revid version - September 7, 1995
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Invited paper prented at the 16th Wittgenstein Symposium, Kirchberg, Austria, August 1993. To appear in R. Casati, B. Smith and G. White (eds.), Philosophy and the Cognitive Sciences , Vienna: Hölder-Pichler-The Ontological Level
Nicola Guarino
1. Introduction
In 1979, Ron Brachman discusd a classification of the various primitives ud by KR systems at that time 1. He argued that they could be grouped in four levels, ranging from the implementational to the linguistic level (Fig. 1). Each level corresponds to an explicit t of primitives offered to the knowledge engineer. At the implementational level, primitives are merely pointers and memory cells, which allow us to construct data structures with no a priori mantics. At the logical level, primitives are propositions, predicates, logical functions and operators, which are given a formal mantics in terms of relations among objects in the real world. No particular assumption is made however as to the natur
e of such relations: classical predicate logic is a general, uniform, neutral formalism, and the ur is free to adapt it to its own reprentation purpos. At the conceptual level, primitives have a definite cognitive interpretation, corresponding to language-independent concepts like elementary actions or thematic roles. Finally, primitives at the linguistic level are associated directly to nouns and verbs.
Brachman noticed an evident gap in this classification: while primitives at the logical level are extremely general and content-independent, at the conceptual level they acquire a specific intended meaning that must be taken as a whole, without any account of its internal structure. He propod the introduction of an intermediate epistemological level, where the primitives allow us to specify “the formal structure of conceptual units and their interrelationships as conceptual units (independent of any knowledge expresd therein)”2.In other words, while the logical level deals with abstract predicates and the conceptual level with specific concepts, at the epistemological level the generic notion of a concept is introduced as a knowledge structuring primitive .Level
Primitives Implementational
Memory cells, pointers Logical
Propositions, predicates, functions, logical operators Epistemological
Concept types, structuring relations Conceptual
Conceptual relations, primitive objects and actions Linguistic Linguistic terms
Fig. 1. Classification of primitives ud in KR formalisms (adapted from Brachman 79).Epistemological level was “the missing level”.
Brachman’s KL-ONE 3 is an example of a formalism built around the notions. Its main contribution was to give an epistemological foundation to cognitive structures like frames and mantic networks, who formal contradictions had been revealed in the famous “What’s in a link?” paper by Bill Woods 4. Brachman’s answer to Woods’ question was that conceptual links should be accounted for by epistemological links , which reprent the structural connections in our knowledge needed to justify conceptual inferences. KL-ONE focud in
particular on the inferences related to the so-called IS-A relationship, offering primitives to describe the (minimal) formal structure of a concept needed to guarantee “formal inferences about the relationship (subsumption) between a concept and another”. This formal structure consists of the sum of the constituents of a concept (primitive concepts and role expressions) and the constraints among them, independently of any commitment as to: (i) the meaning of primitive concepts; (ii) the
meaning of roles themlves; (iii) the nature of each role’s contribution to the meaning of the concept. The intended meaning of concepts remains therefore totally arbitrary: indeed, the mantics of current descendants of KL-ONE is such that – at the logical level – concepts correspond to arbitrary monadic predicates, while roles are arbitrary binary relations. In other words, at the epistemological level, emphasis is more on formal reasoning than on (formal) reprentation: the very task of reprentation, i.e. the structuring of a domain, is left to the ur.
什么是分封制Current frame-bad or object-oriented formalisms suffer from the same problem. For example, the advantage of a frame-bad language over pure first-order logic is that some logical relations, such as tho corresponding to class and slots, have a peculiar, structuring meaning. This meaning is the result of a number of ontological commitments, which accumulate in layers starting from the very beginning of the process of developing a knowledge ba5. For a particular knowledge ba, its ontological commitments are however implicit and strongly dependent on the particular task being considered, since the formalism itlf is in general neutral as concerns ontological choices6.
In this paper I argue against this neutrality, claiming that a rigorous ontological foundation for knowledge reprentation can improve the quality of the knowledge engineering process, making it easier to build at least understandable (if not reusable) knowledge bas. We contrast the notion of f
ormal ontology, intended as a theory of the a priori forms and natures of objects, to that of (formal) epistemology, intended as a theory of meaning connections7. We show in the following how theories defined at the epistemological level, bad on structured reprentation languages like KL-ONE, cannot be distinguished from their "flat" first-order logic equivalents unless we make clear their implicit ontological assumptions by stating formally what it means to interpret a unary predicate as a concept (class) and a binary predicate as a "role" (slot). We need therefore to introduce the notion of ontological level, as an intermediate level between the epistemological and the conceptual one (Fig. 6)8. While the epistemological level is the level of structure, the ontological level is the level of meaning. At the ontological level, knowledge primitives satisfy formal meaning postulates, which restrict the interpretation of a logical theory on the basis of formal ontology, intended as a theory of a priori distinctions:9
•among the entities of the world (physical objects, events, );
•among the meta-level categories ud to model the world (concepts, properties, states, roles, attributes, various kinds of ).
sh命令We focus here on the latter kind of distinctions, showing how the basic dichotomy existing in KR syst
ems between concepts like apple and asrtional properties like red can be understood in terms of the philosophical distinction between sortal and characterising universals10. In ction 2 I prent examples which show the necessity of making such a distinction explicit. In ction 3 I introduce the notion of ontological commitment as a constrained interpretation of a logical theory, and I sketch a basic ontology of meta-level categories of unary predicates. In ction 4 I discuss the role of the ontological level in current knowledge engineering practice.
蔑视拼音2. Reds and apples.
Suppo we want to state that a red apple exists. In standard first order logic, it is straightforward to write down something like ∃x.(Ax ∧ Rx). If we want however to impo some structure on our domain, the simplest formalism we may resort to is many-sorted logic. Yet we have to decide which predicates correspond to sorts: we may write ∃x:A.Rx as well as ∃x:R.Ax (or maybe ∃(x:A,y:R).x=y). All the structured formalisations are equivalent to the previous one-sorted axiom, but each contains an implicit structuring choice. At the epis-temological level, this choice is up to the ur, since the mantics of the primitive “sort” is
the same as its corresponding first-order predicate. At the ontological level, what we want is a formal,
restricted mantic account that reflects the ontological commitment intrinsic in the u of a given predicate as a sort. This means that the choice of a particular axiomatisation is still up to the ur, but its conquences are formalid in such a way that another ur can understand the meaning of the choice itlf, and possibly agree on it on the basis of its -mantics.
In our ca, a statement like ∃x:R.Ax sounds intuitively odd: what are we quantifying over? Do we assume the existence of “instances of redness” that can have the property of being apples? According to Strawson, the difference between the two predicates lies in the fact that apple “supplies a principle for distinguishing and counting individual particulars which it collects”, while red “supplies such principle only for particulars already distinguished, or distinguishable, in accordance with some antecedent principle or method”11. This distinction is known in the philosophical literature as the distinction between sortal and non-sortal (characterising) universals, and is (roughly) reflected in natural language by the fact that the former are common nouns, while the latter are adjectives. The issue is also related to the difference between count and mass terms, and has been a matter of lively debate among linguists and philosophers12. The distinction is implicitly prent in the KR literature, where sortal universals are usually called “concepts”, while characterising universals are called “properties”. The difference between the two is however the result of heuristic considerations,
and nothing in the mantics of a concept forbids any arbitrary unary predicate from acquiring this status.
Our position is that, within a KR formalism, the meaning of structuring primitives as sorts (or concepts, in KR terminology) should be at least specified with formal, necessary conditions at the meta-level, which force the ur to accept their conquences when he/she decides to u a given predicate as a sort. According to our previous discussion, a predicate like red – under its ordinary meaning – will not satisfy such conditions, and should be excluded therefore from being ud as a sort. Notice however that this may be simply a matter of point of view: at the ontological level, it is still the ur who decides which conditions reflect the intended u of the predicate red: a more rigid choice would be distinctive of higher levels, like the conceptual or the linguistic level. For example, compare the statement mentioned above with others where the same unary predicate red appears in different contexts (Fig. 2):
In ca (2), red is still a unary predicate who argument refers to a particular colour instead of a particular fruit; in (3) the argument refers to a particular colour gradation belonging to the t of “reds”, while in (4) the argument refers to a human-being, meaning for instance that he/she is a communist.
Fig. 2. Varieties of predication.
We face here the difference of positions between the stereotype Linguist and Philosopher discusd by Bill Woods in one of the historical papers on knowledge reprentation13: while the Linguist “is interested in characterising the fact that the same ntence can sometimes mean different things”, the Philosopher “is concerned with specifying the meaning of a formal notation rather than a natural language”. Woods goes on by stating that
philosophers have generally stopped short of trying to actually specify the truth conditions of the basic atomic propositions, dealing mainly with the specification of the meaning of complex expressions in terms of the meanings of elementary ones.控制环境
Rearchers in artificial intelligence are faced with the need to specify the mantics of elementary propositions as well as complex ones.
凉拌鸡胗的做法
饮酒诗句
In a knowledge reprentation formalism, we are constantly using natural language expressions within our formulas, relying on them to make our statements readable and to convey meanings we have not explicitly stated: however, since words are ambiguous in natural language, it may be important to “tag” the words with a mantic category, endowed with a suitable axiomatisation, in order to guarantee a consistent interpretation. This is unavoidable, in our opinion, if we want to share theories across different domains14.
狗的忠诚How can we account for the mantic differences in the u of red in the formulas given above? In our opinion, they are not simply related to the fact that the argument belongs to different domains: they are mainly due to different ways of predication, i.e. different subject-predicate relationships. Studying the formal properties of the relationships is a matter of formal ontology.
3. A basic ontology of unary predicate types
A basic ontology, which – according to Strawson’s intuitions – classifies unary predicates on the basis of their ability to supply an identification principle for their
arguments, is prented in Fig. 3.
Following Wiggins15, sortal predicates are here divided into substantial (like apple or human-being) and non-substantial (like food or student), while non-sortal predicates include generic predicates like thing and characterising predicates like red. A preliminary formalisation of such distinctions will be prented in the quel.
But of trying to give a “universal” formal definition of the above categories, we shall pursue here a more modest account: our definitions will be related to a specific knowledge ba described by a standard first order theory, which we are interested in “adding structure”to. This means that the basic knowledge-building blocks are taken as having been already fixed, being the predicates of the theory itlf; our work will be to offer a formal instrument for clarifying their ontological implications for the specific purpos of knowledge bas un-derstanding and reu among urs belonging to the same culture. We assume therefore that interpretations of our specific theory, rather than describing
a real or hypothetical situation in a world that has the same laws of nature of ours16, are states of affairs having an “idealid rational acceptability”17. This choice excludes unwanted metaphysical implications.
Within this framework, let us concentrate on a minimal problem. Suppo we have a first order, non-functional language L with signature Σ=<C, R>, where C is a t of constant symbols, R is a finite t of predicate symbols and P⊆R is a t of monadic predicate symbols. We are interested in some formal criteria for translating L into an order-sorted language L s with signature Σs=<C, S, Q>, where S⊆P is a t of sort symbols called sortals, and Q=R \S is a t of ordinary predicates.
Def. 1. Let L m be the modal extension of a first order language L obtained by adding to L the usual modal operators s and x, and D be a t. A rigid model18 for L m bad on D is a structure M=<W, r, D, f C, f P>, where W is a t of possible worlds sharing the same interpretation f C for constant symbols of L m, r is a binary relation on W, D is a domain common to all possible worlds, and f P is a mapping that assigns to each predicate symbol P of L and world w∈W a unary relation on D.
For a given rigid model M, r is a relation between worlds (i.e., interpretations of L) that may differ in the interpretation of predicates while sharing the same interpretation for constants. We want to give r
the meaning of an ontological compatibility relation: two worlds are ontologically compatible if they describe plausible alternative states of affairs involving the same elements of the domain. r will be in this ca reflexive, transitive and symmetric (i.e., an equivalence relation), and the corresponding modal theory will be S5. Def. 2. Let L be a first order language and D a domain. An ontological commitment for L bad on D is a t C of rigid models for L m bad on D, where the relation r is an equivalence relation. Such commitment can be specified by an S5 modal theory of L m, being in this ca the t of all its rigid models bad on D. A formula Φ of L m is valid under C (C |= Φ) if it is valid in each model M∈C.
Within this modal framework, preliminary reflection on the distinction between sortal and non-sortal predicates reveals that the former cannot be necessarily fal for each element of the domain: they must be natural predicates, in the n of the following definition19:
Def. 3. Let L be a first order language, P a monadic predicate of L, and C an ontological commitment for L. P is called natural under C iff C|= ∃x.x Px.
A more substantial obrvation that comes to mind when trying to formali the nature of the subject-predicate relationship in the examples above, is that the “force” of this relationship is much hi
gher in “x is an apple” than in “x is red”. If x has the property of being an apple, it cannot lo this property without losing its identity, while this does not em to be the ca in the cond example. This obrvation goes back to Aristotelian esntialism, and can be easily formalid as follows20:
Def. 4 A predicate P is ontologically rigid under C iff it is natural under C and C|=∀x.(Px ⊃ s Px).
Ontological rigidity ems a uful property for characterising sortals: stating that apple is rigid and red is not will clarify the intended meaning of the two predicates in the statement (1) of Fig. 2. In this ca, if a∈C, the worlds satisfying (A(a) ∧ R(a)) or (A(a) ∧¬R(a)) will turn out to be mutually compatible, while tho satisfying (R(a) ∧ A(a)) or (R(a)∧ ¬A(a)) are not (due to the constraints impod on r by the rigidity of A). Assuming that rigidity is a necessary property for sortals, we can then exclude both ∃x:R.Ax and ∃(x:A,y:R).x=y from our axiomatisation choices for (1).
Notice that the naturalness condition in the above definition excludes cas where rigidity would be trivially true due to the impossibility of P. On the other hand, ontological rigidity will be trivially satisfied by predicates being necessarily true for each element of the domain, like thing or entity. Yet, according to traditional wisdom they are excluded from being sortals, since no clear distinction criteria are associated with them.Rigidity cannot be therefore be considered as a necessary conditio
西游记情节
n for sortals. We call the “top level”predicates generic predicates21. In the same category other rigid predicates should be included, that, although being not trivially rigid, are still too general to supply a distinction criterion: object, individual, However, a distinctive characteristic of generic predicates is that they are rigid but divisive, in the n that they can hold for parts of their arguments. Various divisivity criteria have been propod in the literature in order to account for the distinction between countable and uncountable predicates22; to the purpos of the prent paper, the following definition will be good enough:
Def. 5. Let P be a natural predicate under C, and < be a “proper part” relation assumed as primitive, satisfying the axioms of classical mereology23. P is divisive under C iff C|=∃x.x(Px ∧ ∃y. y<x) ∧∀x.s(Px ⊃ (∃y.(y<x ⊃ Py))).
In other words, a predicate P is divisive if its arguments can have proper parts, and, necessarily, if its instances have proper parts then P holds for one of the. We are now in a position to give a definition of substantial sortals:
Def. 6. Let P be ontologically rigid under C. It is a generic predicate in C if it is divisive in C, and a substantial sortal in C otherwi.
Within our KR framework, the above definition gives a formal characterisation to the notion of substantial sortals originally introduced by Wiggins, delimiting tho rigid predicates that are sortals. We need now a distinction criterion between non-rigid predicates:
