Generative Ontology of Vaiśesika
Rajesh Tavva1 and Navjyoti Singh2 1,2Center for Exact Humanities, International Institute of Information Technology, Hyderabad, Telangana, India [email protected] and [email protected]
Abstract classes/universals, but also particulars. But since the In this paper we present a foundational as well as generative punctuator has a recursive form, complex graphs can be ontology which is graph-based. We use a form called generated from simple graphs, and each of these graphs punctuator which is non-propositional and also non-set- depicts some or other portion of reality at some or other theoretic to build our system – Neo-Vaiśesika Formal level of granularity. Hence we also have an interpreter to Ontology. The idea is to present an ontological language interpret the generated graphs as portions of reality. which is formal. This language is a set of potentially infinite We take Vaiśesika - one of the Indian philosophical sentences (graphs) whose structure is captured by a finite set schools which focuses on foundational ontology – as of (graph) grammar rules. We also have an interpreter to interpret the graphs generated by this grammar and show formalized in [4,5,6,7] as our base, and present generative that the interpretation of a node as belonging to a particular as well as interpretative grammars for it in this paper. This ontological category is based purely on its structure/form paper's focus is not on defending Vaiśesika description of and nothing else to provide a robust example of a formal, reality or the rationale behind its categorial system1. This foundational and generative ontology. paper is already taking them as given and trying to make implicit formal notions of Vaiśesika explicit. The idea is to Keywords. Formal Ontology, Graph Grammar, Generative show the possibility of an ontological language which can Ontology, Punctuator, Generative Grammar, Vaiśesika be formalized and also generated, and then interpreted. In section 1 we present Vaiśesika ontology in brief. In Introduction section 2 we present Neo-Vaiśesika Formal Ontology, the system we built by formalizing Vaiśesika ontology. In this Most, if not all, of the (computational) ontologies built till section, we give formal definitions of Vaiśesika categories now [12] are either built manually or through automatic in terms of three basic punctuators. In section 3 we show methods of category-extraction from text. There is no how Vaiśesika can be seen as a generative ontology. In notion of generation there since there are no repeating section 3.1, we give the generative grammar of our system structures (each category is different and hence different where we give the production rules to generate graphs, and structure). All the foundational ontologies are presented as in section 3.2 we give interpretative rules using which we diagrams/graphs with finite number of nodes (which stand can interpret these generated graphs to label each node for categories/sub-categories) and edges (which stand for with some or other Vaiśesika category. The generated class-subclass or some other relations). In this paper we graph is considered to be valid if there is at least one present a novel concept of Generative Ontology which interpretation, in terms of Vaiśesika categories, of the presumes Grammar of Reality which, in turn, is based on generated graph. The grammar is considered to be sound if the idea of a recursive ontological form called the it generates only valid Vaiśesika graphs. Though a formal punctuator. A punctuator is a form which is present proof of the soundness of this grammar is not achieved yet, between any two entities and enables us to distinguish one we present in section 4 some results which show that the from the other. It is because of this form that we are able to system is promising. The main focus of the paper is to differentiate various categories as well as different present the idea of a generative ontology and one example instances of the same category. Since a punctuator is found of such an ontology. not only between two classes, but also between two We tried to provide examples and explanation of particulars, or between a class and a particular, the graphs abstract concepts wherever possible but due to space which are based on punctuators contain not only constraints we are unable to get into detailed explanation sometimes. An extended version of this paper with Copyright © 2015 for this paper by its authors. Copying permitted examples, diagrams and detailed explanation of various for private and academic purposes. 1. One can refer to [4] and [14] for that. definitions, axioms etc. presented in this paper is available explicate the system below. at [9]. According to F reality is constituted of only entities and punctuators. Here entities refer to all kinds of beings like objects, events, relations and so on. And any two 1. Vaiśesika Ontology beings/entities are considered two, not one, because there is a separator/boundary/vacuum/non-being between them Vaiśesika, as mentioned in introduction, is one of the many 4 Indian philosophical schools, which focuses on which we call punctuator/point . But a punctuator is not foundational ontology. It classifies all entities of reality only a separator but also a connector/link which links two 2 entities. So here non-being is not an all-encompassing into 6 categories : (1) Substance (e.g: material entities like 5 tables, chairs as well as non-material entities like soul, vacuum whose existence Parmenides was denying but space and time) (2) Quality (e.g: color, weight) (3) Action every non-being is a particular in the sense that it rides on a particular pair of entities/beings which it separates as (e.g: rising up, falling down, motion) (4) Universal (e.g: 6 tableness, chairness, redness) (5) Ultimate Differentiator well as brings together into contiguity in some relational (located in each ultimate substance (explained below) and context. Our idea of punctuator/point is quite similar to, at differentiates one from the other) and (6) Inherence the same time slightly different from, Leibniz’s idea of (explained below). Notice that there is no category like punctum/point and Brentano’s idea of punctiform/point. Relation in this system because any chain constituting a Leibniz [10] tries to explicate this notion by taking a relational context (section 2) can be abstracted into geometrical example like a line segment. He makes a something called a relation. Out of all the categories listed distinction between potential point (euclidean point) and above inherence (Fig. 2.2) is the most significant one for actual point (non-euclidean point). The line segment, us. It gives stability to reality and is the entity constituting according to Leibniz, is not really made up of euclidean the second most pervasive relation found in reality, the first points, they are not parts of the segments the way smaller being self-linking relation (more about it later). We call the segments are. The euclidean points (lengthless, breadthless relational chain constituted of inherence entity as inherence and heightless points) are only potentially there on the relation, and this inherence relation can be found in all the segment but not actually constituting it. The actual points following cases: (1) Universals inhere in substances, are the boundaries or the endpoints of the segment. These, qualities and actions (2) Qualities and actions inhere in Leibniz also calls as punctums because they punctuate the substances (3) Substances (wholes) inhere in other line segment from its neighbourhood (in one-dimensional substances (their parts) and (4) Ultimate Differentiators space). A potential (euclidean) point can become an actual inhere in ultimate substances. Ultimate substances (US) are (non-euclidean) point if one cuts the line segment, say, into those substances which have no parts and hence cannot be two halves. The midpoint which was there before cutting differentiated from each other3. Examples of USs are time, was only potential, but after cutting it becomes actual. So space, souls and atoms (these are different from those of according to Leibniz a punctum is like a boundary that physics, they are the smallest indivisible units according to separates two entities. Brentano [11] also comes up with a Vaiśesika). Since USs have no parts, each of them has an similar idea and he calls it punctiform or point. He says no ultimate differentiator (UD) inhering in it which continuum can be built up by adding one individual point differentiates it from the rest of the entities. The remaining to another. The point exists as a boundary, as a limit. substances which are not USs are automatically Similarly for us, a point is a boundary between two differentiated from others since they are entities, but not constituting those entities. wholes/mixtures/compounds made up of various USs These both – entity and punctuator – are defined in F in which are already differentiated by UDs. We will try to terms of each other. But the mutual dependency of define all these categories formally using only the idea of definitions ends here. All the other definitions are based on inherence. these primitives only. Definition 2.1: An entity is that which is never punctuated from itself. 2. Neo-Vaiśesika Formal Ontology Definition 2.2: A punctuator pr(x|y) has the form
2. A detailed list of Vaiśesika categories and subcategories can be 4. If the separator were also a being, then one would run into infinite found in a tabular format in [13]. regress. Hence one needs to accept a non-being like punctuator. 3. This notion of parthood is quite different from other mereological 5. http://plato.stanford.edu/entries/parmenides/ notions like that of [15]. For detailed explanation of the notion of 6. Two entities in contiguity are related to and yet distinct from each parthood in Vaiśesika please refer to [4]. other. also an entity. relational entity, I (inherence) is necessary to bring two Notice that the punctuator is neither the entities nor the entities x and y into inherence relation. It is called relation between them. It is the form/arrangement of all inseparable because inherence relation itself cannot be these things put together. Though an entity is existentially destroyed without one of its relata being destroyed. independent of its punctuations with other entities, its Various cases where inherence relation is found in reality meaning solely comes from these punctuations. Hence the are given in section 1. form of an entity defines its meaning. In this sense F satisfies Husserl's definition of formal ontology: “eidetic science of the object as such.”7
2.1. Three Basic Punctuators and Abstract of Punctuator From our knowledge of Vaiśesika we have narrowed down Fig. 2.2. Inseparable punctuator to three basic punctuators/forms in F. It seems that these three basic forms are sufficient to derive any other Inherence relation repeats almost everywhere in the complex form in the universe. Their definitions and other universe. Hence it is one of our most fundamental relations details are given below. to derive other complex relations. In fact if we take all the Definition 2.3 (Self-Linking Punctuator): A self- instances of inherence relation in the universe we will get linking punctuator is one whose relational context is the entire synchronic reality or the snapshot of the reality. empty. It is represented as psl(x|y) and its structure is To get diachronic reality, we need to look at the changes
Fig. 2.1. Self-linking punctuator Here the edge being undirected only means that the direction is changeable (between the same pair of entities) in the case of self-linking punctuator. But in the case of inseparable punctuator that is not the case. Definition 2.4 (Inseparable Punctuator or Inherence punctuator): An inseparable punctuator is one which has an inert entity called I (inherence) which inseparably binds Fig. 2.3. Separable punctuator two entities x and y . It is represented as pin(x|y) and it has the structure
To take an example, if color (ei) inheres in table (ek), it 2.3. Fundamental Categories of Vaiśesika Defined can also said to be located in table. So color is the locatee Formally in F 8 a ( e) whereas table is the locus (ae). So the abstract of the We will formally define the fundamental categories of inherence punctuator between color and table gives rise to Vaiśesika in F by doing a functional study of the locational two abstract entities, namely locus and locatee where contiguum of inherence punctuator for which we should locus, the attribute of table, is related to it by self-linking first define something called the Inherence Bifunction. punctuator, and locatee, the attribute of color, is related to Definition 2.8 (Inherence bifunction): The Inherence it by self-linking punctuator. Similarly there can be various e Bifunction, IB(e) = (eL, L) takes an entity e and returns the other abstracts of punctuators like predecessor-successor, locational ranges of e. qualifier-qualificand, expression-expressed etc. which can It is called a bifunction because it works on two be abstracted directly from one of the three basic different things at the same time – locatee range and locus punctuators or from some complex punctuator defined range of e. Locatee range of e refers to all the entities using them. Collection of pairs of each kind forms a which are located in e (by inherence) and locus range of e contiguum like locational contiguum, succession refers to all the entities in which e is located (by contiguum etc. inherence). Now we will define categories by noticing some 2.2. Four Contigua invariances in the output. According to us there are four important contigua to be Definition 2.9: If IB(e) = (eL, Ø) then e is an Ultimate studied formally, to cover a major portion of reality, in the Substance (US) where Ø stands for the empty set. sequence listed below: So if there is an entity which inheres nowhere but has Locational contiguum: This is constituted of all the some other entities inhering in it, then it is said to be an locus-located pairs of abstract entities. Study of this gives ultimate substance (US). In other words US is an unlocated us entire synchronic reality. locus. Some examples are space, time, soul, atoms. (These Succession contiguum: This is constituted of all the are not yet defined in the system, but they are mentioned predecessor-successor pairs of abstract entities. Study of just to give more clarity on the nature of US). this, along with locational contiguum, gives us entire Definition 2.10: If IB(e) = (Ø,1US) then e is an Ultimate diachronic reality. Differentiator (UD) where 1US stands for the unique Qualification contiguum: This is constituted of all the structure
NAC1 NAC2
LHS RHS
Notice that numbers like '1' and '2' are prefixed only to p-labeled-nodes and not q-labeled-nodes. It's because there These ps are supposed to stand for mobile ultimate is no such node in LHS to map with the nodes in RHS or in substances. Once the generation process in this 2nd layer NACs. So the rule, along with its NACs, states that a new terminates i.e. no more h-labeled nodes remain, the whole (q in RHS) inheres in two MUSs (ps) in contact if remaining graphs are TGs of layer 2, and it is on these and only if there is no whole (q in NACs) already inhering graphs that we will apply the rules from layer 3. in one or both of them. These NACs are necessary for us In the 3rd layer we have two rules – one to create because, in our system, no two substantial wholes can contacts among ps and another to create disjuncts among inhere in the same part (Axiom 2.6). So even if one of the them. above NACs is found in that portion of the graph which matched with LHS, then this rule will not be applied. LHS RHS The above rule will produce only wholes inhering in two entities. But if we need wholes inhering in more than 2 entities upto an indefinite number, may be covering all ps (MUSs) in a step-by-step manner we need the following rule. LHS RHS NAC themselves (Axiom 2.3)). The next layer i.e. 6th layer has two rules. They apply only on the TGs of 5th layer. The i-node which we generated in our 1st layer is now used to generate ubiquitous ultimate substances represented by r-labeled- nodes or r-nodes. The two rules here are exactly same as the ones in 2nd layer, only with labels differing in both LHS and RHS. This rule has one NAC which states that no whole (q) should already inhere in 2:p. LHS RHS Notice here that the q-node in NAC is not numbered whereas the q-node in RHS of the rule is numbered. The RHS is saying that an inherence edge should be added from the same q as in LHS while NAC is referring to any q inhering in p, not necessarily the same one as in LHS. LHS RHS The next layer i.e. 5th layer has two rules. These can be applied on both TGs as well as NTGs of layer 4 since the latter (NTGs of layer 4) are also considered as valid intermediate graphs for the process of generation. The first rule states that if two MUSs (ps) are in contact then the wholes (qs) inhering in each of them can also be in contact. In the 7th layer, we have two rules. They apply on the This is to model the fact that the contact between wholes TGs of 6th layer. The first rule states that an ubiquitous can be inferred from the contact between their parts ultimate substance can be in contact with a mobile ultimate (Axiom 2.5). substance, if it is not in disjunct with any. So it has one NAC to specify this negative condition. LHS RHS LHS RHS NAC
The 2nd rule of this layer is to handle disjuncts among The second rule is very similar to the first one. It says wholes. This is like the complement of the first rule. This an ubiquitous ultimate substance can be in disjunct with a says that two wholes can be in disjunct if none of their mobile ultimate substance, if it is not in contact with any. parts are in contact. So this negative condition becomes the NAC of this rule. LHS RHS NAC LHS RHS NAC
Similarly there are 10 more layers of rules in this generative grammar. The layers from 8 to 16 are briefly Till now we had rules which generate mobile ultimate described here, but presented in detail in [9]. substances, contacts or disjuncts among them, wholes The 8th layer is to generate qualities in mobile and inhering in mobile ultimate substances, and contacts or ubiquitous ultimate substances whereas the 9th layer is to disjuncts among them. In the next couple of layers we generate qualities in wholes. The next few layers (10th to introduce rules to generate ubiquitous ultimate substances 15th) have rules devoted to the generation of nodes and contacts or disjuncts - between them and other entities standing for universals. The 16th layer is to generate UDs (since ubiquitous ultimate substances are in contact or in USs – both mobile as well as ubiquitous. disjunct only with mobile ultimate substances, not among The 17th and the final layer is the one where all the LHS RHS NAC label names (except that of contact and disjunct) will be replaced with a common label e. This is a way of erasing all the label-based identities of various nodes, and also creating an occasion to prove later, using the interpretative rules, that the different categories of nodes in the graph can be identified purely based on their structures/forms and not their labels. This layer has only one rule and this is applied only on the TGs of 16th layer.
LHS RHS The 2nd and 3rd rules define a Ubiquitous Ultimate Substance (UUS). A UUS is one which inheres nowhere (NAC1) and is only in contact but not disjunct (NAC2) with any other entity (2nd rule) (or) only in disjunct but not contact (NAC2) with any other entity (3rd rule) (Definition 3.2. Interpretative Rules of Vaiśesika 2.14). They are shown in the next two tables respectively. The difference between generative rules and interpretative rules of Vaiśesika can be thought of as the difference LHS RHS NAC1 NAC2 between syntax and semantics of reality. The graphs generated by the former – generative rules - refer to the syntactic portion of reality – they are forms or arrangements of entities in reality. Given the way entities are arranged we discern their meaning and categorize or classify them accordingly, and that, the semantic portion of reality, is given by the latter – interpretative rules. Interpretative rules are also graph grammars like LHS RHS NAC1 NAC2 generative rules, but we separated both of them because their purposes are different – one parses the graphs which the other generates. So the interpretative rules can be thought of as constituting an interpreter or a parser for the language (graphs) of Vaiśesika which is generated by the generative rules in the previous subsection. So the input to the interpreter are the TGs generated by the last layer (17th layer) of the generative grammar. Any of them can become the start graph for the interpretative Similarly we have interpretative rules to define the rules (IRs). Coming to the alphabet, the edge-labels of remaining categories like UD, U, SW and Q. They are interpretative rules (Ω'E) are same as that of generative presented in [9], the extended version of this paper. rules (ΩE) i.e. Ω'E = ΩE whereas the node-labels of interpretative rules include the node-labels of the TGs of 4. Results GRs, as well as Vaiśesika categories i.e. Ω'V = {e, C, D, U, UD, SW, Q, MUS, UUS}. In some rules nodes are We have constructed a graph grammar software of our own unlabeled. Such anonymous nodes stand for any node with to simulate the above generative as well as interpretative any label. Also the IRs are not prioritized the way GRs rules, and on around 10000 terminal graphs of the last were. IRs can be applied in any order since they are layer of generative rules, we ran the interpretative rules and independent of each other. The process completes when found that we could interpret every node of every graph there is no scope left for any rule to be applied. These IRs there as one of the Vaiśesika categories. This gives us a can also be considered as a graphical way of defining the confidence that the system is not only intuitively sound but categories of Vaiśesika. The rules are given below. also inductively sound, and that the structure of reality, as The first rule defines a Mobile Ultimate Substance described in Vaiśesika, can be discovered, generated and (MUS). An MUS is one which inheres nowhere (NAC) and parsed purely formally. This shows that the categories are is in contact with at least one entity and is in disjunct with differentiated purely based on their formal structures and at least one entity (Definition 2.15). nothing else. The conclusive proof of soundness of the system is part of our future work. 5. Conclusion [4] Singh, N. Comprehensive Schema of Entities: Vaiśesika Category System. 2001. Science Philosophy Interface 5(2): The idea is to present an ontology language which is 1–54. graph-based and which can be formal and which shows the [5] Singh, N. Formal Theory of Categories through the Logic of potential to be scaled up in future to cover many other Punctuator (2002) (unpublished) portions of reality like causation, cognition, language etc. [6] Singh, N., Theory of Experiential Contiguum. 2003. We think we have accomplished that job in this paper by Philosophy and Science: Exploratory Approach to taking Vaiśesika ontology as an example, building Consciousness 111–159, Ramakrishna Mission Institute of Culture, Kolkata. generative as well as interpretative rules to derive the valid sentences (graphs) of Vaiśesika and showed that a rigorous [7] Singh, N. Foundations of Ontological Engineering, Lecture slides of course at IIIT Hyderabad (2008) formal ontology is possible. Since the focus of this paper is on generative and interpretative grammars of Vaiśesika [8] Rozenberg, G., et al. eds. 1997. Foundations. Handbook of Graph Grammars and Computing by Graph and not on the rationale behind its categorial system there Transformation, vol. 1. Singapore: World Scientific. is no scope for comparing this with other existing [9] Tavva, R., and Singh, N. Generative Ontology of Vaiśesika. ontologies since, in our knowledge, there is no other https://sites.google.com/site/vrktavva/resources/Generative_ foundational ontology which has grammar(s) like ours. The Ontology_of_Vaiśesika.extended_version.pdf? only comparison we can draw with other ontologies is that [10] Leibniz, G.W. The Labyrinth of the Continuum: Writings on they are (semi-)manually constructed while ours is the Continuum Problem, 1672-1686. Translated from Latin generated. Here only the grammar is manually written but and French to English by Richard T.W. Arthur. New Haven: the actual ontological graphs are computationally Yale University Press (2001). generated. In other words potentially infinite structures [11] Brentano, F. The Theory of Categories, translated from can be generated using a finite set of rules in our ontology German by Roderick M. Chisholm and Norbert Guterman. whereas in others they need to actually come up with the The Hague, Boston, London: Martinus Nijhoff (1981). potentially infinite structures (semi-)manually. [12] Roberto, P., et al. eds. 2010. Theory and Applications of Ontology: Computer Applications. Springer. [13] Comprehensive list of Vaiśesika Categories. 6. Future Work https://sites.google.com/site/vrktavva/resources/Vaiśesika_C ategories_table.pdf As mentioned earlier this paper is only the first step toward [14] Mukhopadhyay, P.K. 1984. Indian Realism: A Rigorous building a formal ontology which envisages to cover a Descriptive Metaphysics. Calcutta: K P Bagchi & Company. major portion of reality – a long-term project in its own [15] Casati, R., & Varzi, A. C. 1999. Parts and places: The right. We have covered only the locational contiguum, that structures of spatial representation. Cambridge, Mass: MIT too some portion of it, in this paper, and one could see how Press. rigorous the study of even such a small portion can be. This is purely a theoretical work and for any fruitful applications to come out of this kind of work, one needs to formalize the remaining contigua, at least till the 4th one i.e. expression contiguum. Our immediate priority is to prove the soundness of the system we presented till now in a foolproof manner, and then continue to extend the system till we formalize expression contiguum.
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