1. Ontology as a formal one
The language of ontology as the ontology itself: the zero-level language

2. Vasil Penchev
Bulgarian Academy of Sciences: Institute for the Study of Societies and Knowledge: Dept of Logical Systems and Models
Formal Methods and Science in Philosophy Inter-University Centre, Dubrovnik, Croatia May 4-6, 2017

3. Abstract:
“Formal ontology” is introduced first to programing languages in different ways. The most relevant one as to philosophy is as a generalization of “nth-order logic” and “nth-level language” for n=0. Then, the “zero-level language” is a theoretical reflection on the naïve attitude to the world: the “things and words” coincide by themselves. That approach corresponds directly to the philosophical phenomenology of Husserl or fundamental ontology of Heidegger. Ontology as the 0-level language may be researched as a formal ontology

4. Prehistory and background

5. Both “words” and “things”
The “words” and the “things” are not distinguished from each other in the naive attitude to the world
This was restored by Husserl’s phenomenology and Heidegger’s fundamental ontology
Their “phenomenon” can be naturally interpreted linguistically, semiotically, and logically
Furthermore, it makes meaningless the standard criterion of truth as adequacy for it is always and automatically met after the thing and word are the same

6. The “phenomena” Furthermore, they are or as the “true signs”
“Phenomena” are those words, which are the "things themselves“
Furthermore, they are or as the “true signs”
The signifier and the signified are not connected quite loosely and conventionally in those “true signs”
They are linked to each other necessarily
Thus all the three, sign, signifier, and signified coincide with each other,
Therefore they mean the same from three different viewpoints

7. The language of “phenomena”
Furthermore, 'phenomenon' can be also understood as the unit of the language of "zero-level" and its corresponding logic
Indeed, Husserl, a mathematician by education, introduced a new “rigorous” viewpoint to logic initially, and then to psychology, and philosophy
It consisted in the identification of “form and content” definitively in those areas
This approach may be generalized by involving the idea of “formal ontology”

8. The order of logic and the level of language
Logic involves the concept of the "order" of a certain given logic according to whether its symbols may refer only to:
External things ("first-order logic")
Symbols of external things ("second-order logic") as well
The symbols of symbols of external things ("third-order logic") as well
… etc.
Analogically, the level of language is involved

9. The zero-order of logic and the zero-level of language
One natural generalization of that logical conception is the introduction of "zero-order logic" as well as the corresponding "zero-level language“
The symbols do not refer to external things, but they coincide by themselves with those external things
Thus the naive attitude to the world, after which the things and the words coincide by themselves, in a rigorous and logical notion, that of "zero-level language“
So, Husserl’s phenomenology and Heidegger's fundamental ontology can be interpreted as the metaphysics of n-order language

10. About the formal definition of ontology
Then, the concept of ontology admits one exact logical definition as any zero-level language or as all zero-level languages
The properties of the zero-order logic and zero-level language can be naturally transferred and thus interpreted as formal and logical properties of ontology
They allow of the axiomatic definition of "formal ontology" as the category of a certain mathematical object
Furthermore, any “formalizing ontology” can be grounded on that kind of formal ontology

11. Formal and formalized ontology
That fundamental approach is widely applied in the research of software formal and artificial languages created by humans for computers for all of them are logically rigorous and grounded on logic
After that conception of formal ontology, the consideration of any programing language to itself rather than to an external supposed "hardware" or to an external reality supposedly modelled generates a formal or formalizing ontology
On the other hand, meaning a certain hardware or reality for being modelled as granted, a formal software language only repeating them would be their corresponding formal ontology

12. Problem

13. The problem:
May ontology be defined exhaustedly and unambiguously by its language?

14. Thesis

15. The thesis:
(1) Yes, it may if its language is defined in turn as the zero- level language
(2) The fundamental features of ontology according to the philosophical meaning of ‘ontology’ can be deduced from the formal properties of ‘zero-level language’
(3) A few interesting new corollaries are implied by the formal and axiomatic postulating ontology as the zero-level language

16. A few arguments in favor of the thesis

17. The thesis:
(1) Yes, it may if its language is defined in turn as the zero-level language
(2) The fundamental features of ontology according to the philosophical meaning of ‘ontology’ can be deduced from the formal properties of ‘zero-level language’
(3) A few interesting new corollaries are implied by the formal and axiomatic postulating ontology as the zero-level language

18. About (1):
The etymology of “ontology” in ancient Greek addresses the Word which exists, therefore postulating the identity of ‘word’ and ‘thing’. The same idea penetrates Christianity and its theology: what exists is God’s Word
Thus the zero-level language is right God’s language. God’s language was secularized into the “language of nature”, and mathematics was alleged as it in the modern age
However, mathematics unlike God’s language was understood as a “human language”, i.e. not as a zero-level language
The distinction between mathematical model and reality is a fundamental postulate of philosophy of science

19. More about (1):
Husserl’s phenomenology and Heidegger’s fundamental ontology resurrected the concept of the zero-level language in philosophical terms
However, their projects are sometimes interpreted as properly philosophical projects versus (especially Heidegger's) those of philosophy of science or analytic philosophy
In fact, both can be interpreted not less as formal or even formalizing projects in a kind of contemporary neo- Pythagoreanism

20. More about (1):
Consequently, the idea of ontology as the zero-level language is an ancient conception in history of philosophy and theology with many arguments in its favor and discussed during centuries
That understanding of ontology implies its coincidence with its language
If meta-level is postulated as different from the level itself for any language created by humans and thus conventional, this in turn implies the hypothesis of the zero-level language, which is single unlike the plurality of human languages and in which the meta-level of the words coincides with the level the things

21. More about (1):
In particular, the mathematical models of reality should coincide with reality if mathematics as the “language of nature” would be interpreted as that zero-level language
This is an idea, the origin of which can be traced to Pythagoras and his successors even before or simultaneously with the beginning of philosophy in ancient Greece
One can continue Heidegger’s restoration in the origin of philosophy further: from the Pre-Socratic philosophers such as Heraclitus and Parmenides to Pythagoreanism

22. The thesis:
(1) Yes, it may if its language is defined in turn as the zero-level language
(2) The fundamental features of ontology according to the philosophical meaning of ‘ontology’ can be deduced from the formal properties of ‘zero-level language’
(3) A few interesting new corollaries are implied by the formal and axiomatic postulating ontology as the zero-level language

23. About (2):
Ontology is both language and being as the etymology of the word demonstrates. It consists of those words, which are not conventional, but linked to the existing entities necessarily
It is a single one in philosophical definition unlike the definition of formal ontology, after which a given, even usually artificial language and thus conventional is only postulated as an ontology among many others available or possible
All these fundamental features of ontology are implied immediately by its understanding as the zero-level language

24. More about (2):
Furthermore, ontology implies a special conception of truth, which is not that of correspondence: the correspondence of words and things, for the words coincide with the things by themselves
Ontology is necessarily true in the sense of correspondence in definition and thus that conception of truth as adequacy is useless and meaningless being always valid
Heidegger tried to replace “adæquatio” by ἀλήθεια (unhiddenness) as that kind of truth relevant to ontology
Its sense is the ontology itself as truth or as the special kind of philosophical truth
It means one to move to ontology from any kind of non-ontology

25. More about (2):
Any non-ontology is mediated by some language conventional to the signified and therefore the former “hides” the latter
If however one manages to see ontology, Husserl's “things themselves”, or “phenomena”, which “show themselves in themselves by themselves” in Heidegger, the “veil vanishes” and truth appears by itself right as “unhiddennes”, ἀλήθεια
Thus, ontology is the state of being in truth as well as what is seen in that seen for the state and the seen coincide in definition of ontology

26. More about (2):
The zero-level language implies an analogical concept of truth: if many realities have been involved, only a single one is privileged as true, right ontology
Accordingly: if many languages have been involved, only a single one is privileged as true, right the zero-level language therefore necessarily coinciding with ontology
Right in the spirit of mathematical formalization, one can speak of the joint group pf both languages and realities, in which its “zero element” is naturally privileged

27. More about (2): Ontology is totality
In particular, it contains its externality within itself, and any meta-position to it should be internal within itself
For example, the universe can be postulated so in a physical sense of totality and therefore any other universe should contains within it
All those conditions are only extraordinary and seemingly paradoxical, but not contradictory

28. More about (2):
The zero-level language implies a similar, mirror kind of totality
Indeed, any language as a special kind of ‘thing’ will be an element within the zero-level language
For example, totality can be thought as, or even defined as what is single in definition:
Then all totalities (if any) must coincide and thus must be identified as only a single one, particularly, language and reality as “totalities” (an argument from Heidegger’s idea of “fundamental ontology”)

29. The thesis:
(1) Yes, it may if its language is defined in turn as the zero-level language
(2) The fundamental features of ontology according to the philosophical meaning of ‘ontology’ can be deduced from the formal properties of ‘zero-level language’
(3) A few interesting new corollaries are implied by the formal and axiomatic postulating ontology as the zero-level language

30. About (3):
The interpretation of ontology as the zero-level language implies it to be interpreted formally, logically, and mathematically, e.g. as the maximal formalizing ontology
This involves in turn a kind of neo-Pythagoreanism for ontology in the most fundamental sense turns out to be mathematical
The Number of Pythagoras underlies that Word of Heraclitus coinciding with the Thing
Indeed, Pythagoras’s Number had preceded and generated Heraclitus’s Word in the history of philosophy

31. More about (3):
For example, one can introduce the mathematical notion of group for the hierarchy of both languages and realities generalizing the intention of Russell’s theory of types
Then, ontology coinciding with the zero-level language would be the zero (or “unit”) element of that group and thus privileged
That unit being right single can be furthermore considered as the totality containing necessarily all other realities and languages within it just as the ontology of all

32. More about (3):
The discussed above totality of the zero-level language needing the entire group to be represented into its zero element implies both complementarity (e.g. in the sense of Niels Bohr) and mathematical axiom of choice equivalent to the principle of well-ordering
That complementarity of language and reality implies for the linguistic units such as words to be interpreted as ontological “quanta”, in which the counterpart of word is fundamentally inseparable from the counterpart of thing
The language of ontology as the zero-level language consists right of those ontological quanta

33. More about (3):
That zero-level language of ontological quanta being identical to ontology is the fundamental reality as well
Consequently, one can interpret the physical quanta of action determined by the fundamental Planck constant just as the ontological quanta at issue according to the contemporary level of knowledge
Then, quantum mechanics would supply by the necessary zero- level language of ontology, which is furthermore properly mathematical: the formalism of the separable complex Hilbert space

34. More about (3):
That formalism of quantum mechanics implies an internal proof of completeness as what the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker 1968) can be interpreted
That proof confirms indirectly the formalism of quantum mechanics as the zero-level language of ontology as it is supposed to be complete correspondingly to the totality of ontology
Indeed, the research of an alleged physical “theory of all” corresponds directly to the ancient and contemporary philosophical idea of total or fundamental ontology meeting each other on the “bridge of quantum mechanics”

35. More about (3):
This implies furthermore a few corollaries about the consistency and completeness proof of mathematics for it should underlie ontology according any form of neo- Pythagoreanism
The separable complex Hilbert space as a generalization of Peano arithmetic is what is able to ground mathematics
That Hilbert space is able to unifies both arithmetic and geometry and thus mathematics as a whole
Furthermore, that unified mathematics turns out to be the physical world itself by quantum mechanics …

36. Conclusion:
The interpretation of ontology as the zero-level language is useful and fruitful

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