Bulletin of the Section of Logic Volume 37:3/4 (2008), pp. 233–244 Janusz Kaczmarek WHAT IS A FORMALIZED ONTOLOGY TODAY? AN EXAMPLE OF IIC Abstract The paper presents some proposal of formalized ontology. It is based on Meinon- gian ideas (complete and incomplete objects), as well as formally given structures of individuals, kinds and concepts. These structures are used, in the next step, to construct semantics for modal logic S4 thus yielding ontological version of it, i.e. OS4. Some selected theorems of proposed ontology referring to ontological objects are given. The investigation of ontological questions by logical and/or mathemat- ical tools is called formalized ontology. It is a well developed current of contemporary logic and ontology where, for example, logic of action (Bel- nap and Perloff [2]), logic of essence (Fine [4]), researches on general lan- guage of ontology (Leipzig group, Sowa [13], Poli [12]), axiomatic ontology (Meixner [10]), combinational onto/logic (Perzanowski [11]) or situation ontology (Wolniewicz [14], Zalta [15] and [16]) are most representative. The term formalized ontology is used by Poli [12] and Guarino [6], oth- ers use mostly the term formal ontology (Cocchiarella, Sowa), whereas Perzanowski urges the term ontologic or onto/logic meaning “modulo-logic ontology”. Cocchiarellas general formulation is well-aimed: “Formal ontol- ogy is the result of combining the intuitive, informal method of classical ontology with the formal, mathematical method of modern symbolic logic ...”. And “... formal ontology, the result of combining these two methods, is the systematic, formal, axiomatic development of the logic of all forms and modes of being” (Cocchiarella [1]). 234 Janusz Kaczmarek We find different lists of basic concepts in particular approaches. Zalta ([15], [16]) introduces and analyses, for example, ordinary and abstract objects, states of affairs, situations, abstract properties, relations of en- coding and of exemplifying, to be a part, world and possible worlds, etc. Next, Leipzig group (Degen, Heller, Herre and Smith [3]) in their general ontological language (GOL) point at sets, urelements, universalia, indi- viduals, time, space, properties and individual properties, relation of in- statition, identity, event, process, situations, material and formal relations, chronoids, topoids, substantoids and so on. In my opinion, terminological base, considered questions and obtained theorems explain the essence of ontology most suitably. In this paper I will present some of my formalized ontology worked out in [9]. I call this proposal by acronim IIC (Individuals- Ideas-Concepts). Most of the theorems will be presented without proofs. 1. Philosophical Intuitions We meet in our world substances like Socrate, Plato, You and me. We refer to them by the name a men. Why? Because this name has its meaning, we bound with the name a concept. Hence, individuals, language and the domain of concepts lie in the center of our interest. But philosophers also point out a form of individuals or kinds and species that we join with the individuals. They ask: why do we name this different substances by one sign a men? And, why we do not call Bucefal by the name of a men. Because Socrate, Plato, You and me have the same form (substantial form) i.e. we are of the same species. If so, we have to consider a domain of ideas additionally. And then, ontologist must take into account four domains: domains of individuals, ideas, concepts and language. The last is, of course, useful to express ontological truths about individuals, ideas, concepts and their mutual dependences. Individuals are represented in IIC by functions o from T (a set of fea- tures, qualities or atomic properties) into the set {0, 1}, ideas by functions 1 e from T into {0, /2, 1} and concepts by functions C from Q (a set of con- tents) into {0, 1}. However, let ask, what is a language of ontology? It is, mostly, a natural language, but in the case of formalized ontology it is a language of a mathematical theory or a logic. My intension is simple and is grounded on Meinongian idea that there are two kinds of objects: What is a Formalized Ontology Today? An Example of IIC 235 complete an incomplete. The former correspond with the function of type 1 e, the latter with that of type o. Moreover, if a function has a value 0, /2 or 1 for feature t, then I interpret t as, respectively, a negative, accidental or positive feature. In the case of representing an idea, I suppose additionally that every 1 function e has the value /2 for at least one t. It corresponds to Meinongian formulation, i.e. that incomplete objects are objects such that for some atomic properties t they neither have t nor the complement of t. For example, the idea of a man neither has the feature just nor the feature unjust. Moreover, unlike Meinong and according to the spirit of Ingarden’s ontology, I interpret the features t from T as the elements of a content of idea. The features t are not the properties of idea but the properties of an individual that exemplifies that idea. 2. Individuals, Ideas, Concepts Let us remind: a partially ordered set hX, ≤i is to be called pseudo-tree iff for any x ∈ X the set O(x) = {y : y ∈ X & y ≤ x} is a chain. In the class of pseudo-trees one can distinguishe a subclass of trees. Namely, a pseudo-tree hX, ≤i is a tree iff any set O(x) is a well-founded set. Example: the set of all functions from {0, 1, . , n} to {0, 1}, for n ∈ ω, with relation of inclusion ⊆ is called a binary tree. ∅ is a minimal object referred to as a root of the tree. 2.1. Construction of PTS, CS and U. Let T be an uncountable (and infinite) set and T 0 its infinite subset. The 0 set T can be represented as a sequence (t0, t1, t2,...). Let us consider a set 0 1 BT (binary tree) of all functions e : T → {0, /2, 1} such that e(t) ∈ {0, 1} 1 0 for t ∈ {t1, . , tn} and e(t) = /2 for t ∈ T − {t0, t1, . , tn}, and let ≤ 0 0 be a relation fulfilling a condition: e ≤ e iff e ⊆01 e , where ⊆01 refers to inclusion on the sets of these pairs ht, ki of e and e0 for which k = 0 or k = 1. It is a fact, that hBT, ≤i is a tree. Moreover, hBT, ≤i is isomorphic with the binary tree (on a set of natural numbers). Fact 1. Let ∅ 6= BTFIN ⊆ BT and BTFIN be a set fulfilling the following 0 conditions (1) there exists e ∈ BTFIN such that for any e ∈ BTFIN : e ≤ 236 Janusz Kaczmarek 0 0 00 00 0 0 00 e , (2) for any e ∈ BTFIN : if there exists e such that e 6= e and e ≤ e , 000 000 00 000 0 then there exists e ∈ BTFIN such that: (a) e 6= e , (b) e 6= e , (c) 00 000 000 00 0 000 ∼ (e ≤ e or e ≤ e ) and (d) e ≤ e . Than (BTFIN , ≤) is a tree. Of course the Condition (1) forces one root and Condition (2) that after any object e (in the sense of ≤) there exist 0, 2, . , n elements (but not 1), for n ∈ ω. The structure hBT, ≤i is pararel with the structure of genera and species that is due to the philosophers like Plato, Aristotle and others. Thus the relation ≤ may be called over and its convers under. By its means the idea that one species (e.g. a men) is under another (e.g. an animal) may be grasped. Definition 1. hBTFIN , ≤i is a simplified Porphyrian tree structure, shortly: PTS. An element e is to be called an incomplete object (in Meinongian terms). If e(t) = 1 or e(t) = 0, the feature t is an essential positive (or negative) fea- 1 ture, respectively. Next, if e(t) = /2, then t is an accidental or unspecified feature. It is worth pointing out that I do not treat these properties (fea- tures) as the properties of incomplete objects (of genera or species). They are properties of complete objects (individuals) generated by the first (see Definition 4). Following Ingardens ontological solutions we can tell that these properties belong to the content of an idea and there are no idea’s properties. They are properties of what exemplifies the idea. Definition 2. A relation ≤ on a set PTS is signed by over and the converse of over by under. Definition 3. Let e ∈ PTS and for any e0 6= e : ¬(e0 under e). Then I call e natural species in PTS. If NS is a set of all natural species of PTS and G = PTS − NS, then I call elements of G proper genera of PTS. Of course, any two species of PTS are incomparable with regard to under. Definition 4. Let e ∈ NS, De be a domain of e and Ue be a set of all functions o from De into {0, 1} such that e ⊆01 o. Then any function o ∈ Ue is called a complete object generated by species e. Next, a set S UPTS = Ue, for any e ∈ NS, is a universum (or domain) of PTS. e What is a Formalized Ontology Today? An Example of IIC 237 Remark. A set of complete objects generated by e can be interpreted as a set of individuals (like Socrate) with the same essence (species). Evidently, Socrate has a positive and/or negative properties fixed in the species as positive and/or negative, respectively, but additionally, all accidental prop- 1 erties of species (with value /2) are positive and/or negative (i.e with value 1 or 0) in case of individual - Socrate.