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Ontological aspects of logical databases
/n/orni S.vsrrms Vol. Prmted in the b.S.A
IO. No. 3. pp 331-338.
198.’
ONTOLO~ICAL
ASPECTS OF LOGICAL DATABASES
RONALD M. LEE College and Graduate School of Business, The University of Texas at Austin. CBA 5.202, Austin, Texas 78712, U.S.A. RONALD STAMPER Department of Statistical and Mathematical Sciences, London School of Economics. Houghton Street, London WC2A 2AE. England (Received 4 April 1984)
Abstract-The practical advantage of logical databases is the inferencing capability they provide. An important theoretical effect is the clarifkation of semantic issues, through logic to model theory.
A. IDENTIFICATIONVSPREINCATION
typically has the form
Relational databases have a st~gh~o~~d representation in logic programming as ground clauses. For instance, the relational database:
EMPLOYEE (name, rank. SMITH CLERK
= elements of D Ftindividual constant Ft I-pIace predicate constant) = subsets of D Ftn-plane predicate constant) = relations on D
sex,
marital status)
MALE
MARRIED
1
JONES DRIVER FEMALE DIVORCED becomes, in logic programming: EMPLOYEE (SMITH. CLERK, MALE, MARRIED) EMPLOYEE (JONES, DRIVER, FEMALE, DIVORCED). That is, the tuples of a database relation are interpreted as predicate assertions on the data values in each tuple. The notation here and throughout is that individual and predicate constants are in upper case. Variables will be in lower case and. unless otherwise indicated, are presumed to be universally quantified at the front of the expression. Note that attribute names are represented implicitly as predicate places; a separate schema is therefore not needed. This relatively minor change in syntax highlights an important shift in semantic perspective between database modeling and logic modeling, in particular in the emphasis given to the distinction between identification and predication. The prevailing view of semantics for logical languages is of course Tarskian model theory[U, 261. A model, or interpretation of a frost-order language is of the form: M = (D, F)
where D is a universal set, the “domain of individuals,” and F is an interpretation function, mapping symbols and expressions in the language to D. This
That is, the function, F, gives the denotation of the terms and expressions of the logical language, i.e. its “denotational semantics.‘* Here we are concerned principally with the denotational semantics of the open (nonlogical) vocabulary of the logic. (Denotational semantics can also be provided for the logical vocabulary, i.e. connectives, quantifiers, other logical operators. See for example [S, chapters 3-4 for a discussion.) Tarski[ZSJ was mainly concerned with the characteristics of a logical language that held for all of its many possible interpretations, leading to a concept of logical truth as “true in all possible models.” In database applications, by contrast, we are concerned with one specific interpretation only. This raises the issue of ontology; that is, the basic sorts of objects assumed in the set D. In short, “What is there?” Following the standard model theoretic approach, the implicit ontology of a logical database would be the things denoted by individual constants in the system or, alternatively, the sorts of things in the range of variables in the deductive rules.
R. M. LEE and
332
(“To be is to be the value of a bound variable,“[lB]). From this perspective, most actual databases present a rather confusing mix. In the foregoing example, for instance, the underlying ontology apparently consists of such things as people (e.g. SMITH, JONES), ranks (e.g. CLERK, DRIVER), sexes (MALE, FEMALE), and status (e.g. MARRIED, DIVORCED). In the usual applications of first-order logic, there is a separation of semantic function: individual constants are used for identification of entities in D, predicates are used for predication, i.e. to ascribe properties and relationships to these entities. In the foregoing example, this separation is not upheld, e.g. CLERK, DRIVER, MALE, FEMALE seem to be properties rather than names of individual entities. This sort of phenomenon is to be found in many
R. STAMPER To illustrate the problem, suppose that in addition to the foregoing EMPLOYEE relation, we have another relation about students: STUDENT
represented
(name,
sex,
marital status)
ADAMS
FEMALE
MARRIED
BAKER
MALE
SINGLE
by the ground clauses:
STUDENT
(ADAMS, FEMALE,
STUDENT
(BAKER,
(x) + EMPLOYEE
(x, y, z, SINGLE).
ELIGIBLE
(x) + EMPLOYEE
(x, y, z, DIVORCED).
ELIGIBLE
(x) + STUDENT
(x, y, SINGLE).
ELIGIBLE
(x) + STUDENT
(x, y, DIVORCED).
semantic functions of ider@cation and predicatiori. This is because the relational model is a relational model of data. The relations defined are
relations among data items, rather than entities in the environment. Stated otherwise, the basic ontology of the relational model consists of data objects, e.g. character strings and numbers. Clearly, these have an implicit real world significance. For example, the fUnctional dependencies that guide the normalization process are not observed from actual data, but are assumed, based on background knowledge of what the data signifies. However, the semantic correspondence between data and real world objects is not made explicit. The pragmatic response is “So what?” For EMPLOYEE STUDENT
The basic criticism is that, in order to make inferences on properties embedded as relational attributes, the user not only has to know the correct predicate name, but also where this predicate might appear as a value of a relational attribute. Likewise, the properties of being an EMPLOYEE or STUDENT are not cleanly represented. Rather than indicating these simple properties of an individual, the foregoing relations convey a mixture of properties. Here the converse problem arises: to make rules or queries involving these properties, one must know which places are used for identification purposes. These ambiguities can be avoided by enforcing the discipline that predication (properties, relationships) are represented exclusively as predicates, whereas identification is conveyed by individual constants or variables. Continuing the foregoing example, this might be done using rules such as:
(x) + EMPLOYEE (x)
MALE, SINGLE).
Now, suppose for purposes of arranging an office party, we want to invent the concept “eligible” as being either single or divorced. This would involve the following rules:
ELIGIBLE
actual database designs. This stems from the fact that in the relational model (and other data models as well) there is no explicit distinction between the
+ STUDENT
(x, y. Z, w). (x, y, 2).
CLERK (x)
+ EMPLOYEE
(x, CLERK,
DRIVER (x)
+- EMPLOYEE
(x, DRIVER, y, z).
MALE (x)
+ EMPLOYEE
(x, y. MALE, z).
MALE (x)
+ STUDENT
databases used only for retrieval, this semantic distinction does not seem to matter much. However, when used as the basis for inferencing, the motivation is much stronger. This is because firstorder deductions reflect the interdependence of properties, i.e. predication.
MARRIED).
y, z).
(x, MALE, z).
and so on. The foregoing remarks may be summarized in the observation that the relational model is ambiguous in its representation of predication, i.e. properties are conveyed either as relation names or as attribute values. For database inferencing pur-
Ontological aspects of logical databases
poses, it is useful to maintain the semantic distinction between identification and predication in the syntactic roles of individual constants and predicate constants. (Note that it is not necessary to re-organize the underlying database to do this; a ‘view’, e.g. as produced by the above rules, is sufficient.) 9. WHY ONTOLOGY?
The foregoing example was of course carefully chosen to arrive at a nice, tidy solution. The resulting ontology (the range of individual constants and variables) was L) = the set of people. The interpretation function, F, would be a mapping from symbols in the logic to actual people in the world, e.g. F(SMITH)
= Smith
F(JONES)
= Jones
F(CLERK) = (Smith, Jones} and so on. Here the names used on the right are symbols in a mefalanguage, e.g. English. Ideally, we would point to the real objects themselves, but this is rather difficult in a written text.3 This leads to a recogni~on of the real purpose of an ontology. If two or more people are to make use of a symbol system, e.g. a database, as a medium of communication and reasoning, it is essential that they mutually understand the denotation of the primitive symbols. Typically, this is done by means of a natural language explication of these primitives. On the other hand, logic, especially in database applications, is often used to refine a certain area of natural language discourse. The logical vocabulary is used in a more disciplined, restricted way than its natural language counterpart. To be confident that the users of the logic mutually understand its elementary vocabulary, one would prefer to use nonlinguistic explanations, e.g. pointing to the basic objects being denoted.8
333
This would suggest an ontology consisting of discrete physical objects. Indeed, such an ontology is implicit in many database applications, and is probably among the “safest” to ensure mutual understa~dab~ity. On the other hand, any ontology is adequate provided the users of the logic mutually recognize the underlying domain of objects. The most outstanding examples are mathematical theories, for instance assuming a domain of real numbers. It is difficult to define such objects by pointing to them. Indeed, it might be argued that mathematical domains are defined proof theoretic~y-by their axiomatizations, That is, nobody cares what numbers “really are,” apart from the things that satisfy the axioms. Similar remarks might be said of games, e.g. board games, card games. Clearly the physical representations have no importance since these are now often played using computer screens. One might say that, like mathematics objects, they are defined by the rules controlling their symbolic manipulation. Nobody cares what a knight or queen of hearts “really is,” apart from its role in the rules of the game. It is noteworthy that much of the initial work in artificial intelligence research has focused on domains of this type (mathematics, games), where the ontology was implicit in the system of rules. Database applications, by contrast, are detinitely nor of this type. Indeed, this is perhaps their distinguishing characteristic: a database does not define its objects, implicitly, in its rule structure; rather, it only describes objects which have a separate, independent existence. The recognition of such objects is established by the people in the organization who make the appropriate divisions of the world into pieces that facilitate their business activitiesl21, 221. c. SOMEOhTOLoGICAL ISSUES
In the previous section, an ontology of discrete physical objects was suggested. Assuming this as a starting point, we proceed to investigate various issues that arise in actual database contexts, espe$ Tbc superficial similarity between object language tokens, e.g. SMITH. JONES and metalanguagetokens, cially in organizational settings. These are are.g. Smith, Jones, is often a point of confusion. The spell- ranged, roughly, in order of difficulty. ing in the object language is merely a convenient memory aid. Rather than SMITH we could have used X275I? for instance, with the same logical effect, as long as: F(X2751) = Smith.
The rne~~gu~e tokens are, by contrast, an accomodation to using a typographic medium to explain the logic. Here the spelling does matter since these are assumed to be direct labefs to the objects themselves, e.g. like the name tags used by conference attendees. I But pointing already suppose that the objects can be individuated from their spatiaVtemporal context. For example, imagine two people standing on a shore watching the waves. In trying to discuss individual waves they have a twofold problem: a) there is no clear-cut boundary of a
1. Multiple names An immediate problem is that in many administrative contexts, an individual person or object may have multiple names or identification codes. Further, these names may not necessarily uniquely
wave from the surrounding sea and b) the form, hence boundary of the wave, is constantly changing.This is one aspect of the so-called “individuation problem”; see e.g. Strawson[24]. Wittgenstein[28] and Quinel171 raise other epistemological issues relating to ostensive (pointing) definitions.
334
R. M. LEE and R. STAMPER
and metalanguage. Alternative identifiers identify the ~dividu~. For example, in the U.S., a l~age person typically has a frrst, middle and last name of this sort are clearly extensions to the object lanas well as a social security number, assigned by the guage. However, the sorts of things denoted remain federal Social Security Administfation, He/she may the same; we simply have a richer vocabulary for in addition have an employee number, assigned by referring to them. the employing comply, a health insane numOn the other hand, these ~temative identifiers ber, assigned by an insurance company, a driver’s arise mainly due to relationships with other parties, license number, assigned by the state granting the e.g. an SEC number is of interest because of translicense, various credit card numbers, assigned by actions involving the Securities Exchange Commission. Hence, alternative identifiers tend to the c@dit card companies, etc. Similar remarks can be made of organizations imply the involvement of other organizational entthemselves: they have a company name and per- ities, namely the institutions that assigned the idenhaps a corporate code assigned by the state of in- tifying codes. This suggests that the recognition of alternative corporation. If their stock is publicly traded, they will have an SEC code, assigned by the Securities identifiers affects the ontolo~ by forcing the inExchange Commission. In addition, companies .clusion of these other organizations in the domain, typically record their transactions with other com- D. Insofar as this only affects the cardinality of D, panies by ass~n~g codes for in-house use, e.g. a it is not an ontological problem. However, we have now a new ontological problem: organ~ation~ entsupplier number, customer number. Vehicles may also have multiple identifiers, e.g. ities such as (corporations, public institutions and a car may have one code on the engine block, an- agencies) do not fit neatly into the assumed ontolother on the frame, both assigned by the manufac- ogy of discrete physical objects. This point is returer. If used on public roads, it must also have a considered ia the last section. vehicie license number, assigned by the state of residence. If used in interstate or intematioual transAs remarked earlier, fo~~tions of matheport, other numbers may be assigned. (Kent{91 suggests numerous additions examples of the naming matical domains typically imply an ontology involving numeric entities, e.g. the integers, real problem in data processing.) Some of these identifiers, in particular names of numbers, etc. The issue considered in this section people. and companies, have no g-tee of is the role of quantitative data in a database ontoluniqueness. In the other cases, where an identifing ogy. Again, there is no absolute answer. 09 the code was assigned, the uniqueness claim is relative other hand, it is a reasonable claim that databases, to the scope of the institution assigning the code. as used in organizational applications, are seldom For instance, one person’s employee number could if ever about numbers per se. Rather, numbers appear in the context of ~~~~~~e~e~~, i.e. they are correspond to another’s credit card number. In first-order logic, the problem of multiple iden- used to ascribe properties to other (real world) tifiers is usually iguored. Each entity in the don&n, objects. D, is assumed to have just one logical namer which Some observations from measurement theory is unique within the scope of the logic, (see, for example&, 6, 10, 19,20 (chapter 3)1)have An approach might be to disting~s~ between in- bearing here. A meas~ement is a ~o~es~nden~e ternal YS external identifiers. An internal identifier between two relational systems: an empirical rewould be an arbitrarily assigned code, serving as a lational system (relationships among real world oblogical name. Other names and codes are consid- jects), and a numeric relational system (hong nuered external, for aiding various ext.emaI pafties to meric objects). For example, an empirical relational identify the entity. system might be: For instance, for an entity known internally as E = (D, RAFTER-THAN) X2751, we might use the notation: LAST-NAME (X2751) = “ SOCIAL-SECUR~Y-NUMBER
(X2751) = “512-27-6219”
EMPLOYEE”NUMBER (X But what are these expressions in double quotes? Obviously they are character strings, but do they represent a new kind of entity in the ontology? The answer here could go either way. One might admit them into the ontology, regarding them as a kind of linguistic entity, which denote themselves. This, however, tends to once again confuse object
where D, as before, is tbe set of discrete physical objects. These are ordered by the relationship, ROARER-THAN (x, y) ~dicating that x is redder than y. A corresponding numeric system might be: N = (I, >) where f is the set of integers, and > is greater than.
33s
Ontological aspects of logical databases
The correspondence between two such relational systems is a homomorphism mapping from the empirical to the numeric system. For example, REDNESS
(x) = n
might be such a homomorphism, mapping objects in D to objects in n such that the relationship REDDER-THAN is preserved in >. Thus, where such a correspondence exists, arithmetic deductions may be substituted for logical deductions. A meusurement scale consists of the triple,
lbs” (where “5 lbs” is a denominate number distinct from the “pure number,” 5), whereas the functional representation would paraphrase these reports into pure number form. e.g. as “lb (x) = 5,” where lb (x) is the weight-inpounds function whose values are pure real numbers. In banishing denominate number ontological categories, the functional representation accomplishes an ontological reduction which is conceptually important.
The ontological reduction in terms of the real world corresponds to the recognition of a specific object as the standard unit of measurement. Returning to database considerations, a typical relation containing numeric data might be:
M = (E, A’, H)
PERSON where E is an empirical relational system, N is a numeric relational system, and H is a homomorphism from the first to the second. Measurement scales are usually classified based on the nature of the numeric relational system employed. For example, scales involving N = (Re, Z-=)
are called ordinal Scales involving
scales
(Re = real numbers).
N = (Re, z=,
+)
are.called ratio scales. A central problem of measurement theory, the representation problem, is to axiomatize the conditions that must hold in the empirical system for the measurement scale to be valid. For example, in the case of ratio scales (also called extensive measurement), some correspondent to arithmetic addition must exist in the empirical system. For instance in weight measurement, one physically adds weight to a balance scale. In current measurement theory, differences in measurement units (e.g. pounds vs kilograms) amount to transformations on the homomorphism, H. For example, two homomorphisms might be: WEIGHT-IN-LBS
(x) = n
WEIGHT-IN-KGS
(x) = m
the transformation
between them being:
WEIGHT-IN-LBS
(x) = 2.2 * WEIGHT-IN-KGS
Adams[l,
(name, weight-in-kgs) SMITH
80
JONES
60
In a logical syntax this might be: WEIGHT-IN-KGS
(SMITH)
= 80.
WEIGHT-IN-KGS
(JONES)
= 60.
Note that the relational attribute containing quantitative data corresponds to a measurement scale homomorphism. This, we claim. is the typical role of quantitative data in databases!; Supposing this to be the case, what perspective does this offer concerning the ontology of databases? The arguments here are somewhat analogous to those for multiple identifiers. While numbers could be regarded as a separate sort of entityaxiomatized, perhaps, using a multisorted logicthis does not seem to be necessary for database formalizations. Quantitative data in databases has a role much like that of predicates, i.e one of predication. The point is that such measurements permit an alternative mode of inferencing-arithmetic rather than deductive-but the underlying entities, the subject matter of the inferences, is the same. Thus, from the database perspective, quantitative measurement amounts to an extension of the logical language, but does not affect the underlying ontology. 3. Collective objects The ontology assumed thus far consists of discrete physical objects. This implies that each such object might be assigned a distinct logical name.
(x).
p. 2101 comments:
It is important to stress to students unfamiliar with the modem fincrional representation of measurement that fundamental measurement theory adopts this representation, and in so doing “paraphrases” or translates more traditional metrical language (which is still widely prevalent in the sciences) into unfamiliar and initially unituitive forms. In particular, metrical data are traditionally reported in denominate number form, e.g. as “x weighs 5
I A research issue for logic programming applications to databases is to explore the interrelationship between logical and arithmetic inference from a measurement theory perspective. At present, the IS operator in PROLOG appears almost as an afterthought. The point is that certain types of inferences, e.g. on order relationships, extensive measurement, can be done using resolution, based on empirical relationships, or arithmetically, using numeric relationships. To the end user, the choice should be transparent, it being a question of inferential efficiency. Can this choice be automated?
336
R. M. LEE and R. STAMPER
However, in administrative applications there are many cases where this would be impractical. Inventories, for example, often contain objects that are too small or unimportant to name individually, e.g. bins of bolts, boxes of nails. In the extreme are granular objects like wheat or flour, and liquids or gaseous objects, e.g. oil, propane. The problem here is that these populations are effectively infinite, making them infeasible for resolution-based inferencing methods. On the other hand, organizations themselves seldom, if ever, regard these objects at such a detailed level of granularity. Rather, such objects are generally handled in some sort of c&&her, e.g. cartons, drums, boxcars. These are “middle-sized” objects which are, typically, assigned an identifying name. Entities having these characteristics might be called “collective objects” in that they represent granular populations collected into a larger, discretely identifiable aggregate. The cardinality of a collective object is simply another type of measurement (“absolute measurement”). Here the interaction of deductive vs arithmetic modes of inference is a requirement. Inferences concerning the collective objects, e.g. their location, transfer of ownership, take the form of deductions on any other type of discrete physical object. Inferences regarding their detailed composition, e.g. total stock on hand, inventory costing, are donearithmetically. 4. Time In contrast to the preceding cases, time (consequently, change) presents a fundamental and difficult issue in database ontology.” A basic difficulty is in distinguishing the identity of an entity at different points in time. To illustrate, consider the old philosophical chestnut, commonly called the Boat of Theseus. We have a wooden boat. We replace one of its planks and lay it aside. Is it the same boat? Replace another plank, and another, and another. Is it still the same boat? Continue until all the planks are replaced. Still the same boat? Now, with the old planks that have been removed, construct a different boat of the same design. Which, now, is the original boat? (Note that an entire navy of identical boats could be constructed by iterating this process.) Analogous problems of individuation arise, for instance, in the replacement of parts on vehicles and equipment. The problem, essentially, relates to the distinction alluded to earlier between observing vs deciding the denotations of terms. The temporal extent of an individual entity is often a matter of decision, relative to the purposes of the discourse or logic. ’ See Bolour et a1.[2] for a useful survey of temporal issues in information processing. Lee et a1.[14] presents a proposal specific to logical databases.
A related problem is that the entities typically described in databases have a life, i.e. they are born (created) and later die (are destroyed). The issue, then, is that the universal set, D, apparently has different membership at different points in time. The ontological issue here can be considered differently, depending on how the database is used. A common use of databases in organizations is rather like a movie film-the database is like an ongoing series of snapshots, portraying the state of the organization at the corresponding state in time. Under this view, the fact that the domain of individuals, D, changes from one time to the next is irrelevant, since inferences are made only within a given snapshot. However, if the use of the database is to include cross-temporal inferences, e.g. comparing the current state of affairs with the past, a different ontology is required. Initially, this would require extending the underlying model with a time dimension, i.e. M = (T, D, F)
where T is a time line, D is the set of’entities (discrete physical objects), and F is an interpretation function mapping terms and expressions in the logic into T x D. Assertions in the logic could then be regarded as temporally dependent. For example, a proposal of Rescher and Urquhartll81 uses the notation: R(r): p. to
express that the assertion p “is realized” at time,
t.
In business, the usual conception of time is as a continuous dimension. For purposes of mechanical inferencing, however, it is preferable to assume a discrete time. This apparent conflict can be resolved by recognizing only certain time intervals, e.g. the dates of the calendar, explicitly in the logic. These time intervals function conceptually rather like the containers discussed for collective objects. However, quite independent from whether time is considered to be continuous or discrete, is the issue about the composition of the domain of individuals, D. If we consider appplications involving historical comparisons, D must clearly contain more than those entities existing at present. For example, past production may have created objects that have since been consumed. This suggests an ontology consisting of objects both present and past. In such a view, a temporal predicate of existence is required, e.g. R(r):
EXISTS (x).
indicates that x exists at time, t.
Ontological aspects of logical databases
The idea of a domain, D, consisting of objects both past and present may seem a bit awkward. For instance, one might ask what these entities are doing when they are not existing. Likewise, recognizing existence as a predicate gives rise to such constructions as:
337
options, etc., all of which represent contractual relationships. The most universal type of contractual object is money. Originally, printed currency represented a promise for the delivery of gold (or silver). Now that the gold standard is no longer used, one wonders what gives money its value. (The same ques3x R(r): -EXISTS (x). , tion might, indeed, be asked about gold itself.) The answer typically given is couched in terms of read that there exists an x such that at time t, it “faith” or “confidence.” Domestically, this indidoes not exist. This, however, is an artifact of the cates a sort of universal promise of the people usual reading of the existential quantifier. The se- among themselves. On international markets, a mantic interpretation of the existential quantifier is country’s currency has an even stronger resemblance to contracts: it is a general purpose I.O.U. to refer to some element in D. The existence predfor goods and services of that country. icate, as suggested here, is a temporal predicate, Earlier, it was remarked that the concept of an relating elements in D to elements in T. Given the revised ontology suggested here, this interpretation organization (e.g. corporation or government is perfectly legitimate[l8, Chap. 201. agency) did not tit neatly into an ontology of disNote that here we have proposed that D contain crete physical objects. An organization is more than only present and past entities. Considering that the its assets; also it is more than a loose affiliation of time dimension normally is conceived as extending people. In the U.S. at least, the principal type of into the future, one is led to consider a correspondprivate organization is the corporation. A corpoing extension to D, i.e. to include future entities as ration might also be regarded as a type of contracwell. The problem with this is that future entitie tual object. Specifically, it is a permission, granted are likewise hypothetical entities. We do not know by the state of residence, to engage in commerce, about future entities (or events) in any factual sense negotiate financing, with limited liability to the as we know the past. The past is history, the future shareholders. Importantly, the corporation itself is is speculation. Our concern here, being limited to the agent of its contracts, not its shareholders or its databases, is with the factual: Predicting and hymanagers. A corporation might thus be viewed as pothesizing about the future, while outside the presa sort of “second-order” contractual entity: a conent scope, are nonetheless vital concerns to mantract permitting contracting. agement. Ontologically speaking, this enters into By the same token, governmental agencies-at the area of so-called “possible worlds semantics,” least in democratic societies-have a contractual a subject of considerable current interest and character. The agency is formed by a charter. specifying the services it is to fulfill. It is this charter, debate.+ rather than any particular assets or personnel, that 5. Contractual objects is the essence of the agency. While we might exclude future or hypothetical But what is a contractual object? In each of the entities as not belonging to an ontology for factual abovementioned cases, its essence seems to be databases, there is an area that seems factual, yet something nonphysical, namely an obligation (or which has hypothetical aspects about it as well, perhaps permission) between certain individuals or namely, contracts. organizations to others. On the other hand, obligation and permission are both future-oriented conConsider, for example, that much of the contents cepts-they apply to future behavior rather than to of administrative databases deals with accounting data. Among this data are references to such things actual circumstances. One may of course simply speczfi contractual a receivables, payables, leases, notes, bonds, inobjects as part of the ontology. This, however, surance, stock, options, etc. AlI of these are exseems to negate the basic purpose of ontology, amples of what might be called a “contractual obnamely to identify sets of referrents that are unject.” A contract is a kind of relationship between ambiguously recognized and identifiable by all two parties obliging them to perform certain actions perhaps only under certain conditions, before a cerusers of the logical language. Various forrnalizations of obligation and pertain time, etc. mission have been proposed as so-called “deontic Yet at the same time, these “relationships” are logics” (see, e.g. 17, 81). Contracts may be viewed often, in business, regarded as objects. Major exas an application of deontic logic[l2, 13, 33. On the amples are the securities exchanges which deal in other hand, attempts to provide a formal semantics the buying and selling of stocks, bonds, futures, for deontic logic are typically based on an ontology involving possible worlds. Von Wright[27], for int Useful discussions of possible world semantics are stance, draws a parallel between deontic logic and found in Rescher and Urquhart[H], Dowty et al.[S], Cresmodal logic, and uses the concept of a “deontically swell[4], Kripkef 1I]. McDermott[ IS] proposes a futureoriented temporal logic involving hypothetical reasoning. possible world” in his semantics. The difficulty, we
338
R. M. LEE and R. STAMPER
find, is that possible worlds are too esoteric a concept to be used as the basis for consensual understanding by a community of database users. An alternative course, perhaps more t?uitfLl, is analogous to that described for mathematics and various sorts of games: the ontology is described implicitly by the rules controlling it. By this viewpoint, a contractual object is that thing that satisfies the rules of contract law.
D.
CONCLUJIING REMARKS
The purpose, here, has been to sketch the outlines of a theory of what databases are about; the elementary sorts of entities that databases, in particular databases in organizations, refer to. The concept of a logical database provides a useful context to study these issues, by linking to the rich literature on the semantics of first-order languages. Tar&an model theory provides a mathematically crisp conception of semantics in the notion of a model or interpretation assigned to the language. This raises the issue of ontology-choosing an underlying domain of entities that is consensually recognized by all users of the language. The view presented here is most appropriate for those aspects of business relating to physical systems, e.g. manufacturing, distribution systems. It is less satisfactory, however, for business Cmctions dealing with phenomena of a more social nature, e.g. marketing, management of personnel, contract negotiations. It was suggested that such things as contractural objects might be defined proof theoretically, i.e. that which satisfies the axioms of contract law, rather than hypothesizing the existence of abstract, “contractual objects.” Further consideration along these lines might lead to a completely different conception of semantics. As opposed to the view presented here, where symbols denote real world objects having an independent existence, an alternative view might define objects relative to their social function[23]. For example, a knife and fork are better understood relative to the social norms of eating, rather than by their physical characteristics alone. This view will be developed more fully in a subsequent paper.
REFERENCES
[II E. W. Adams: Measurement theory. In Current Research in Philosoohv of Science. DD. 207-227. (Edited by P. D. Asquith and H. E. Kyburg, Jr.). East Lansing, Ml. Philosophy of Science Association (1979). PI A. Bolour, T. L. Anderson, L. J. Dekeyser and H. K. T. Wang: The role of time in information Drocessing: A &vey. S&nod Letters, (April I&):I31 H. N. Casteneda: The logical structure of legal systems: A new perspective. In Deontic Logic, Computational Linguistics and Lrgul Information Systems, Vol. II. (Edited by A. A. Martino). NorthHolland, New York (1982).
[41 M. J. Cresswell: Logics and Languages. Nethuen Co. Ltd., London (1973). I51 D. R. Dowty, R. W. Wall and S. Peters: Introduction to Montague Semantics. Reidel, Boston (1981). WI B. Ellis: Basic Concepts of Measurement. Cambridge University Press, Cambridge (1966). r71 R. HilDinen (Ed.): Deontic Logic: Introductotv and Systematic Readings. D. Reidel, Dordrecht
IO. No. 3. pp 331-338.
198.’
ONTOLO~ICAL
ASPECTS OF LOGICAL DATABASES
RONALD M. LEE College and Graduate School of Business, The University of Texas at Austin. CBA 5.202, Austin, Texas 78712, U.S.A. RONALD STAMPER Department of Statistical and Mathematical Sciences, London School of Economics. Houghton Street, London WC2A 2AE. England (Received 4 April 1984)
Abstract-The practical advantage of logical databases is the inferencing capability they provide. An important theoretical effect is the clarifkation of semantic issues, through logic to model theory.
A. IDENTIFICATIONVSPREINCATION
typically has the form
Relational databases have a st~gh~o~~d representation in logic programming as ground clauses. For instance, the relational database:
EMPLOYEE (name, rank. SMITH CLERK
= elements of D Ftindividual constant Ft I-pIace predicate constant) = subsets of D Ftn-plane predicate constant) = relations on D
sex,
marital status)
MALE
MARRIED
1
JONES DRIVER FEMALE DIVORCED becomes, in logic programming: EMPLOYEE (SMITH. CLERK, MALE, MARRIED) EMPLOYEE (JONES, DRIVER, FEMALE, DIVORCED). That is, the tuples of a database relation are interpreted as predicate assertions on the data values in each tuple. The notation here and throughout is that individual and predicate constants are in upper case. Variables will be in lower case and. unless otherwise indicated, are presumed to be universally quantified at the front of the expression. Note that attribute names are represented implicitly as predicate places; a separate schema is therefore not needed. This relatively minor change in syntax highlights an important shift in semantic perspective between database modeling and logic modeling, in particular in the emphasis given to the distinction between identification and predication. The prevailing view of semantics for logical languages is of course Tarskian model theory[U, 261. A model, or interpretation of a frost-order language is of the form: M = (D, F)
where D is a universal set, the “domain of individuals,” and F is an interpretation function, mapping symbols and expressions in the language to D. This
That is, the function, F, gives the denotation of the terms and expressions of the logical language, i.e. its “denotational semantics.‘* Here we are concerned principally with the denotational semantics of the open (nonlogical) vocabulary of the logic. (Denotational semantics can also be provided for the logical vocabulary, i.e. connectives, quantifiers, other logical operators. See for example [S, chapters 3-4 for a discussion.) Tarski[ZSJ was mainly concerned with the characteristics of a logical language that held for all of its many possible interpretations, leading to a concept of logical truth as “true in all possible models.” In database applications, by contrast, we are concerned with one specific interpretation only. This raises the issue of ontology; that is, the basic sorts of objects assumed in the set D. In short, “What is there?” Following the standard model theoretic approach, the implicit ontology of a logical database would be the things denoted by individual constants in the system or, alternatively, the sorts of things in the range of variables in the deductive rules.
R. M. LEE and
332
(“To be is to be the value of a bound variable,“[lB]). From this perspective, most actual databases present a rather confusing mix. In the foregoing example, for instance, the underlying ontology apparently consists of such things as people (e.g. SMITH, JONES), ranks (e.g. CLERK, DRIVER), sexes (MALE, FEMALE), and status (e.g. MARRIED, DIVORCED). In the usual applications of first-order logic, there is a separation of semantic function: individual constants are used for identification of entities in D, predicates are used for predication, i.e. to ascribe properties and relationships to these entities. In the foregoing example, this separation is not upheld, e.g. CLERK, DRIVER, MALE, FEMALE seem to be properties rather than names of individual entities. This sort of phenomenon is to be found in many
R. STAMPER To illustrate the problem, suppose that in addition to the foregoing EMPLOYEE relation, we have another relation about students: STUDENT
represented
(name,
sex,
marital status)
ADAMS
FEMALE
MARRIED
BAKER
MALE
SINGLE
by the ground clauses:
STUDENT
(ADAMS, FEMALE,
STUDENT
(BAKER,
(x) + EMPLOYEE
(x, y, z, SINGLE).
ELIGIBLE
(x) + EMPLOYEE
(x, y, z, DIVORCED).
ELIGIBLE
(x) + STUDENT
(x, y, SINGLE).
ELIGIBLE
(x) + STUDENT
(x, y, DIVORCED).
semantic functions of ider@cation and predicatiori. This is because the relational model is a relational model of data. The relations defined are
relations among data items, rather than entities in the environment. Stated otherwise, the basic ontology of the relational model consists of data objects, e.g. character strings and numbers. Clearly, these have an implicit real world significance. For example, the fUnctional dependencies that guide the normalization process are not observed from actual data, but are assumed, based on background knowledge of what the data signifies. However, the semantic correspondence between data and real world objects is not made explicit. The pragmatic response is “So what?” For EMPLOYEE STUDENT
The basic criticism is that, in order to make inferences on properties embedded as relational attributes, the user not only has to know the correct predicate name, but also where this predicate might appear as a value of a relational attribute. Likewise, the properties of being an EMPLOYEE or STUDENT are not cleanly represented. Rather than indicating these simple properties of an individual, the foregoing relations convey a mixture of properties. Here the converse problem arises: to make rules or queries involving these properties, one must know which places are used for identification purposes. These ambiguities can be avoided by enforcing the discipline that predication (properties, relationships) are represented exclusively as predicates, whereas identification is conveyed by individual constants or variables. Continuing the foregoing example, this might be done using rules such as:
(x) + EMPLOYEE (x)
MALE, SINGLE).
Now, suppose for purposes of arranging an office party, we want to invent the concept “eligible” as being either single or divorced. This would involve the following rules:
ELIGIBLE
actual database designs. This stems from the fact that in the relational model (and other data models as well) there is no explicit distinction between the
+ STUDENT
(x, y. Z, w). (x, y, 2).
CLERK (x)
+ EMPLOYEE
(x, CLERK,
DRIVER (x)
+- EMPLOYEE
(x, DRIVER, y, z).
MALE (x)
+ EMPLOYEE
(x, y. MALE, z).
MALE (x)
+ STUDENT
databases used only for retrieval, this semantic distinction does not seem to matter much. However, when used as the basis for inferencing, the motivation is much stronger. This is because firstorder deductions reflect the interdependence of properties, i.e. predication.
MARRIED).
y, z).
(x, MALE, z).
and so on. The foregoing remarks may be summarized in the observation that the relational model is ambiguous in its representation of predication, i.e. properties are conveyed either as relation names or as attribute values. For database inferencing pur-
Ontological aspects of logical databases
poses, it is useful to maintain the semantic distinction between identification and predication in the syntactic roles of individual constants and predicate constants. (Note that it is not necessary to re-organize the underlying database to do this; a ‘view’, e.g. as produced by the above rules, is sufficient.) 9. WHY ONTOLOGY?
The foregoing example was of course carefully chosen to arrive at a nice, tidy solution. The resulting ontology (the range of individual constants and variables) was L) = the set of people. The interpretation function, F, would be a mapping from symbols in the logic to actual people in the world, e.g. F(SMITH)
= Smith
F(JONES)
= Jones
F(CLERK) = (Smith, Jones} and so on. Here the names used on the right are symbols in a mefalanguage, e.g. English. Ideally, we would point to the real objects themselves, but this is rather difficult in a written text.3 This leads to a recogni~on of the real purpose of an ontology. If two or more people are to make use of a symbol system, e.g. a database, as a medium of communication and reasoning, it is essential that they mutually understand the denotation of the primitive symbols. Typically, this is done by means of a natural language explication of these primitives. On the other hand, logic, especially in database applications, is often used to refine a certain area of natural language discourse. The logical vocabulary is used in a more disciplined, restricted way than its natural language counterpart. To be confident that the users of the logic mutually understand its elementary vocabulary, one would prefer to use nonlinguistic explanations, e.g. pointing to the basic objects being denoted.8
333
This would suggest an ontology consisting of discrete physical objects. Indeed, such an ontology is implicit in many database applications, and is probably among the “safest” to ensure mutual understa~dab~ity. On the other hand, any ontology is adequate provided the users of the logic mutually recognize the underlying domain of objects. The most outstanding examples are mathematical theories, for instance assuming a domain of real numbers. It is difficult to define such objects by pointing to them. Indeed, it might be argued that mathematical domains are defined proof theoretic~y-by their axiomatizations, That is, nobody cares what numbers “really are,” apart from the things that satisfy the axioms. Similar remarks might be said of games, e.g. board games, card games. Clearly the physical representations have no importance since these are now often played using computer screens. One might say that, like mathematics objects, they are defined by the rules controlling their symbolic manipulation. Nobody cares what a knight or queen of hearts “really is,” apart from its role in the rules of the game. It is noteworthy that much of the initial work in artificial intelligence research has focused on domains of this type (mathematics, games), where the ontology was implicit in the system of rules. Database applications, by contrast, are detinitely nor of this type. Indeed, this is perhaps their distinguishing characteristic: a database does not define its objects, implicitly, in its rule structure; rather, it only describes objects which have a separate, independent existence. The recognition of such objects is established by the people in the organization who make the appropriate divisions of the world into pieces that facilitate their business activitiesl21, 221. c. SOMEOhTOLoGICAL ISSUES
In the previous section, an ontology of discrete physical objects was suggested. Assuming this as a starting point, we proceed to investigate various issues that arise in actual database contexts, espe$ Tbc superficial similarity between object language tokens, e.g. SMITH. JONES and metalanguagetokens, cially in organizational settings. These are are.g. Smith, Jones, is often a point of confusion. The spell- ranged, roughly, in order of difficulty. ing in the object language is merely a convenient memory aid. Rather than SMITH we could have used X275I? for instance, with the same logical effect, as long as: F(X2751) = Smith.
The rne~~gu~e tokens are, by contrast, an accomodation to using a typographic medium to explain the logic. Here the spelling does matter since these are assumed to be direct labefs to the objects themselves, e.g. like the name tags used by conference attendees. I But pointing already suppose that the objects can be individuated from their spatiaVtemporal context. For example, imagine two people standing on a shore watching the waves. In trying to discuss individual waves they have a twofold problem: a) there is no clear-cut boundary of a
1. Multiple names An immediate problem is that in many administrative contexts, an individual person or object may have multiple names or identification codes. Further, these names may not necessarily uniquely
wave from the surrounding sea and b) the form, hence boundary of the wave, is constantly changing.This is one aspect of the so-called “individuation problem”; see e.g. Strawson[24]. Wittgenstein[28] and Quinel171 raise other epistemological issues relating to ostensive (pointing) definitions.
334
R. M. LEE and R. STAMPER
and metalanguage. Alternative identifiers identify the ~dividu~. For example, in the U.S., a l~age person typically has a frrst, middle and last name of this sort are clearly extensions to the object lanas well as a social security number, assigned by the guage. However, the sorts of things denoted remain federal Social Security Administfation, He/she may the same; we simply have a richer vocabulary for in addition have an employee number, assigned by referring to them. the employing comply, a health insane numOn the other hand, these ~temative identifiers ber, assigned by an insurance company, a driver’s arise mainly due to relationships with other parties, license number, assigned by the state granting the e.g. an SEC number is of interest because of translicense, various credit card numbers, assigned by actions involving the Securities Exchange Commission. Hence, alternative identifiers tend to the c@dit card companies, etc. Similar remarks can be made of organizations imply the involvement of other organizational entthemselves: they have a company name and per- ities, namely the institutions that assigned the idenhaps a corporate code assigned by the state of in- tifying codes. This suggests that the recognition of alternative corporation. If their stock is publicly traded, they will have an SEC code, assigned by the Securities identifiers affects the ontolo~ by forcing the inExchange Commission. In addition, companies .clusion of these other organizations in the domain, typically record their transactions with other com- D. Insofar as this only affects the cardinality of D, panies by ass~n~g codes for in-house use, e.g. a it is not an ontological problem. However, we have now a new ontological problem: organ~ation~ entsupplier number, customer number. Vehicles may also have multiple identifiers, e.g. ities such as (corporations, public institutions and a car may have one code on the engine block, an- agencies) do not fit neatly into the assumed ontolother on the frame, both assigned by the manufac- ogy of discrete physical objects. This point is returer. If used on public roads, it must also have a considered ia the last section. vehicie license number, assigned by the state of residence. If used in interstate or intematioual transAs remarked earlier, fo~~tions of matheport, other numbers may be assigned. (Kent{91 suggests numerous additions examples of the naming matical domains typically imply an ontology involving numeric entities, e.g. the integers, real problem in data processing.) Some of these identifiers, in particular names of numbers, etc. The issue considered in this section people. and companies, have no g-tee of is the role of quantitative data in a database ontoluniqueness. In the other cases, where an identifing ogy. Again, there is no absolute answer. 09 the code was assigned, the uniqueness claim is relative other hand, it is a reasonable claim that databases, to the scope of the institution assigning the code. as used in organizational applications, are seldom For instance, one person’s employee number could if ever about numbers per se. Rather, numbers appear in the context of ~~~~~~e~e~~, i.e. they are correspond to another’s credit card number. In first-order logic, the problem of multiple iden- used to ascribe properties to other (real world) tifiers is usually iguored. Each entity in the don&n, objects. D, is assumed to have just one logical namer which Some observations from measurement theory is unique within the scope of the logic, (see, for example&, 6, 10, 19,20 (chapter 3)1)have An approach might be to disting~s~ between in- bearing here. A meas~ement is a ~o~es~nden~e ternal YS external identifiers. An internal identifier between two relational systems: an empirical rewould be an arbitrarily assigned code, serving as a lational system (relationships among real world oblogical name. Other names and codes are consid- jects), and a numeric relational system (hong nuered external, for aiding various ext.emaI pafties to meric objects). For example, an empirical relational identify the entity. system might be: For instance, for an entity known internally as E = (D, RAFTER-THAN) X2751, we might use the notation: LAST-NAME (X2751) = “ SOCIAL-SECUR~Y-NUMBER
(X2751) = “512-27-6219”
EMPLOYEE”NUMBER (X But what are these expressions in double quotes? Obviously they are character strings, but do they represent a new kind of entity in the ontology? The answer here could go either way. One might admit them into the ontology, regarding them as a kind of linguistic entity, which denote themselves. This, however, tends to once again confuse object
where D, as before, is tbe set of discrete physical objects. These are ordered by the relationship, ROARER-THAN (x, y) ~dicating that x is redder than y. A corresponding numeric system might be: N = (I, >) where f is the set of integers, and > is greater than.
33s
Ontological aspects of logical databases
The correspondence between two such relational systems is a homomorphism mapping from the empirical to the numeric system. For example, REDNESS
(x) = n
might be such a homomorphism, mapping objects in D to objects in n such that the relationship REDDER-THAN is preserved in >. Thus, where such a correspondence exists, arithmetic deductions may be substituted for logical deductions. A meusurement scale consists of the triple,
lbs” (where “5 lbs” is a denominate number distinct from the “pure number,” 5), whereas the functional representation would paraphrase these reports into pure number form. e.g. as “lb (x) = 5,” where lb (x) is the weight-inpounds function whose values are pure real numbers. In banishing denominate number ontological categories, the functional representation accomplishes an ontological reduction which is conceptually important.
The ontological reduction in terms of the real world corresponds to the recognition of a specific object as the standard unit of measurement. Returning to database considerations, a typical relation containing numeric data might be:
M = (E, A’, H)
PERSON where E is an empirical relational system, N is a numeric relational system, and H is a homomorphism from the first to the second. Measurement scales are usually classified based on the nature of the numeric relational system employed. For example, scales involving N = (Re, Z-=)
are called ordinal Scales involving
scales
(Re = real numbers).
N = (Re, z=,
+)
are.called ratio scales. A central problem of measurement theory, the representation problem, is to axiomatize the conditions that must hold in the empirical system for the measurement scale to be valid. For example, in the case of ratio scales (also called extensive measurement), some correspondent to arithmetic addition must exist in the empirical system. For instance in weight measurement, one physically adds weight to a balance scale. In current measurement theory, differences in measurement units (e.g. pounds vs kilograms) amount to transformations on the homomorphism, H. For example, two homomorphisms might be: WEIGHT-IN-LBS
(x) = n
WEIGHT-IN-KGS
(x) = m
the transformation
between them being:
WEIGHT-IN-LBS
(x) = 2.2 * WEIGHT-IN-KGS
Adams[l,
(name, weight-in-kgs) SMITH
80
JONES
60
In a logical syntax this might be: WEIGHT-IN-KGS
(SMITH)
= 80.
WEIGHT-IN-KGS
(JONES)
= 60.
Note that the relational attribute containing quantitative data corresponds to a measurement scale homomorphism. This, we claim. is the typical role of quantitative data in databases!; Supposing this to be the case, what perspective does this offer concerning the ontology of databases? The arguments here are somewhat analogous to those for multiple identifiers. While numbers could be regarded as a separate sort of entityaxiomatized, perhaps, using a multisorted logicthis does not seem to be necessary for database formalizations. Quantitative data in databases has a role much like that of predicates, i.e one of predication. The point is that such measurements permit an alternative mode of inferencing-arithmetic rather than deductive-but the underlying entities, the subject matter of the inferences, is the same. Thus, from the database perspective, quantitative measurement amounts to an extension of the logical language, but does not affect the underlying ontology. 3. Collective objects The ontology assumed thus far consists of discrete physical objects. This implies that each such object might be assigned a distinct logical name.
(x).
p. 2101 comments:
It is important to stress to students unfamiliar with the modem fincrional representation of measurement that fundamental measurement theory adopts this representation, and in so doing “paraphrases” or translates more traditional metrical language (which is still widely prevalent in the sciences) into unfamiliar and initially unituitive forms. In particular, metrical data are traditionally reported in denominate number form, e.g. as “x weighs 5
I A research issue for logic programming applications to databases is to explore the interrelationship between logical and arithmetic inference from a measurement theory perspective. At present, the IS operator in PROLOG appears almost as an afterthought. The point is that certain types of inferences, e.g. on order relationships, extensive measurement, can be done using resolution, based on empirical relationships, or arithmetically, using numeric relationships. To the end user, the choice should be transparent, it being a question of inferential efficiency. Can this choice be automated?
336
R. M. LEE and R. STAMPER
However, in administrative applications there are many cases where this would be impractical. Inventories, for example, often contain objects that are too small or unimportant to name individually, e.g. bins of bolts, boxes of nails. In the extreme are granular objects like wheat or flour, and liquids or gaseous objects, e.g. oil, propane. The problem here is that these populations are effectively infinite, making them infeasible for resolution-based inferencing methods. On the other hand, organizations themselves seldom, if ever, regard these objects at such a detailed level of granularity. Rather, such objects are generally handled in some sort of c&&her, e.g. cartons, drums, boxcars. These are “middle-sized” objects which are, typically, assigned an identifying name. Entities having these characteristics might be called “collective objects” in that they represent granular populations collected into a larger, discretely identifiable aggregate. The cardinality of a collective object is simply another type of measurement (“absolute measurement”). Here the interaction of deductive vs arithmetic modes of inference is a requirement. Inferences concerning the collective objects, e.g. their location, transfer of ownership, take the form of deductions on any other type of discrete physical object. Inferences regarding their detailed composition, e.g. total stock on hand, inventory costing, are donearithmetically. 4. Time In contrast to the preceding cases, time (consequently, change) presents a fundamental and difficult issue in database ontology.” A basic difficulty is in distinguishing the identity of an entity at different points in time. To illustrate, consider the old philosophical chestnut, commonly called the Boat of Theseus. We have a wooden boat. We replace one of its planks and lay it aside. Is it the same boat? Replace another plank, and another, and another. Is it still the same boat? Continue until all the planks are replaced. Still the same boat? Now, with the old planks that have been removed, construct a different boat of the same design. Which, now, is the original boat? (Note that an entire navy of identical boats could be constructed by iterating this process.) Analogous problems of individuation arise, for instance, in the replacement of parts on vehicles and equipment. The problem, essentially, relates to the distinction alluded to earlier between observing vs deciding the denotations of terms. The temporal extent of an individual entity is often a matter of decision, relative to the purposes of the discourse or logic. ’ See Bolour et a1.[2] for a useful survey of temporal issues in information processing. Lee et a1.[14] presents a proposal specific to logical databases.
A related problem is that the entities typically described in databases have a life, i.e. they are born (created) and later die (are destroyed). The issue, then, is that the universal set, D, apparently has different membership at different points in time. The ontological issue here can be considered differently, depending on how the database is used. A common use of databases in organizations is rather like a movie film-the database is like an ongoing series of snapshots, portraying the state of the organization at the corresponding state in time. Under this view, the fact that the domain of individuals, D, changes from one time to the next is irrelevant, since inferences are made only within a given snapshot. However, if the use of the database is to include cross-temporal inferences, e.g. comparing the current state of affairs with the past, a different ontology is required. Initially, this would require extending the underlying model with a time dimension, i.e. M = (T, D, F)
where T is a time line, D is the set of’entities (discrete physical objects), and F is an interpretation function mapping terms and expressions in the logic into T x D. Assertions in the logic could then be regarded as temporally dependent. For example, a proposal of Rescher and Urquhartll81 uses the notation: R(r): p. to
express that the assertion p “is realized” at time,
t.
In business, the usual conception of time is as a continuous dimension. For purposes of mechanical inferencing, however, it is preferable to assume a discrete time. This apparent conflict can be resolved by recognizing only certain time intervals, e.g. the dates of the calendar, explicitly in the logic. These time intervals function conceptually rather like the containers discussed for collective objects. However, quite independent from whether time is considered to be continuous or discrete, is the issue about the composition of the domain of individuals, D. If we consider appplications involving historical comparisons, D must clearly contain more than those entities existing at present. For example, past production may have created objects that have since been consumed. This suggests an ontology consisting of objects both present and past. In such a view, a temporal predicate of existence is required, e.g. R(r):
EXISTS (x).
indicates that x exists at time, t.
Ontological aspects of logical databases
The idea of a domain, D, consisting of objects both past and present may seem a bit awkward. For instance, one might ask what these entities are doing when they are not existing. Likewise, recognizing existence as a predicate gives rise to such constructions as:
337
options, etc., all of which represent contractual relationships. The most universal type of contractual object is money. Originally, printed currency represented a promise for the delivery of gold (or silver). Now that the gold standard is no longer used, one wonders what gives money its value. (The same ques3x R(r): -EXISTS (x). , tion might, indeed, be asked about gold itself.) The answer typically given is couched in terms of read that there exists an x such that at time t, it “faith” or “confidence.” Domestically, this indidoes not exist. This, however, is an artifact of the cates a sort of universal promise of the people usual reading of the existential quantifier. The se- among themselves. On international markets, a mantic interpretation of the existential quantifier is country’s currency has an even stronger resemblance to contracts: it is a general purpose I.O.U. to refer to some element in D. The existence predfor goods and services of that country. icate, as suggested here, is a temporal predicate, Earlier, it was remarked that the concept of an relating elements in D to elements in T. Given the revised ontology suggested here, this interpretation organization (e.g. corporation or government is perfectly legitimate[l8, Chap. 201. agency) did not tit neatly into an ontology of disNote that here we have proposed that D contain crete physical objects. An organization is more than only present and past entities. Considering that the its assets; also it is more than a loose affiliation of time dimension normally is conceived as extending people. In the U.S. at least, the principal type of into the future, one is led to consider a correspondprivate organization is the corporation. A corpoing extension to D, i.e. to include future entities as ration might also be regarded as a type of contracwell. The problem with this is that future entitie tual object. Specifically, it is a permission, granted are likewise hypothetical entities. We do not know by the state of residence, to engage in commerce, about future entities (or events) in any factual sense negotiate financing, with limited liability to the as we know the past. The past is history, the future shareholders. Importantly, the corporation itself is is speculation. Our concern here, being limited to the agent of its contracts, not its shareholders or its databases, is with the factual: Predicting and hymanagers. A corporation might thus be viewed as pothesizing about the future, while outside the presa sort of “second-order” contractual entity: a conent scope, are nonetheless vital concerns to mantract permitting contracting. agement. Ontologically speaking, this enters into By the same token, governmental agencies-at the area of so-called “possible worlds semantics,” least in democratic societies-have a contractual a subject of considerable current interest and character. The agency is formed by a charter. specifying the services it is to fulfill. It is this charter, debate.+ rather than any particular assets or personnel, that 5. Contractual objects is the essence of the agency. While we might exclude future or hypothetical But what is a contractual object? In each of the entities as not belonging to an ontology for factual abovementioned cases, its essence seems to be databases, there is an area that seems factual, yet something nonphysical, namely an obligation (or which has hypothetical aspects about it as well, perhaps permission) between certain individuals or namely, contracts. organizations to others. On the other hand, obligation and permission are both future-oriented conConsider, for example, that much of the contents cepts-they apply to future behavior rather than to of administrative databases deals with accounting data. Among this data are references to such things actual circumstances. One may of course simply speczfi contractual a receivables, payables, leases, notes, bonds, inobjects as part of the ontology. This, however, surance, stock, options, etc. AlI of these are exseems to negate the basic purpose of ontology, amples of what might be called a “contractual obnamely to identify sets of referrents that are unject.” A contract is a kind of relationship between ambiguously recognized and identifiable by all two parties obliging them to perform certain actions perhaps only under certain conditions, before a cerusers of the logical language. Various forrnalizations of obligation and pertain time, etc. mission have been proposed as so-called “deontic Yet at the same time, these “relationships” are logics” (see, e.g. 17, 81). Contracts may be viewed often, in business, regarded as objects. Major exas an application of deontic logic[l2, 13, 33. On the amples are the securities exchanges which deal in other hand, attempts to provide a formal semantics the buying and selling of stocks, bonds, futures, for deontic logic are typically based on an ontology involving possible worlds. Von Wright[27], for int Useful discussions of possible world semantics are stance, draws a parallel between deontic logic and found in Rescher and Urquhart[H], Dowty et al.[S], Cresmodal logic, and uses the concept of a “deontically swell[4], Kripkef 1I]. McDermott[ IS] proposes a futureoriented temporal logic involving hypothetical reasoning. possible world” in his semantics. The difficulty, we
338
R. M. LEE and R. STAMPER
find, is that possible worlds are too esoteric a concept to be used as the basis for consensual understanding by a community of database users. An alternative course, perhaps more t?uitfLl, is analogous to that described for mathematics and various sorts of games: the ontology is described implicitly by the rules controlling it. By this viewpoint, a contractual object is that thing that satisfies the rules of contract law.
D.
CONCLUJIING REMARKS
The purpose, here, has been to sketch the outlines of a theory of what databases are about; the elementary sorts of entities that databases, in particular databases in organizations, refer to. The concept of a logical database provides a useful context to study these issues, by linking to the rich literature on the semantics of first-order languages. Tar&an model theory provides a mathematically crisp conception of semantics in the notion of a model or interpretation assigned to the language. This raises the issue of ontology-choosing an underlying domain of entities that is consensually recognized by all users of the language. The view presented here is most appropriate for those aspects of business relating to physical systems, e.g. manufacturing, distribution systems. It is less satisfactory, however, for business Cmctions dealing with phenomena of a more social nature, e.g. marketing, management of personnel, contract negotiations. It was suggested that such things as contractural objects might be defined proof theoretically, i.e. that which satisfies the axioms of contract law, rather than hypothesizing the existence of abstract, “contractual objects.” Further consideration along these lines might lead to a completely different conception of semantics. As opposed to the view presented here, where symbols denote real world objects having an independent existence, an alternative view might define objects relative to their social function[23]. For example, a knife and fork are better understood relative to the social norms of eating, rather than by their physical characteristics alone. This view will be developed more fully in a subsequent paper.
REFERENCES
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