the other hand, to prove that FM is a proper subcollection of OB we used (58), which depends on (7) as well as on (49), which in turn depends on classical logic and the axiom (U).
Therefore the axiomatic background of the proof is as follows: classical logic, PTB axioms (A1) and (A2), the unification axiom (U), Leibniz’s Principle (LP) and both axioms of extensionality: (CE) and (IE).

8.51 Notice that by (62) models of the theory outlined above, a fortiori models of ETB, are infinite.

8.52 Axioms of extensionality, which play so essential role in our proof, usually are considered as reducing the size of their domain.
Therefore, the ontological universe seems to be infinite also without extensionality axioms, provided that forms are still present in it.
Query: Prove or disprove this hypothesis.

Final Remarks

8.53 The being, i.e. the universe of beings is very broad indeed. Hence its idea is also very general. Too general to be nonempty.
But some ideas are nonempty. Moreover, ideas are interrelated by means of logical relations between appropriate conditions. Hence in the realm of nonempty ideas some ideas are more general than others.
Therefore we can and should search after regions of the being which are metaphysically coherent, i.e., such that their objects participate in common (or compatible) ideas.

8.54 From a metaphysical point of view, a possible world is any region of the being which is maximal with respect to participation in ideas.

8.55 I would like to finish the present study of being by asking certain simple questions which are, for sure, easier to ask than to answer:
What is the idea or ideas delineating the real world?
Is it the idea of reality, or the idea of existence, or both?
Are these two ideas different?
Which is their relation to the only nonempty idea among the basic ontological ideas: the idea of universality?

9. Parmenidean Statements Reconsidered and Classical Questions Answered

9.1 Parmenidean statements concern the status of being and nonbeing: being is, being isn’t, etc.
Previously three variants of being were distinguished. A particular being, say a being x: Bx, the being: B and Being:B. Similarly for nonbeing: a nonbeing x (x), the nonbeing , and Nonbeing .
My plan is to use the above distinctions to express in our formal language different versions of the Parmenidean statements and next to discuss them.

9.2 Variables denote objects. Beings are qualified objects, whereas nonbeings are objects without qualities. I.e., the variable x varies over the full ontological universe OB. Its restriction to the (improper) subuniverse of beings is made by means of the objectivum B, whereas the restriction to the (empty) subuniverse of nonbeings is made by means of the objectivum .
The remaining relevant concepts: B, , B and are four objects distinguished within the realm OB.

9.3 B is, above all, an objectivum. It is made from the verb “to be” by the gerundial construction: being = be + ing.
According to the anylysis provided in chapters 4, 5 and 6 it has several readings. Bx, in its primary use, means: x is, which through paraphrasing becomes: x is be-ing or, as I prefer to say, x is in being. This, in qualitative ontology, is clarified by: x is a being.

9.4 The basic principle of qualitative ontology is therefore:

(QB) x is iff x is in being iff x is a being

9.5 Analogously, Bx means: x is the being, whereas the standard reading if Bx is: x is Being.

9.6 Similarly for negative phrases.
x means, primarily, that x isn’t. Via paraphrases analogous to those made in the case of B, it is also read: x is in nonbeing or x is a nonbeing.
x means: x is the nonbeing, whereas x: x is Nonbeing.

9.7 With this key it is easy to find formulas expressing Parmenidean statements.
“Being is” is, depending which kind of being is taken into
account, either BB or BB or BB.
“Being isn’t” is either B or B or B.
“Nonbeing is” is either B or B or B.
“Nonbeing isn’t” is either or or .

9.8 By the above paraphrases we have the following unfolding of the basic Parmenidean statements:

A being is iff A being is in being
iff A being is a being
A being isn’t iff A being is in nonbeing
iff A being is a nonbeing
A nonbeig is iff A nonbeing is in being
iff A nonbeing is a being
A nonbeing isn’t iff A nonbeing is in nonbeing
iff A nonbeing is a nonbeing

The being is iff The being is in being
iff The being is a being
The being isn’t iff The being is in nonbeing
iff The being is a nonbeing
The nonbeing is iff The nonbeing is in being
iff The nonbeing is a being
The nonbeing isn’t iff The nonbeing is in nonbeing
iff The nonbeing is a nonbeing

Beig is iff Being is in being
iff Being is a being
Being isn’t iff Being is in nonbeing
iff Being is a nonbeing
Nonbeing is iff Nonbeing is in being
iff Nonbeing is a being
Nonbeing isn’t iff Nonbeing is in nonbeing
iff Nonbeing is a nonbeing

9.9 Observe the metaphysical tenor of the equivalences made above and, in turn, some virtues of our formalism.
The formalism of PTB and ETB indeed enables us to answer which of the statements under scrutinization is true. On the other hand, it also makes for fluency in dealing with these rather obscure metaphysical formulas.

9.10 To see this, let us scrutinize the positive Parmenidean statement “Being is”. As we know, this has at least three variants, one for each of the three sorts of being.
In addition, two versions of it should be distinguished: ontological and logical.

9.11 In the ontological version the predicate “is” has a purely ontological meaning: to be means to be in being, which is expressed by means of the objectivum B.
But, by (7), we know that everything is: “xBx.
Therefore, by (QP), everything is in being or everything is a being.
A fortiori, being of each sort is a being. Among others, BB: being itself is or being is in being or being is a being.
Furthermore, BB: The being is, or the being is a being. Also, BB: Being is , for it, like everything, is in being or is a being.
And, in the case of nonbeing, B: nonbeing is, B: the nonbeing is, and B: Nonbeing is (sic!)

9.12 The logical, or quantifier version says: Whatever is, is. Or: Whatever is, is in being. Or: Whatever is, is a being.
This is a bit ambiguous. It either says “Whatever is a being, is (a being)”: “x(BxBx).
Or it could be understood as “Whatever is an object, is (a being)”: “xBx.
Both versions, in fact, are true: the first – by classical logic, whereas the second – by PTB.

9.13 Consider now forms using B and B instead of B.

9.13.1 As we know, Bx means: x is the being.
It is also ambiguous. Its ontological reading, developed in PTB says: x is in the being, which, by axiom (C), is equivalent to Bx, i.e., x is a being.
Hence, again by (7), everything is in the being: “xBx. A fortiori, BB: being is in the being, BB: the being is in the being, and BB: Being is in the being.
Similarly for the three sorts of nonbeing.

On the other hand, the logical version of Bx stresses the identity-reading. I.e., x is the being means x is identical with the being: Bx iff x=B.
This is so, because B is a distinguished object of the universe.
It is easy to see that in the logical version the claim that everything is the being as well as the claim that each being is the being, etc implies a weak monistic thesis: there is at most one object.
Indeed, the first claim is simply “x(x=B); whereas the second one is “x(Bx  x=B), which, by (7) is also equivalent to “x(x=B).
In conclusion, the logical version of the statement “x is the being” can easily be used to entail a monistic thesis.

9.13.2 As regards Bx, which means: x is Being, notice first that its ontological explication says that x participates in Being.
In ETB, by (49), we have that nothing participates in Being, hence all suitable positive Parmenidean statements are false.
The logical reading of Bx says: x is the same as Being. Therefore, claims like “Everything is Being” or “Each being is Being”, interpreted logically, imply monism; wheras under the ontological reading they imply contradiction.

9.14 Passing to the negative statements:

9.14.1 In PTB, nothing isn’t: ¬$xx, because, by (7), everything is: “xBx.
A fortiori, all negative Parmenidean statements under the ontological interpretation are false.

9.14.2 The logical version of “Nonbeing isn’t” is: Whatever isn’t, isn’t.
This is a quantifier statement which, in virtue its inherent ambiguity, can be formalized in one of the following three ways:
(i) Whatever is not an object, isn’t: “x(x is not an object  ¬Bx).
The notion of an object is, however, the most general of all, i.e., everything is an object. Therefore, the antecedent of (i) is false, hence, by classical logic, the implication (i) is valid.
(ii) Any object which isn’t, isn’t: “x(¬Bx¬Bx).
It, like the first formalization of the appropriate positive formula (cf.§9.12), is a classical tautology.
And, finally, a purely logical reading:
(iii) For no x, x isn’t: ¬$x¬Bx, which is equivalent to “xBx, i.e., UB.
This formalization, like the second one in §9.12, is a theorem of PTB.

9.15 The positive and the negative Parmenidean statements in their logical versions are formalizable in two equivalent ways. They are either classically true or equivalent to the theorem of PTB: UB.
This equivalence was recognized, among others, by Leibniz, but not by Kant. Recall that Leibniz was using [10] only the positive version of the logical Parmenidean principle, indicating it as one of the few basic principles of human knowledge. About eighty years later Kant used [11] both versions introducing them as two different axioms of his precritical metaphysics.
However, by classical logic, Kant’s distinction is only verbal, because both versions are equivalent.

9.16 Finally, I am going to answer the classical ontological questions of chapter 5.

Ad Q1. Yes: Being is, in each version and variant considered in the present paper: In particular: Beings are.
Also: The nonbeing is. And, Nonbeing is, But, nonbeings aren’t.

Ad Q2. Yes: Each variant of being has content, for everything has a content: “x BQ(x).
N.B., nonbeings have empty content by definition, but there is no nonbeing.

Ad Q3. Yes: The quality B “to be in being” is universal.

Ad Q4. Yes: Some objects are universal. For example, B is universal: UB.

Ad Q5. Emphatically yes. Being is well-defined and, as the theory shows, a nontrivial object.

9.17 As regards the argument against categoremacity of being, pointed out in §5.12, observe that:
The most general notion in given ontological framework, in our case the notion of an object, must be syncategorematic. Indeed, it is used like a general variable. Hence it can be defined only contextually, by use. An essential definition is not available, because any such definition must refer to at least one more general, or at least one foreign, term. This, however, is impossible in the ontological framework, due to its extreme generality and, in consequence, to the most general status of its basic notion.
Someone can try to apply the same argument to other universal notions, among others to the notion of being. It indeed, by (7), is universal.
In vain, however. Universality is not generality. The notion of a being is in virtue of its definition not the most general notion.It was defined by means of the notion of ontological connection, which in the framework of PTB is more primitive, hence a more general, notion.
The notion of a being is universal, i.e., it qualifies any object. It was found to be such by means of a calculation, relying on classical logic and referring to the primitive notion of PTB, i.e., the notion of ontological connection.
Therefore, the notion of a being as well as the two subsequent notions of being are categorematic, though universal.

10. Summary

10.1 In the main, the present essay is an exercise in qualitative ontology.
To solve questions arising in traditional ontology two formal theories of being were introduced: PTB devoted to the quality “to be a being” considered as an object; and its extension ETB, introducing formalism necessary to deal with abstract concepts of being.

10.2 The chief result of PTB is that everything is a being, hence nothing is a nonbeing.
As regards ETB, its remarkable contribution is, I think, the characterization of ideas and participation, with the main observation that nothing participates in ideas which are too general, that their extensions are empty. In particular, nothing participates in the idea of being, in spite the fact that everything is a being.
As by-product we obtain that forms are coherent, hence nonempty, ideas and, in consequence, that the ontological universe is infinite.

10.3 As regards Parmenidean statements it was shown inter alia:
That beings are in being.
That the being, as well as the nonbeing, is a being; hence both of them are in being.
That Being, as well as Nonbeing, is a being; hence both of them are in being.

10.4 Hence, in the light of logic, the way of truth is:

Being is and Nonbeing is;
beings are, but nonbeings aren’t.

Acknowledgments

The paper covers the first two of the five topics of my lectures “A Theory of Qualities” read at the International Summer School in Philosophy, “Formal Ontology”, Bolzano, Italy, July 1-5, 1991. Its first version was presented at the University of Trento in May 1990.
The first draft of the paper was prepared during my stay at Universidade Federal da Paraiba, Jo_o Pessoa, PB, Brasil. The support of the CNPq grant no.30.0095/90 is thankfully acknowledged. Its final version was written during my stay at the Internationale Akademie für Philosophie im Fürstentum Liechtenstein, prepared as a part of a project sponsored by the Swiss National Foundation for Scientific Research on the topic “Formal Ontological Foundations of Artificial Intelligence Research”. Thanks go both to the SNF and the IAP for their support and hospitality.
I would like particularly to thank Prof.Matias F.Dias and Prof.Barry Smith for their kind invitations and providing me in both places with excellent conditions to work.
I also owe thanks to a number of scholars, in particular C.Gorzka, A.Pietruszczak, R.Poli, P.Simons and B.Smith for inspiring remarks, as well as I.Bodnr, R.Lüthe, B.Mezei and R.Olvera-Mijares for discussions concerning linguistic mysteries of the verb “to be”.
Last, but not least, I like to thank Cz.Porêbski for his friendly help.

References

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Notes:

[1] Cf.Diels [7], Parmenides B2.3. Notice a rather subtle problem connected with the translation of this claim (see Bodnár [1]). Inter alia, the following translations have been offered: Diels [7] : “..dass [das Seiende] ist und dass es unmöglich nicht sein kann..”, Bormann [4]: “..dass [das Seiende] ist und das Nicht-Seiende ist nicht..”, Kirk and Raven [13]: “..that it is and that it cannot not-be..”, Burnet [6]: “..It is, and..it is impossible for it not to be..”, Tarán [36]: “..it is and to not be is not..”,Manheim in [9]: “..it is, and..nonbeing is impossible..”.
[2] For Parmenides and Zeno cf. Kirk and Raven [13], for Plato cf. Parmenides and particularly the Sophist in [29].
[3] Cf.Tatarkiewicz [37]
[4] Cf. Sophist, 258 b-d.
[5] On the differentiation between being, thought and language ontologies cf. [22] and [23].
[6] Cf. Kisielewicz’s modification of ZF into the double extension set-theory [12].
[7] In a way resembling, to some extent, the manner of Bolzano in his [2]. Cf.also [3]. I owe this reference to P.Simons.
[8] Cf.Owens [19] and Poli [30]. The Aristotelian position was defended forcefully by Twardowski in [38].
[9] For a very illuminating discussion of the development of ontological terminology in Greek and Latin cf, Kahn [10].
[10] Cf. Leibniz [14]
[11] Cf.Kant [11]

JERZY PERZANOWSKI