=Paper=
{{Paper
|id=Vol-1517/JOWO-15_FOfAI_paper_5
|storemode=property
|title=Generative Ontology of Vaisesika
|pdfUrl=https://ceur-ws.org/Vol-1517/JOWO-15_FOfAI_paper_5.pdf
|volume=Vol-1517
|dblpUrl=https://dblp.org/rec/conf/ijcai/TavvaS15
}}
==Generative Ontology of Vaisesika==
https://ceur-ws.org/Vol-1517/JOWO-15_FOfAI_paper_5.pdf
Generative Ontology of Vaiśesika
Rajesh Tavva1 and Navjyoti Singh2
1,2
Center for Exact Humanities, International Institute of Information Technology, Hyderabad, Telangana, India
1
vrktavva@research.iiit.ac.in and 2navjyoti@iiit.ac.in
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 system 1. 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
Vaiśesika, as mentioned in introduction, is one of the many is a separator/boundary/vacuum/non-being between them
Indian philosophical schools, which focuses on which we call punctuator/point4. But a punctuator is not
foundational ontology. It classifies all entities of reality only a separator but also a connector/link which links two
into 6 categories2: (1) Substance (e.g: material entities like entities. So here non-being is not an all-encompassing
tables, chairs as well as non-material entities like soul, vacuum whose existence Parmenides5 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
(e.g: rising up, falling down, motion) (4) Universal (e.g: a particular pair of entities/beings which it separates as
tableness, chairness, redness) (5) Ultimate Differentiator well as brings together into contiguity6 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 where x and y are entities punctuated from each other
Vaiśesika ontology, due to Kanada [1], Prasastapada [2] in some relational context R. This R refers to all the chains
and Udayana [3] has been formalized by Navjyoti [4,5,6,7] of entities (along with punctuators between them)
which resulted in Neo-Vaiśesika Formal Ontology. From connecting x and y where each chain looks like C = p(x|e1)
now we will refer to this formal system as F for short. We – p(e1|e2)... - p(ei|ei+1) - ...p(en-1|en) – p(en|y) where each ei is
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
>. which occur in it for which we need to move to the next
A self-linking relation is one in which one of the relata punctuator.
itself acts as the relation. The relata don't need a third Definition 2.5 (Separable Punctuator): A separable
entity to bring them together. The relation is called self- punctuator pcd(x|y) with structure where X is either of the relational
by itself to other entities. The punctuator corresponding to entities C called conjunct (or contact) or D called disjunct.
the self-linking relation is called the self-linking C (or D) binds x and y in such a way that they are
punctuator. It can be represented as in Fig. 2.1. associated (or dissociated). pcd(x|y) switches between two
structures that can be represented in short as and
since no two entities can be in both conjunct and
disjunct at the same time. It is shown in Fig. 2.3.
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
two entities x and y . It is represented as pin(x|y) and it has Fig. 2.3. Separable punctuator
the structure . The relation between C (or D) and its relata is
This I is nothing but the inherence category of inherence. If two entities x and y are in conjunct (or
Vaiśesika discussed above. It can be represented as in Fig. disjunct) then the conjunct (or disjunct) with y inheres in x
2.2. Here the chain connecting x and y via I can be and vice-versa. The conjunct and disjunct which we are
abstracted into a relation called inherence relation, and the referring to here are completely different from the logical
entity I can be said to be playing a relational role only operations – conjunction and disjunction – which are used
when it is part of such a chain. This chain, along with both in propositional logic. They only refer to the relation of
x and y constitutes the inherence punctuator between x and two entities being in contact (as in when we clap both our
y. Hence a relation and punctuator are different. hands are in contact) or not in contact.
Unlike self-linking relation where one of the relata One can notice that the 2nd and 3rd punctuators are
itself is playing the role of the relation here a third entity, a recursive in nature in the sense that the separable
punctuator is made of two inherence/inseparable
7. Edmund Husserl, Ideen zu einer reinen Phänomenologie §10, punctuators while the inherence/ inseparable punctuator is
Husserliana 3/1: 26–7
made of two self-linking punctuators put together. In this
sense, self-linking punctuator is the most fundamental of related issues like synonymy, polysemy etc. come at this
these 3 punctuators, but we will use all three of them as stage.
building blocks of our system since their patterns repeat We think that the four contigua listed above need to be
everywhere and each of them plays a role in deriving more studied formally in the specified order since each
complex punctuators. contiguum is dependent, for its study, on its previous one.
We need to look at some more definitions and axioms This is a long-term project and this paper can be
before we proceed further. considered as the first step in that direction. Here we are
Definition 2.6 (Abstract of punctuator): abs(p) maps doing a formal study of only the first contiguum and that
relational context of punctuator to two abstract entities ae too a portion of it. Formal study of the remaining portion
and ae which are attributes of the two punctuated entities ei of it, as well as formal study of other contigua is part of the
and ek respectively. future work.
Now we are all set to define the fundamental categories
abs(p(ei |ek )) → ae ∈ F : psl (ei | ae) & ae ∈ F : psl (ek | ae) of Vaiśesika formally in our system F.
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
(ae) 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
Bifunction, IB(e) = (eL, eL) 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 .
qualifier-qualificand pairs of abstract entities. The A UD is one in which nothing is located, but it itself is
qualifier-qualificand relation is usually found in cognition. located in one and only one US. The role of UD is to
For instance when we see a white cloth, the structure of our differentiate one US from another since the USs are
cognition is of the form - qualifier (white) and qualificand partless and cannot differentiate themselves from each
(cloth) - whereas the form of entities in reality as such is other.
that of locus (cloth) and located (white). Study of this Axiom 2.1: Any US cannot have more than one UD
contiguum, along with the previous two, is the study of located in it because a single UD in a locus is sufficient to
how world is cognized, and the notions of knowledge, distinguish its locus from the rest of the entities, and the
reasoning, truth etc. come at this stage. second UD only becomes redundant.
Expression contiguum: This is constituted of all the Definition 2.11: If IB(e) = (Ø, 2+) then e is a
expression-expressed pairs of abstract entities. Study of Universal (U).
this contiguum, along with the previous three, is the study
of the relation between language and reality. All language 8. Detailed explication of these definitions along with diagrams and
examples can be found in [9].
This says that a universal (U) is something in which These MUSs are its parts. For instance a table inheres in all
nothing inheres, but it itself inheres in 2 or more (hence its parts and hence is present in each of its part. That is the
2+) loci i.e. its instances. A universal can inhere not only reason why the entire table moves even when a part of it is
in US but also in non-US, hence 2+ is not subscripted with moved, or the table as a whole is cognized even when a
US. part of it is perceived. And what inheres in a table can be
Axiom 2.2: Given any two universals only one of the its qualities or universals like tableness, substanceness etc.
following two relations is possible between them: (1) Both Axiom 2.4: Given two parts of a whole there has to be
of them are mutually exclusive i.e. both of them have no at least one chain of contacts connecting them directly or
instances in common or (2) One of them subsumes the indirectly.
other i.e. all the instances of one universal are also the Axiom 2.5: Two SWs are in contact with each other if
instances of the second universal whereas the second one or more parts of one SW are in contact with one or
universal has some more instances which the first doesn't more parts of the other SW.
have. This gives rise to subclass-superclass structure Axiom 2.6: Two different SWs cannot have
among universals. overlapping parts i.e. there can be no MUS in which two
Definition 2.12: If IB(e) = (1+, 1) then e is a Quality different SWs can inhere.
(Q). Axiom 2.7: An SW cannot have qualities inhering in it
This says that a quality is that entity in which one or if none of its parts has quality/qualities inhering in it.
more entities inhere, and it itself inheres in only one entity. For instance a cloth cannot be white if none of its
As of now we are not making a distinction between threads has whiteness in it.
Vaiśesika categories – Quality and Action – in our formal Given the above definitions let's have a look at a
system F. That distinction is part of our future work, and sample valid graph of F, say, that of a substantial whole. It
for the time being we will refer to both as Qualities (Q). is depicted in Fig. 2.4.
Now we'll further analyze US for which we need to
define the following function.
Definition 2.13: The function, Locality of any Ultimate
Substance is defined as LUS(eu) = (JC, JD) where JC is the
set of structures conjoint with eu and JD is the set of
structures disjoint with eu.
Given eu (a US) as input, the above function returns the
ordered pair – the set of all entities in conjunct with eu, and
the set of all entities which are in disjunct with eu. We will
again look for invariances in the output to further analyze
US category.
Definition 2.14: If LUS(eu) = (JC,Ø) or if LUS(eu) =
(Ø,JD) then eu is called a Ubiquitous Ultimate Substance
(UUS).
This says that those ultimate substances which are only
in conjunct (and not disjunct) relation with other entities or
Fig. 2.4. Ontology of a Substantial Whole
only in disjunct (and not conjunct) relation with other
entities are defined as ubiquitous ultimate substances. It's easy to understand this graph if we start with the
Definition 2.15: Those ultimate substances which are focal point of it which is a substantial whole (SW). This
not ubiquitous ultimate substances are called Mobile SW is inhering in two mobile ultimate substances (MUS)
Ultimate Substances (MUS) i.e. if eu is an MUS then (it can inhere in more than two MUSs as well) which are in
LUS(eu) = (JC,JD) where neither JC nor JD is Ø i.e. they contact with each other10. Now there are also two UUSs
are in contact with some while in disjunct with some which are in contact with each of the MUSs. Each of the
others. ultimate substances – two MUSs and two UUSs – has an
Axiom 2.3: UUSs are in contact only with MUSs, not ultimate differentiator (UD) inhering in it. Each of the
among themselves9. substances – SW, MUS and UUS – has one quality (more
Now we get back to inherence bifunction to define one than one is also possible) inhering in it, and also one
last category of Vaiśesika i.e. Substantial Whole (SW). universal (more than one is also possible) in it. Each of the
Definition 2.16: If IB(e) = (1+, 2+MUS) then e is a qualities has one universal (more than one is also possible)
Substantial Whole (SW). in it. The universals are shown with dangling edges, it is to
This says that an SW is that in which one or more
entities inhere, and it itself inheres in two or more MUSs. 10. Each bidirectional edge here is a short form for the separable
punctuator, in contact position. The contact entities are currently hidden
for aesthetic purposes. Similarly the unidirectional edges stand for
9. Refer to [4] for the rationale behind this axiom. inherence punctuators and the inherence entities are hidden as well.
show that the universals (can) inhere in more entities its NACs matches with any subgraph of the host graph.
which are not shown in this graph. Notice that no UD has a For a detailed introduction to graph grammars please
universal inhering in it because it would violate its refer to [8].
definition (Definition 2.10).
We can take a particular substantial whole, say, a table 3.1. Generative Rules of Graph Grammar of
to exemplify the above graph. In this example each of the Vaiśesika
above categories stand for the following:
Any generative grammar will have a start symbol, some
• SW – table
alphabet and some production rules involving the symbols
• MUSs – atoms of table of the alphabet. Similarly our grammar will also have a
• UUSs – space and time start graph, an alphabet which consists of a set of node-
• Universals inhering in table – tableness, labels (ΩV) and a set of edge-labels (ΩE), and production
substanceness rules to generate various graphical structures from these
• Universals inhering in MUSs – mobile-ultimate- symbols.
substanceness, substanceness One can treat the generative rules of Vaiśesika as the
• Qualities inhering in table – color, size syntactic portion and its interpretative rules as the semantic
• Universals inhering in these qualities – colorness, portion. The generative rules generate the graphs as pure
sizeness, qualityness symbolic structures without any meaning as such whereas
• Qualities inhering in atoms of table – color, touch the interpretative rules add meaning to these symbolic
structures by labeling each node with some category of
• Universals inhering in these qualities – colorness,
Vaiśesika.
touchness, qualityness In our grammar, the node-labels are Ω V = {g, C, D, h, i,
• Universals inhering in UUSs – ubiquitous- p, q, r, s, u, v, e} and edge-labels are ΩE = {sl, in, con, dis}.
ultimate-substanceness, substanceness Each of the node-labels stands for the following: g – start
• Qualities inhering in ultimate substances – size, node (this is the only node in the start graph), C – conjunct
number entity, D – disjunct entity, h, i, p, q, r, s, u, v – are all
• Universals inhering in these qualities – sizeness, various node labels used in the process of generation. At
numberness, qualityness the end of the generation all of them will be replaced by a
The no. of entities (Qs, Us, MUSs etc.) presented in common label, e, to show that the nodes they were labeling
this example don't match exactly with those presented in can be interpreted later based purely on their structures and
Fig. 2.4. The above figure is only a kind of template to not on their labels. And the edge labels stand for the
understand this example. following: sl – self-linking relation, in – inherence relation,
In the next section we present generative rules - to con – conjunct relation, dis – disjunct relation. But in the
generate the structures as in Fig. 2.4 – as well as rules below we have differentiated edges based on their
interpretative rules - to label the nodes in these structures arrows instead of their labels for aesthetic purposes. A self-
with the categories of Vaiśesika. linking relation has no arrows (though it's asymmetric its
direction is changeable), an inherence relation has one
arrow (it's asymmetric and its direction is fixed), conjunct
3. Vaiśesika as a Generative Ontology relation has two arrows and a thick line whereas disjunct
A graph grammar is a generalization of string grammars relation has two arrows and a dashed line (both are
and tree grammars. The Left-Hand-Side (LHS) and Right- symmetric relations).
Hand-Side (RHS) of a production/transformation rule in No two entities have more than one edge (of any type)
graph grammar are both graphs instead of strings or trees. between them. That is a default NAC for every rule and
The rules modify a host graph into a different graph by hence not being specified with each rule.
replacing a subgraph of it which is matching with LHS, Currently the rules are generated keeping Vaiśesika
with an incoming graph - RHS. The below categories and their corresponding invariant structures in
production/transformation rules will be self-explanatory mind. Intuitively they seem to generate valid Vaiśesika
except, may be, for Negative Application Condition graphs but the proof of soundness is necessary to prove it
(NAC). It is defined below. formally. That is not achieved yet and is one of our future
Definition 3.1: Rules can have exceptions - it may not goals.
be likely to apply a rule in some particular cases. Those We give below the rules as well as the rationale in
cases/conditions are called Negative Application coming up with them. If we consider USs to be the bottom
Conditions (NACs). NACs can also be depicted in the form of universe (since they inhere nowhere else i.e. not located
of graphs. Hence each rule may (or may not) have one or anywhere) and universals and UDs to be the top of
more NACs. So a rule will be applied only when its LHS universe (since nothing inheres in them i.e. nothing is
matches with some subgraph of the host graph, and none of located in them), then we are trying to generate all the
graphs from bottom to top. The rules are prioritized by LHS RHS
dividing them into layers - once you are in n th layer, you
cannot apply any rules from layer 1 to n-1. But if there are
multiple rules in a given layer they can be applied in any
order.
We start with the first layer. It has only one rule in Here the numbers '1:' and '2:' are used to map a
which we replace the start node g with two other nodes – h particular node of LHS with a particular node of RHS (and
and i. NAC, if there is one). In our system this mapping is
injective i.e. no two nodes in LHS can map to the same
LHS RHS node in RHS or NAC and vice-versa. Once we get all the
TGs of this layer, we will move to 4th layer.
Now in the 4th layer we will have rules for generating
wholes, represented by q-labeled nodes. It has 2 rules, we
will apply both these rules only on the TGs of 3rd layer,
not others. In the first rule we state that if two ps (MUSs)
Here h is intended to generate mobile ultimate are in contact, then let a new entity q (whole) inhere in
substances whereas i is intended to generate ubiquitous both of them.
ultimate substances. Those rules follow in coming layers.
The second layer also has two rules. In the first rule, h LHS RHS
is replaced by itself and another node p. This is intended to
generate as many ps as one wants and in the second rule of
this layer, h is replaced just with p. This is to terminate the
process of generation of ps.
We define the graphs generated in a particular layer to
be the terminal graphs (TGs) of that layer if no more
graphs can be generated from them using the rules of that
layer, otherwise we will call them non-terminal graphs
This rule has 2 NACs. These NACs state that the ps in
(NTGs).
which q is inhering shouldn't already have a whole (q)
inhering in them. They are depicted below.
LHS RHS
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 (NAC 2) 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
GRs, as well as Vaiśesika categories i.e. Ω' V = {e, C, D, U, 4. Results
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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