What Is a Formalized Ontology Today? an Example of Iic

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 mathematical 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 language 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], others 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 ontology 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, individuals, 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 features, 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 contents) 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 (features) 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.

0 0 Fact 2. Let e, e ∈ PTS be two species and e 6= e . Then Ue ∩ Ue0 = ∅.

0 0 Proof. Let o and o be any objects such that o ∈ Ue and o ∈ Ue0 . 0 0 0 Then e ⊆01 o and e ⊆01 o . If e 6= e , then there exists t ∈ T such that 0 e(t) = 1 and e (t) = 0, or conversely. So, for any o ∈ Ue, o(t) = 1 and 0 0 for any o ∈ Ue0 , o (t) = 0, (and in the latter case, respectively). Hence, Ue ∩ Ue0 = ∅.

Definition 5. A relation under (over) extends on a set PTS ∪ UPTS, i.e. for any a, b ∈ PTS ∪ UPTS : a under b iﬀ b ⊆01 a. Fact 3. 1. Any two individuals, o and o0, are incomparable. 2. A set hPTS ∪ U, overi is a tree.

Definition 6. Let Q be a countable infinite set, Q ∩ T = ∅ and QFIN = {QF : QF ⊂ Q & QF is a finite set} be a family of finite subset. By a concept system (CS) I understand a set of (not necessary all) functions C : QF → {0, 1}, for any QF ∈ QFIN , with a relation ⊆01 given by 0 0 the condition: C ⊆01 C iff C ⊂ C . I will write: hCS, ⊆01i. One may interprete the functions C,C0,C00,... from CS as concepts and the elements q1, q2,... of Q as (atomic) contents (or constituents) of concepts.

Fact 4. The relation ⊆01 on CS is a partial-ordered relation. Definition 7. An incomplete object e ∈ PTS generates object o ∈ U iﬀ he, oi ∈≤PTS∪U , which obviously means that o exempliﬁes e.

Definition 8. For any e ∈ PTS, a set of all o such that he, oi ∈≤PTS∪U may be called an extension of incomplete object e, shortly: Ext(e). Commentary. The extension of e is deﬁned as a subset of universum U. This universum is ﬁxed by natural species, so the extension of any uncom- pleted object e is equal to the union of all extensions of s where s is a natural species and e ≤PTS s. 238 Janusz Kaczmarek

Now, let us consider some relations between: P T S, CS or U. To this aim, we take into account a one-to-one function I from T onto Q. This function ﬁxes one-to-one correspondence between elements of T (features) and Q (atomic contents). Then we consider notions of extension of concept or of adequacy between CS and PTS.

Definition 9. Let hCS, ⊆01i be a concept system and C ∈ CS. Let, additionally C∗ be a function from T in {0, 1} such that: hq, ki ∈ C iff hI−1(q), ki ∈ C∗, for q ∈ Q, k ∈ {0, 1}. Then any object o ∈ U, i.e. an complete object belonging to the universum U, is to be called a designator ∗ of C ∈ CS at I iff C ⊂01 o. Definition 10. The set of all designators of C at I is an extension of C at I and I call it CExt(I)(C). Example. Let us take into account a concept of a man as a feath- erless biped. We have two atomic contents: being a feather (q1), being a biped (q2). So, the concept of a man is a function C such that: C(q1) = 0 and C(q2) = 1. Let us assume that the function I satisfies the conditions: I(feather) = feather and I(biped) = biped, where the under- lined term refers to a feature and the term written in italic to a content. Then, if Socrate has these two features, he is an element of CExt(I)(C). However, if I satisfies the conditions: I(being a lawyer) = feather and I(being a teacher of Aristotle) = biped, then Socrate is not an element of CExt(I)(C) because every element of CExt(I)(C) must be a lawyer that taught Aristotle. The above example would not have seemed to be so peculiar, if we had formulated an adequate conceptualization of our world. Definition 11. A concept C is extensionally adquate with an object e at I iff CExt(I)(C) = Ext(e). We will write shortly: C = ea(e). Definition 12. A concept C is strongly content-adequate with e at I ∗ ∗ iff C ⊆01 e ⊆01 C i.e. if contents of concept correspond to essential properties in the uncomleted object. Shortly: C = sca(e). Definition 13. Let CS0 ⊆ CS. CS0 is an extensional adequate system of concepts at I with PTS iff for any e ∈ PTS exists C ∈ CS0 such that Ext(e) = CExt(I)(C). We will write: CS0 = ea(PTS). CS0 is strongly content- adequate system of concepts with PTS at I iff for any e ∈ PTS there exists C ∈ CS0 such that C = sca(e). Then we write: CS0 = sca(PTS). What is a Formalized Ontology Today? An Example of IIC 239

At these assumes the following facts are true: Fact 5. If C = sca(e), then C = ea(e). If CS0 = sca(PTS), then CS0 = ea(PTS). If CS0 = sca(PTS) and CS0 is cardinally equivalent with PTS then CS0 is a tree. Moreover, sca(PTS) is similar to the structure PTS.

2.2. Some types of hP T S, ≤PTSi

I presented a simplified Porphyrian tree structure. In the widened version (comp. Kaczmarek [2008]) I considered different types of trees admitting that, among others, a set of positive and negative properties of e is countable, or the set T is finite, or a tree has many roots. Hence, I introduced different types of trees. Definition 14. The trees that have, as their elements, functions defined on infinite and countable set T and such that the set of positive and negative properties is finite are trees with finite characteristics and infinite domain. The simplified Porphyrian tree structure was constructed basing on the binary tree and so it is a tree with finite characteristics and infinite domain. Moreover, it is a tree fulfilling the condition: (QC) If e ∈ P T S, {e1, . . . , en} ⊂ PTS and for any ei, i = 1, . . . , n, ei −1 under e, then for every ei 6= ej, i, j = 1, . . . , n, there exists t ∈ (ei) ({0, 1}) −1 ∩ (ej) ({0, 1}) such that: ei(t) 6= ej(t). A tree satisfying this condition is a quasi-classical tree. For this type of trees the following fact holds: Fact 6. Let PTS be a quasi-classical tree. For every o ∈ U there exists one and only one species e ∈ PTS such that he, oi ∈ ≤PTS&U . Proof. Suppose that PTS is quasi-classical but there exists o ∈ U having two or more species e1, . . . , en such that (1) hei, oi ∈ ≤PTS&U , for i = 1, 2, . . . , n and n > 1. Then, by (QC) and construction of a PTS, there exists e ∈ PTS such that e ≤PTS&U ei and for every ei 6= ej, i, j = 1, . . . , n, −1 −1 there exists t ∈ (ei) ({0, 1}) ∩ (ej) ({0, 1}) such that: ei(t) 6= ej(t). From (1) we admit: (2) hei, oi ∈ ≤PTS&U and (3) hej, oi ∈ ≤PTS&U . Assume that ei(t) = 1 and ej(t) = 0 (the case: ei(t) = 0 and ej(t) = 1 is analogical). Then, by (2), (3) and Definition 4: o(t) = 1 6= 0 = o(t). Contradiction. 240 Janusz Kaczmarek

Commentary. In philosophical interpretation it means that every individual is univocally characterized by one species. In the case of trees that are not a quasi-classical ones it is possible that an individual exempliﬁes two or more species (what is philosophically objectionable).

Fact 7. If hP T S, ≤PTSi is a tree with finite characteristic and infinite domain, then there exists CS0 ⊆ CS such that CS0 = sca(PTS). Proof. One has to construct a set CS0 included in CS fulfilling the conditions of Definition 13 (definition of strong content adequacy). By assumption and Definition 14 the set of essential properties of e ∈ PTS is finite. If so, then denote e|01 = {ht, e(t)i ∈ e : e(t) ∈ {0, 1}}. Since I : T → Q, one has to find a concept C fulfilling the condition e|01(t) = C(I(t)). If so, then define: Ce = {hI(t), e(t)i : ht, e(t)i ∈ e|01} and a map h : PTS → CS such that h(e) = Ce. Naturally, the set CS0 = {h(x): x ∈ PTS} is strongly content-adequate with PTS and an order on CS0 is settled by 0 0 equality: h(e) ≤CS h(e ) iff e ≤PTS e .

2.3. Modal logic OS4

The essence of the formalized ontology must be here limited to an outline of a semantically given modal logic. It is a version of S4 described on propositional modal language with a set CONST = {a1, a2,...} of individual constants, a set PRED1 = {P1,P2,...} of 1-ary predicates and a set PREDS = {S1,S2,...} of 1-ary father predicates, that I interpret as species predicates and call OS4 (ontological version of S4). The language LOS4 of OS4 is described as: α =df P (a)|S(a)|¬α|α ∧ β|α, for P ∈ PRED1 and S ∈ PREDS. Formulas with ∨, →, ↔ and ♦ are deﬁned in standard way as: (α ∨ β) =df (¬(¬α ∧ ¬β)); (α → β) =df (¬α ∨ β); (α ↔ β) =df (α → β) ∧ (β → α); (♦α) =df (¬¬α).

A model for LOS4 is a triple:

M = hW, R, | |i = hhPTSw,Uwiw∈W , R, | |i, What is a Formalized Ontology Today? An Example of IIC 241 where: W - is any nonempty set of possible worlds; every w ∈ W is a pair hPTSw,Uwi, where PTSw is a simplified Porphyrian tree structure, Uw is a set of individuals generated by species of PTSw; let us notice, S in addition, that U = Uw is a set of all individuals of the world w; w R - is a relation of accessibility on W given by the condition: wRw0 iff NSw ⊂ NSw0 , where NSw is a set of all natural species of w; as you can see R is reflexive and transitive relation, hence we obtain a model for S4;

| | - is a function on LOS4 such that | |: CONST → U, for individual constants, | |: PRED → U n, for predicates, and for every formula α, | α | ∈ {0, 1}; I will also use a sign J for | | on the set CONST ∪ PRED, where PRED = PRED1 ∪ PREDS; Additionally, the function || ||: (CONST ∪ PRED) in CS is used to calcu- late a value of | | for CONST ∪ PRED.

The following conditions describe interpretation for LOS4: (1.1) For every a ∈ CONST, || a ||= ∅.

(1.2) For every a ∈ CONST, J (a) = o such that o ∈ U& ∅ ⊂01 o.

Then on the set CONST we deﬁne a function Jw, for w ∈ PTS ∪ U, by formula: Jw(a) = J (a), if J (a) ∈ Uw Jw(a) is not described, if J (a) 6∈ Uw (we admit then: Jw(a) = #, where # 6∈ U). Remark. It means that the constant Socrate has a denotation in this world, where the object J (Socrate) exists. A function Jw is partial.

(2.1) For every P ∈ PRED1, || P ||∈ CS. ∗ (2.1) For every P ∈ PRED1, o ∈ J(P ) iff (|| P ||) ⊂01 o (comp. Defini- ∗ tion 9 and C ). Then o ∈ Jw(P ) iff o ∈ J(P ) ∩ Uw. (3.1) For every P ∈ PREDS, || P || ∈ CS, where || P || is an object such that for some e ∈ NS : e|01=|| P || (I), (3.2) For every P ∈ PREDS, o ∈ J (P ) iff e |01⊂01 o & o ∈ U. Then we define: o ∈ Jw(P ) iff o ∈ J (P ) ∩ Uw & e is a species of PTSw. 242 Janusz Kaczmarek

(1) | P (a) |w= 1 iff Jw(a) ∈ Jw(P ). (2) | ¬α |w= 1 iff | α |w= 0 (3) | α ∧ β |w= 1 iff | α |w= 1 & | β |w= 1 0 0 (4) | α |w= 1 iff for every w such that wRw :| α |w0 = 1 Remark. In the case of (4), if α is true at w, then complete objects mentioned in α exist at w and then they exist at each w0 accessible from w. The following simple facts that are interesting from ontological point of view hold in LOS4: (O1) S(a) → S(a) (O2) P (a) → P (a) Proof. Assume, on the contrary, that for some w in the model M: (1) w k– P (a) & (2) w k6– P (a). If (2), then for some w0 such that wRw0: (3) w0 k6– P (a). Then (4) J w0(a) 6∈ 0 J w (P ). Next, from (1) we admit that Jw(a) ∈ Jw(P ) and this means that: Jw(a) ∈ Jw(P ) = {u : u ∈ Uw & u ∈ CExt(I)(|| P ||)}. Let us remark that wRw0. Due to the definition of R, every species form w 0 is a species from w . Hence, if Jw(a) ∈ Uw, then Jw(a) ∈ Uw0 . By the fact that the concept || P || is rigid we admit: (5) Jw(a) ∈ Jw0 (P ). Moreover, 0 remark that if wRw , then Jw(a) = Jw0 (a) and by (5) Jw0 (a) ∈ Jw0 (P ). This, however, is contradictory to (4). (O3) (S(a) ∧ S0(b) ∧ S(b)) → S0(a), (O4) (S(b) ∧ S0(b)) → (S(a) ↔ S0(a))

3. Theorems of ontology

The variety of ontological questions, formal and informal character of ontology make us formulate the theorems of ontology in natural and formal language. Hence, it is worth distinguishing the following types of theorems: 1) formal theorems of type I that are formulated in formal language of some mathematical theory (comp. e.g. Facts 1 – 7 given above), 2) formal theorems of type II that are formulated in formal language of a logic (like theorems of OS4, e.g. O1 – O4 of part 2.3) and 3) the so called interpreting (interpretational) theorems expressed in natural language. What is a Formalized Ontology Today? An Example of IIC 243

Formal theorems of type I and II are comprehensible and clear. They are valid in the domain of mathematical theory or in logic. But some ontological theorems may be rendered only in natural language as formal language seems to be too poor and the interpretational theorems are of that kind; for example: a) conclusion from O1, i.e. that the accidental features of an incomplete object are necessary features of an individual that exempliﬁes that incomplete object; and next that not only species features are necessary, b) conclusion from Fact 6; to be characterized univocally by species every object has to be generated by species in quasi-classical tree; it means that univocal species characteristics of complete objects demands such and such assumptions; this statement, of course, cannot be expressed by any formal theorem. Another example of interpretational theorems is a sequence of the following utterances that are derived from formally given concepts and obtained facts: Individuals, generated by species, the ontologists speak about, are real objects. The species alone are unknown. If we had known the species, we would have known (in accordance with our interpretation) the essence of objects. Hence, we construct concepts structures that can be strongly content or extensionally adequate with a structure of species and genera. As we know, there exists only one system that is strongly content adequate with PTS. Such a system is a system with concepts that grasp the essence of individuals adequately. Perhaps this system and the so-far unknown structure of PTS, are the aim of our ontological considerations, our Holy Grail. This is, of course, a gloss that goes beyond the interpretational theorems.

References

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Department of Logic University ofL´od´z, Poland e-mail: kaczmarek@ﬁlozof.uni.lodz.pl