Ontology Theory Christopher Menzel1 Abstract. a mathematical framework, akin to number theory or modern anal- Ontology today is in many ways in a state similar to that of anal- ysis, that enables us to characterize the notion of an ontology for- ysis in the late 18th century prior to arithmetization: it lacks the sort mally and develop accounts of their properties and the various ways rigorous theoretical foundations needed to elevate ontology to the in which one ontology can be related to another. Note also that the level of a genuine scientific discipline. This paper attempts to make framework itself might not be used as it stands for any actual ontol- some first steps toward te development of such foundations. Specif- ogy integration work. It is in the respect analogous to computabil- ically, starting with some basic intuitions about ontologies and their ity theory. No one actually programs turing machines (except as a content, I develop an expressively rich framework capable of treating heuristic exercise). Rather, the notion provides a model of computa- ontologies as theoretical objects whose properties and logical inter- tion that serves as a foundation for both theoretical and, therefore, connections — notably, potential for integration — we can clearly indirectly, applied computer science. define and study. In this brief paper we can only make some first halting steps to- ward a general ontology theory. The bulk of this paper will be to ar- gue for, and lay out in varying degrees of detail, a formal framework 1 Introduction with the representational horsepower adequate for a robust ontology Ontology today is in a state similar to that of analysis in the late theory. 18th century. The practical power of the calculus had been convinc- ingly demonstrated in the work or Newton and his great successors. 2 Intuitions Moreover, the field of real analysis itself had seen an explosion of creativity, exemplified most notably in the work of Euler. However, I begin with some intuitions to motivate the design of a framework Euler’s own work also revealed worrisome foundational problems. for ontology theory. For techniques used with great success in one instance to prove deep 1. Ontologies consist of propositions. and dramatic theorems in another instance could lead to absurdities, 2. Propositions are not sentences, they are what sentences express; e.g., that the value of certain monotonically increasing infinite series different sentences in different languages (or possibly the same was −1. Such results led to a conceptual crisis — how can any results language) can express the same proposition. be trusted when the methods that generate them can lead to error? 3. Propositions can be equivalent without being identical. This crisis was addressed, and successfully eliminated, by the de- 4. Propositions and ontologies are objects, things we can talk about. velopment of rigorous foundation for analysis — widely known as 5. The content of an ontology consists of the propositions involv- the arithmetization of analysis — by Cauchy, Weierstrass, Bolzano, ing concepts in the ontology that are entailed by the constituent and others in the early 19th century. Building on the sound foun- propositions of the ontology. dation of number theory, mathematicians replaced the intuitive but 6. Ontologies are comparable in terms of their content. In particular, undefined notions of analysis — limit, continuity, series, integra- two ontologies are equivalent if they have the same content. tion, real number etc. — with clearly defined counterparts (e.g., the now-familiar , δ definition of limit) and banished unruly notions like that of an infinitesimal altogether.2 With these solid underpinnings in 3 Desiderata place, mathematicians were able to identify clear conditions of ap- In developing a general ontology theory our concern is describe the plicability for their analytic methods that prevented the derivation of phenomenon, just as in the development of number theory or real absurdities without limiting their ability to prove desirable results. analysis or, for that matter, computability theory. We therefore place A similar foundation is needed in the study of ontologies. As with no computational restrictions on expressiveness, and hence will avail analysis prior to arithmetization, the potential of ontologies is evi- ourselves of at least full first-order logic.3 dent, but the fundamental notions remain largely intuitive; notably, However, we will need quite a lot more than that to satisfy the there is no precise characterization of the notion of an ontology, nor intuitions in the preceding section. Notably: what it is for two ontologies to be intergrated. What we need, then, is our own “arithmetization” — in a nutshell, we need ontology theory: • Re (1) above, we need formal notions of ontology and proposition, 1 Department of Philosophy, Texas A&M University, College Station, TX and a notion of the relation between ontologies and the proposi- 77840, email: cmenzel@tamu.edu tions they consist of. 2 Ironically, the very foundational work that began with the arithmetization • Re (2), we need a notion of proposition that is independent of any of analysis and led to the development of mathematical logic ultimately particular language. resurrected the notion of an infinitesimal and an alternative foundation for analysis built thereon — so-called “nonstandard” analysis. See, e.g., [4] 3 As with both number theory and analysis, of course, we may want to explore Chapter 3. computationally more tractable subtheories of our theory. • Re (3), we need a notion of proposition robust enough to allow for 4.1.2 Grammar distinct logically equivalent propositions. We define formulas and terms by a simultaneous recursion: • Re (4), we need to be able to name and quantify over propositions and ontologies; i.e., ontologies and propositions must be “first- 1. Any constant or variable (individual or predicate) is a term. class citizens” in ontology theory. 2. If π is an n-place predicate and τ1 , . . . , τn are any terms, n ≥ 0, • Re (5), we need to be able to represent the notion of content, and then π(τ1 , . . . , τn ) is an (atomic) formula of L. π is said to oc- hence (i) a notion of entailment that can hold between ontologies cur in predicate position, and each τi in argument position, in and propositions and (ii) a notion of the concepts within an ontol- π(τ1 , . . . , τn ). In the case where n = 0, we omit the empty paren- ogy. theses and say that π standing alone is an atomic formula. • Re (6), we need to be able to define notions of comparability in 3. If ϕ, ψ are formulas, so are ¬ϕ, 2ϕ, and (ϕ → ψ). terms of ontological content. 4. If ϕ is any formula and ν1 , . . . , νn any variables, then (∀ν1 . . . νn )ϕ is a formula. I will satisfy these desiderata by developing a first-order theory of 5. If ϕ is a formula containing no occurrences of 2, no bound vari- structured relations, of which propositions will be one species. On- ables occurring in predicate position, and no bound predicate vari- tologies will be identified with 1-place relations, which for most pur- ables, and ν1 , . . . , νn are any individual variables that do not occur poses can play the role of classes. This theory will satisfy desiderata free in any term occurring in ϕ, then [λν1 . . . νn ϕ] is an n-place (1), (2), (3), and (4). By “structured” I mean that, although they will predicate. not be identified with formulas, relations will have a decomposable 6. Nothing else is a term or formula of L logical form similar to formulas. Together with a primitive modality, the structured nature of relations in turn will enable us to define a The usual definitions of ∧, ∨, ↔, and ∃ will be assumed. notion of entailment for propositions that will enable us to define a There are two particularly distinctive features of L. First, although notion of content for ontologies, and hence to satisfy desiderata (5) the language of L contains so-called “higher-order” variables, unlike and (6). standard higher-order languages, these variables, and n-place pred- icates generally, are considered terms; they can occur as arguments 4 A Formal Framework for Ontology Theory to other predicates. Semantically speaking, as we will see explic- itly below, this means that our universe is type-free — everything In this section I will define a language with appropriate expressive is an object; the quantifiers of the language will range over every- power for ontology theory and a corresponding semantics. thing alike. Note this does not mean that there is no distinction be- tween kinds of things. Notably, as noted already, our basic ontology 4.1 Syntax includes relations as well as ordinary individuals. Rather, in accor- dance with intuition (4), it simply means that all of these things are To accomodate the narrow columns of the ECAI 2-column format, as in the universe of discourse, i.e., the range of the quantifiers. All en- much as anything, I will simply use the basic apparatus of standard tities — individuals, propositions, properties, and relations alike — Principia Mathematica-style first-order language, augmented with a are first-class logical citizens that jointly constitute a single domain number of useful constructs. I will call the language “L”. of quantification. As such, properties and relations can themselves Note that the unfriendliness of such languages in regard to com- have properties, stand in relations, and serve as potential objects of puter processability is no more to the point here than it is with re- reference. spect to group theory or computability theory. Our goal is theoreti- Perhaps the strongest linguistic evidence for type freedom is the cal — a mathematical theory of ontologies. Such work, of course, if phenomenon of nominalization, whereby any verb phrase can be sound, should lead to developments wherein computer processable transformed into a noun phrase of one sort or another, most com- languages are critical, but at this point processability is not an issue. monly, a gerund. So, for example, the verb phrase ‘is famous’ indi- cates a property that can be predicated of individuals, as in ‘Quentin is famous’. Its gerundive counterpart, however, ‘being famous’, 4.1.1 Lexicon serves to denote a subject of further predication, as in, e.g., ‘Being The lexicon consists of a countable set of individual constants, a de- famous is all Quentin thinks about’. Intuitively, the verb phrase indi- numerable set of individual variables, for each n ≥ 0, a countable cating the property predicated and the gerund indicating the object of set of n-place predicate constants and a denumerable set of n-place Quentin’s thoughts (i.e., the object possessing the property of being predicate variables (jointly called n-place predicates), the reserved thought by Quentin) are the very same thing, the property of being logical symbols ¬, ∧, ∨, →, ↔, ∀, ∃, λ, and 2, and parentheses famous. and brackets. Individual variables will consist of lower case letters, In L, this “dual role” of properties and relations — thing predi- typically x, y, z, possibly with numerical subscripts. n-place pred- cated vs. object of predication — is reflected in the fact that the same icate variables will consist of upper case letters with numerical su- constant can play both traditional syntactic roles of predicate symbol perscripts (suppressed where context serves to indicate the arity of and individual constant. Thus, in L, we can write both an n-place predicate), typically F n , Gn , and H n , possibly also with numerical subscripts. For purposes here, constants will consist of al- (1) Famous(quentin) phanumeric strings — other than the single-character strings already and in use for the variables — beginning with an upper or lower case let- ter; dashes are also permitted to join alphanumeric strings. Typically, (2) (∀F )(ThinksAbout(quentin, F ) ↔ (F = Famous)) I will use a strings beginning with a lower case letter for constants that are intended to denote individual objects and strings beginning (L retains no representation of the grammatical distinction between with an upper case letter for constants intended to denote relations. verb phrases — e.g., ‘is famous’ — and their gerundive counterparts 2 — e.g., ‘being famous’. One could be added easily enough, of course, Clause (5) imposes a number of other restrictions on the formation but as there is no semantic difference between verb phrases and their of terms that are, in fact, dispensable in the sense that we could in fact gerunds on a type free conception, any such representation would be provide a reasonable semantics for them. Specifically: semantically otiose.) Because all objects are of the same logical type, it follows that • The requirement that λ-bound variables all occur free in ϕ rules any property can be predicated of any property and, in particular, out such terms as [λxy P x] that contain vacuous λ-bound vari- a property can be predicated of (and, indeed, can exemplify) itself. ables;4 Again, this comports with natural language; the property of being a • The restriction on free occurrences of λ-bound variables property, for instance, is a property, and hence exemplifies itself. This within complex terms occurring in ϕ rules out such terms as is naturally represented in L in the obvious way: [λxy P [λzQxz]y];5 • The restriction on bound occurrences of predicate variables (3) Property(P roperty) within complex terms occurring in ϕ rules out such terms as [λy (∃F 1 )y = F 1 ];6 It must be emphasized that the fact that we will be quantifying over properties, propositions, and relations generally does not in and However, the terms that would be permitted without these restric- of itself mean that L is higher-order. For that, one’s semantics must tions are inessential to our purposes here and hence allowing them involve higher-order quantifiers whose range includes a power set would introduce unnecessary technical complexity. construction of some ilk over a domain of logical individuals. In our While the restrictions to non-modal formulas in the formation of semantics, there is no such construction; there is but a single domain terms is, like the two above, also inessential, it has a certain intuitive over which a single type of quantifier ranges. warrant. For, unlike the three restrictions above, this restriction re- The second distinctive feature of L, and arguably the most promi- flects an important feature of the intended domain guiding the devel- nent, is the presence of complex terms [λν1 . . . νn ϕ]. Intuitively, opment of the current framework. Specifically, we are formulating a these terms denote complex relations. For instance, the term theory of first-order ontologies, that is, ontologies whose constituent propositions are expressible by sentences in a non-modal first-order (4) [λx Enjoys(x, salmon)∧Prefers(x, red _wine, white_wine)] language (hence in any weaker sublanguage thereof). This is, of indicates the property of enjoying salmon and preferring red wine to course, not to say that there are no modal (or higher-order) ontolo- white. Terms with no bound λ-variables indicate 0-place relations, gies. However, the vast majority of existing ontologies are first-order, i.e., propositions. In this case the λ can be dropped. Thus, and it seems quite unlikely that this will change with the development of the Semantic Web if the expressiveness of its basic language is to (5) [∀x(Planet(x) → Larger (sun, x))] be on the order of DAML+OIL. Therefore, to provide the capacity to express modal propositions, at this point, seems unwarranted. indicates the proposition that the sun is larger than all of the planets. A theory of ontologies, however, does need this expressive power. This feature of L is particularly important, as ontologies in the pro- Specifically, modality is useful for characterizing the nature of on- posed theory will be characterized roughly as classes of propositions, tologies and their logical connections. Most notably, perhaps, as will and the logical connections between ontologies will be expressed in be seen explicitly in Section 7 below, the modal component of the terms of logical relations between propositions. λ-terms enable us language of our theory enables us to define a robust notion of en- to talk about the propositions in a given ontology explicitly. And as tailment which, in turn, can be used to formulate a correspondingly we will see, they are also extremely useful for defining a variety of robust notion of ontological content. important auxiliary notions. 4.2 Semantics 4.1.3 On Syntactic Restrictions on Term Formation In this section I will build upon work by Bealer [1], Zalta [7], and Clause (5) in the grammar for L imposes a number of restrictions Menzel [5] to develop a rich “meta-ontology” of structured relations. on the formation of complex terms. The most noteworthy of these is the restriction permitting only individual variables to be bound by the λ operator in complex terms. This restriction avoids the Russell 4.2.1 Model Structures paradox, as without that restriction the term [λF ¬F (F )] — indicat- ing, intuitively, the property of non-self-exemplification — would be A modelS structure M for L is a 5-tuple hD, W, dom, Op, exti. Here D = {A, R} is the domain of M, and consists of the union of two legitimate. The grammar would then permit the construction of the mutually disjoint sets A and R. A is the set of individuals of D and R atomic formula [λF ¬F (F )]([λF ¬F (F )]), which, by the logical is the set of relations, of which we consider propositions a species. R principle of λ-conversion ((10) below), could be proven equivalent itself can be partitioned in two significant ways. First, R is the union to its negation. However, the restriction that prevents the paradox is of two mutually disjoint nonempty sets Rp and Rc , intuitively, the not ad hoc. Its justification — which will become clear in Section 4.2 sets of logically primitive and logically complex relations, respec- — is that there is simply no intuitive logical operation that yields re- tively. Additionally, R is the union of denumerably many nonempty lations whose logical form corresponds to such terms, and hence no sets R0 , R1 , . . . , each Rn being, intuitively, the class of n-place rela- warrant for permitting them. The avoidance of Russell’s paradox falls tions. We let Rn p and Rnc be Rp ∩ Rn and Rc ∩ Rn , respectively. W out as a consequence of this restriction, and hence is explained rather 4 Such terms are easily accommodated by means of a further set of logical than merely avoided: the paradox arises from a theoretically unwar- ranted assumption about the structure of complex relations, much as operations Vac i that insert vacuous argument places into the ith “slot” in the argument structure of a relation. the corresponding paradox of self-membership arises from a theoret- 5 See Menzel [5] for an account of the semantics of such terms and surround- ically unwarranted assumption about the nature and structure of sets ing philosophical issues. 6 Again, see [5] for the semantics of such terms. (see, e.g., [2]). 3 is a nonempty set, intuitively, a set of “possible worlds” or “possible • ext(Conv ij (r), w) = situations.” More formally, W provides us with a model of modal- {ha1 , . . . , ai−1 , aj , . . . , aj−1 , ai , . . . , an i : ha1 , . . . , an i ∈ ity that enables us to represent entailment and other logical relations ext(r, w)} between ontologies. Accordingly, dom is a function that maps every • ext(Refl ij (r), w) = element w of W to a subset dom(w) of D representing, intuitively, {ha1 , . . . , aj−1 , aj+1 , . . . , an i : ha1 , . . . , an i ∈ the set of things that “exist” in the possible world w. ext(r, w) and ai = aj } The next element of a model structure, Op is a set of five sets of logical operations: a set of predication operations, {Pred n i1 ...ik : Constituency and Logical Form The intuitive picture here is a 0 < i1 < . . . < ik ≤ n; k ≥ 07 }; a set of two boolean operations, “quasi-constructive” one similar to the intuitive picture that under- {Neg, Impl }; a set of universalization operations, {Univ i1 ...ik : lies the iterative conception of sets. We begin with a set A of in- 1 ≤ i1 < . . . < ik ≤ n; }; a set of conversion operations, dividuals and a set Rp of logically simple relations. The logically {Conv ij : 1 ≤ i < j < ω}; and a set of reflection operations, simple relations are thought of as the meanings of the primitive pred- {Refl ij : 1 ≤ i < j < ω}. These operations “construct” logically icates in an ontology. The predication functions applied to primitive complex relations from individuals and less complex relations in D. relations and individuals yield basic atomic relations — notably, ba- Specifically, for all n: sic atomic propositions — and the remaining logical operations ap- plied to these yield logically complex relations. These in turn, can • Pred ni1 ...ik : Rn × D k −→ Rn−kc (1 ≤ i1 < . . . < ik ≤ be arguments to further applications of the logical operations, yield- n; k ≥ 1); c ing an “iterative hierarchy” of relations of increasing complexity. In- • Neg : Rn −→ Rn ; c tuitively, then, relations in R are either primitive or are “built up” • Impl : Rn × Rm −→ Rn+m ; c from individuals and other relations via the logical operations, and • Univ i1 ...ik : Rn −→ Rn−k , for k ≤ n. the manner in which a relation is so built up can be thought of as • Conv ij : Rn −→ Rn c (1 ≤ i < j ≤ n); i c its logical form. So, for example, our example proposition (5) — • Refl j : Rn −→ Rn−1 (1 ≤ i < j ≤ n); [∀x(Planet(x) → Larger (sun, x))] — that the sun is larger than every planet would be built up from the property of being a planet, the We stipulate that Pred n (i.e., Pred n i1 ...ik for k = 0) is just the 2-place relation of being larger than, and the sun as follows. Pred 21 identity relation on Rn ,8 and that Rc is Sjust the union of theSranges applied to the larger-than relation and the sun yields the property of the logical operations, i.e., Rc = { Range(f ) : f ∈ Op}. To capture fine-grainedness, it is assumed that all of these operations (6) [λy Larger (sun, y)] are one-to-one and that the ranges of all of the operations are pairwise disjoint — similar to their syntactic counterparts, the “logical forms” of being something that the sun is larger than.10 The boolean “ma- of relations formed from these operations are all distinct from one terial implication” operator Impl applied to the property of being a another. planet and (6) yields the relation S Finally, let Dn be the set of all n-tuples over D and let D∗ = n 0 (truth) b. The reflection operation Refl 12 applied to (7) “collapses” its two and the latter as the truth value ⊥ (falsity). argument places into one to yield the property The behavior of ext is constrained further by the logical opera- S tions in Op. Some notational conventions will be helpful for stat- (8) [λx Planet(x) → Larger (sun, x)] ing these constraints. For A ⊆ Dn , let A be Dn −A. Where s _ s0 is of being something such that if it is a planet, then the sun is larger the concatenation of two sequences (tuples) s, s0 , for subsets A, B of than it. Finally, application of the “quantification” operator Univ 1 Dn and Dm , respectively, let A _ B = {a _ b : a ∈ A, b ∈ B}. i ...i yields our desired proposition (5). In a single equation, then, we have Where 1 ≤ i1 < . . . < ij ≤ n, we let hb1 , . . . , bn ia11 ...ajj be result of replacing each bik with ak , and we let hb1 , . . . , bn ii1 ...ij (9) (5) = Univ 1 (Refl 12 (Impl (Planet, Pred 21 (Larger , sun)))). be the result of deleting each bik from hb1 , . . . , bn i. Given this, let r ∈ Rn , q ∈ Rm ; then: The manner in which a relation is built up from individuals and other relations can be thought of as its logical form. We can make this • ext(Pred n i1 ...ik (r, a1 , . . . , ak ), w) = idea rigorous as follows. Say that a constituency tree for an element i ...i {hb1 , . . . , bn ii1 ...ik : hb1 , . . . , bn ia11 ...akk ∈ ext(r, w)};9 r ∈ R is any labeled ordered tree T whose nodes are in D and • ext(Neg(r), w) = ext(r, w); whose root node is r, such that, for every node e of T , the daughter S • ext(Impl (q, r), w) = ext(q, w) _ Dn ∪ Dm _ ext(r, w) nodes e1 , . . . , ej of e are such that, for some operation F ∈ Op, • ext(Univ i1 ,...,ij (r), w) = F (e1 , . . . , ej ) = e. A constituency tree T for r is complete iff every {ha1 , . . . , an ii1 ...ij : ∀b1 , . . . , bj ∈ dom(w), leaf node o of T is an individual or a primitive relation, i.e., iff o ∈ i ...i A ∪ Rp . Given the constraints on our logical operations it is easy to ha1 , . . . , an ib11 ...bjj ∈ ext(r, w)} show that every r ∈ R has exactly one complete constituency tree, which we can therefore identify with the logical form of r. We define 7 If k = 0, then i . . . i is the null sequence, which we want to allow here. 1 k 8 This stipulation will yield as logical truths all instances of π = an object o ∈ D to be a constituent of a relation r ∈ R just in case [λν1 . . . νn π(ν1 , . . . , νn )], for all n-place predicates π. 10 I am of course using the term ‘[λy Larger (sun, y)]’ here, not mentioning 9 Note that by the definition of the Pred functions, we always have k ≤ n in it; I am not talking about the term itself, but rather the property it intuitively Pred n i1 ...ik . denotes under the standard English meanings of the constituent constants. 4 e is a node in the complete constituency tree for r. o is a primitive complex terms that ensure fine-grainedness, e.g., that no universal- constituent of r iff o is a constituent of r and o ∈ A ∪ Rp . The ization is an implication or a negation, that predications are identical notion of constituency will be important for defining the concept of iff they are predications of the same relation of exactly the same ob- ontological content in Section 7 below. jects, and so on. More relevant for ontology theory, however, is a principle of λ- 4.2.2 Denotations, Interpretations, and Truth conversion that takes as axioms all instances of: Denotations for the terms of L relative to a model structure M are (10) [λν1 . . . νn ϕ](τ1 , . . . , τn ) ↔ ϕτν11,...,τ ,...,νn n , determined by partitioning the class of complex terms according to their syntactic form. In brief, where τ is [λν1 . . . νn ϕ], if the order where ϕτν11,...,τ ,...,νn n is the result of replacing every free occurrence of νi of the λ-bound variables in ϕ does not correspond to ν1 , . . . , νn , in ϕ with τi . This principle lets us move freely between statements then τ is the conversionij of an appropriate term τ 0 . Otherwise, if one about individuals and the attribution of complex properties and rela- of the λ-bound variables occurs free more than once in ϕ, then τ is tions to those individuals, e.g., a reflectionij of an appropriate τ 0 . Otherwise, τ is classified as the [λx Enjoys(x, salmon) ∧ Prefers(x, wine, beer )](jo) universalizationi1 ,...,ij , implication, negation, or predicationn i1 ,...,ik (11) ↔ Enjoys(jo, salmon) ∧ Prefers(jo, wine, beer ) of the appropriate sort depending on the logical form of ϕ. Complex terms of the form [λν1 . . . νn π(ν1 , . . . , νn )], for any predicate π — Notably, as we will see below, this principle will give us the ability i.e., those of the form predicationn — are said to be trivial, as they to move from talking about the propositions in an ontology to using indicate no more logical complexity than the constitutive predicate them in logical inferences. Like the other axioms of our theory, (10) π. is easily shown to be valid relative to the semantics above. Given a model structure M, let d be a function assigning elements of the domain D of M to the individual constants and variables of L and elements of Rn to the n-place predicates of L. Such a d is 6 The Logic of Constituency known as a denotation function for L relative to M. Denotations for The notion of constituency enables us to capture the intuitive fact that complex terms are then assigned by extending d in an obvious way different ontologies contain different concepts: the concepts in an on- that exploits the close parallel between the syntactic form of complex tology are simply the properties and relations that are constituents of terms and the logical forms of complex relations: the propositions of that ontology. Our fine-grained, structured notion of properties, relations, and propositions gives us a rigorous foun- • If τ is the conversionij of τ 0 , then d(τ ) = Conv ij (d(τ 0 )). dation for analyzing and exploiting the notion of constituency. We • If τ is the reflectionij of τ 0 , then d(τ ) = Refl ij (d(τ 0 )). have characterized constituency model theoretically above in Section • If τ is the universalizationi1 ,...,ij of τ , then d(τ ) = 4.2.1. In this section we capture the notion axiomatically. We begin Univ i1 ,...,ij (d(τ 0 )). with a schema:12 • If τ is the implication of τ 0 and τ 00 , then d(τ ) = Impl (d(τ 0 ), d(τ 00 )). Const(τ, τ 0 ), where τ 0 is a nontrivial complex term and (12) • If τ is the negation of τ 0 , then d(τ ) = Neg(d(τ 0 )). τ occurs free in τ 0 0 • If τ is the predicationn i1 ,...,ij of τ of τ1 , . . . , τj , then d(τ ) = Pred n (d(τ 0 ), d(τ 1 ), . . . , d(τ j )). That is, any term occurring free within a complex term indicates a i1 ,...,ij constituent of the relation denoted by the complex term. We say that a denotation function d0 for L relative to M is a ν- Next, we note that the constituency relation is a strict partial or- variant of d, for any variable ν, just in case, for all variables µ 6= ν, dering, i.e., it is transitive and asymmetric (hence also irreflexive): d0 (µ) = d(µ).11 An interpretation A of L is a pair hM, di consisting of a model (13) (Const(p, q) ∧ Const(q, r)) → Const(p, r) structure M = hD, W, dom, Op, exti and a denotation function d (14) Const(q, r) → ¬Const(r, q) for L relative to M. For any variable ν, a ν-variant of A = hM, di is any interpretation A0 = hM, d0 i such that d0 is a ν-variant of d. Finally, we can define an object to be primitive just in case it has Let A = hM, di be an interpretation, where M = no constituents: hD, W, dom, Op, exti. Truth at a world w ∈ W in A for the for- mulas of L is defined in the standard sort of way: (15) Prim(x) =df ¬(∃q)Const(q, x) • π(τ1 , . . . , τn ) is true at w in A iff hd(τ1 ), . . . , d(τn )i ∈ This reflects the model theoretic fact that the ranges of the logical ext(d(π), w). operations (other than the “trivial” Pred n operations) are all subsets • ¬ϕ is true at w in A iff ϕ isn’t. of Rc . • (ϕ → ψ) is true at w in A iff either ϕ isn’t or ψ is. • ∀νϕ is true at w in A iff ϕ is true at w in all ν-variants of A. 7 Content • 2ϕ is true at w in A iff ϕ is true at w0 in A, for all w0 ∈ W . As indicated, content is best cashed out in terms of some notion of entailment. In classical first-order logic, entailment is usually under- 5 Proof Theory stood model theoretically. Ciociou and Nau [3] have taken some steps The proof theory for this semantics is an extension of classical first- in this direction in developing a formal notion of intertranslatability order logic with identity. Notably, there are principles of identify for between ontologies. For them, ontologies are understood as sets of 11 Thus, as it is often informally put, d0 differs from d at most in what it 12 Recall that a trivial complex term is of the form assigns to µ. [λν1 . . . νn π(ν1 , . . . , νn )]. 5 sentences, and the content of an ontology is understood in terms of Thus, combining (19) and (20), we have the notion of entailment we its formal models: the content of an ontology O consists in the set of are after: its semantic consequences, i.e., the set of sentences that are true in all the models of O. This approach thus can yield a robust notion of (21) StrEntails(O, F 0 ) =df Entails(O, F 0 ) ∧ ShPrim(O, F 0 ). common ontological content across different languages in terms of shared models. That is, an ontology O strongly entails a proposition F 0 just in case This approach is clear and insightful, but suffers from two short- O entails F 0 and F 0 and O share primitives; that is, intuitively, if O comings. First, though a notion of common content is possible on entails F 0 and F 0 is “built up” from the same pool of concepts and this approach, the notion of ontology is still language-dependent; an objects — the same “conceptual vocabulary” — as the propositions ontology is a set of sentences in some language. This violates intu- in O. We will sometimes write “O ⇒ F 0 ” for “StrEntails(O, F 0 )”. itions 1 and 2 above, which jointly imply that ontologies are classes The content of an ontology, then, can be thought of as all of the of language-independent propositions. More seriously, however, the propositions that it strongly entails. As it happens, we cannot strictly approach — as a basis for a general theory of ontologies — is un- define the content of an ontology as an object. However, for theo- wieldy. Content is understood in terms of the models of a theory. retical purposes, strong entailment appears to be all we need. For Hence, on this approach, one has to import the full apparatus of first- example, we can say that one ontology O subsumes another O0 just order model theory — basic set theory, formal languages, interpre- in case the content of O0 is included in that of O, i.e., just in case O tations, model theoretic truth and entailment, etc — just to define a strongly entails every proposition that O0 does: reasonable notion of ontological content. Moreover, the model theo- Subsumes(O, O0 ) =df Ont(O) ∧ Ont(O0 ) ∧ retic approach makes for a rather austere and formal notion of content (22) (∀p)(O0 ⇒ p → O ⇒ p) — to identify the meaning of a sentence with a set of models is rather far removed from ordinary semantic notions. Ontologies can then be said to be equivalent just in case they sub- Though the present approach has a strong model theoretic com- sume each other: ponent, that component serves only to ground a first-order theory of ontologies and their content; no linguistic or model theoretic entities, (23) Equiv (O, O0 ) =df Subsumes(O, O0 ) ∧ Subsumes(O0 , O). properties, or relations are introduced into the theory.13 Rather, it de- velops an account of ontologies and their content that is language Subtler metrics for comparison are of course also possible, e.g., independent and grounded in the intuitive notion of propositions — two non-equivalent ontologies might nonetheless share all or some rather than the austere and abstract notion of a model — as the basic of their primitives. More generally, the notions defined above pro- semantical unit of meaning. vide a rich framework for analyzing a wide variety of notions rel- To get at the relevant notion of entailment in our theory, recall evant to understanding the nature of, and logical relations between, once again that, intuitively, ontologies can be thought of as classes of ontologies. propositions. The notion of a proposition is easily defined in terms of our “higher-order” quantifiers: 8 Integration (16) Proposition(p) =df (∃F 0 )p = F 0 A major extension of this work that goes far beyond the current scope Understanding classes as properties, we can now define an ontology will consist in developing a theory of integration. At a purely abstract to be a nonempty class of propositions: level, integration is fairly straightforward. One can import several ontologies into the language L of our approach by creating a separate Ont(O) =df (∃F 1 )O = F 1 ∧ (∃x)F 1 (x) ∧ namespace for the terms in each ontology and translating them from (17) ∀p(F 1 (p) → Proposition(p)) the language of the ontology into L. There will thus be, initially, no possibility of name conflicts. Because the principle [ϕ] ↔ ϕ is The notion of entailment we are after involves both modality and valid, it will be possible to move seamlessly back and forth between our notion of constituency. We first define a constituent of an ontol- using the axioms of a given ontology to investigate its properties and ogy O to be a constituent of one of the propositions in O: talking more generally about the ontology and its content. (18) OntConst(x, O) =df Ont(O) ∧ (∃p)(O(p) ∧ Const(x, p)) Because distinct ontologies are imported with separate names- paces, there is no danger of logical inconsistency arising from incom- Next, say that an ontology O entails a proposition F 0 just in case F 0 patible ontologies. Integration can proceed by identifying or other- must be true if all the propositions in O are true: wise logically connecting the concepts (objects, properties, relations, and propositions) expressed across ontologies. Thus, for instance, it Entails(O, F 0 ) =df Ont(O)∧ might be postulated that two concepts (properties) from different on- (19) 2((∀G0 )(O(G0 ) → G0 ) → F 0 ) tologies are identical; or that one concept subsumes the other; or that for every instance a of one there are two instances of the other that Now say that F 0 and O share primitives if every primitive con- bear some relation to a; and so on. In this way the logical connec- stituent of F 0 is a constituent of O: tions between ontologies can be mapped clearly and rigorously and ShPrim(F 0 , O) =df Ont(O) ∧ with ever greater precision. (20) (∀x)((Prim(x) ∧ Const(x, F 0 )) → OntConst(x, O)) However, while this account of integration is theoretically ade- quate as far as it goes, a complete treatment will have to include a 13 Though of course we define a model theory for the language L of our theory of languages that connects sets of sentences with the ontolo- theory, but that’s just a matter of our own metatheoretic housekeeping: it simply provides a proper theoretical foundation for the language we are gies that they express, and which should lead to more practical ap- using to express our theory; the model theory for L is not itself a part of plications of the theory. Investigating integration at this more applied ontology theory. level will be the next phase of this project. 6 9 Conclusion It will be possible for ontology to make significant progress toward the lofty goals workers in the area are pursuing only if it has proper theoretical foundations. For such goals can be reached only if there is a clear, generally shared understanding of the subject matter of ontology, one that makes it possible clearly to define the scope of the discipline, to identify its subject matter, and chart a course to- ward the resolution of its outstanding problems. The approach in this paper shows promise for providing these essential theoretical under- pinnings REFERENCES [1] Bealer, G., Quality and Concept, Clarendon Press, 1982. [2] Boolos, G., “The Iterative Conception of Set,” Journal of Philosophy 68, 215-231. 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