Most natural language terms do not have precise universally agreed definitions that fix their meanings. Even when conversation participants share the same vocabulary and agree on taxonomic relationships (such as subsumption and mutual exclusivity, which might be encoded in an ontology), they may differ greatly in the specific semantics they give to the terms. We illustrate this with the example of 'forest', for which the problematic arising of the assignation of different meanings is repeatedly reported in the literature. This is especially the case in the context of an unprecedented scale of publicly available geographic data, where information and databases, even when tagged to ontologies, may present a substantial semantic variation, which challenges interoperability and knowledge exchange. Our research addresses the issue of conceptual vagueness in ontology by providing a framework based on supervaluation semantics that explicitly represents the semantic variability of a concept as a set of admissible precise interpretations. Moreover, we describe the tools that support the conceptual negotiation between an agent and the system, and the specification and reasoning within standpoints.
Since the shift in philosophy of language from logical positivism to behaviourism and
pragmatism, it is widely accepted that most natural language terms do not have precise
universally agreed definitions that fix their meanings. Even when conversation
participants share the same vocabulary and agree on taxonomic relationships (such as
subsumption and mutual exclusivity, which might be encoded in an ontology), they may differ
greatly in the specific semantics they give to the terms in a particular situation.
Moreover, except for certain technical terms, individuals do not hold permanent and precise
interpretations of the meaning of terms themselves [
Humans generally cope with these imprecisions of language by using context and
other pragmatic information to narrow the semantic variability of the terms. We say, they
‘take a standpoint’ on the semantics of the terms. If conflicts occur during the
conversation, participants may adapt dynamically their standpoints in order to maintain the
cooperation principles (Grice [
To support human-machine interactions, ontologies are aimed at providing a
common vocabulary in which shared knowledge can be represented [
In this paper we take the example of negotiating the interpretation of the term ‘forest’, and propose a framework based on supervaluation semantics to allow for semantic variability within the ontology. We first introduce what we understand as vagueness and how it affects geographical objects in general, followed by the specific case of forest. We then summarise different approaches to vagueness and how they relate to our research. We propose a framework based on standpoint semantics and its tools for supporting share of information, concept negotiation and querying and reasoning within an agent’s standpoint.
Vagueness is pervasive in language [
Moreover, the geographic domain is particularly characterised by objects lacking
bona fide boundaries and by the inherent vagueness in the definition of geographic
descriptors, frames of reference and context [
While the value of ontologies is now well established [
1Occurs when ontologies, schemas or datasets of the same domain present differences in meaning and interpretation of categories and/or data values, thus challenging interoperability.
In this research we examine the term ‘forest’, for which a broad range of definitions
(more than 600 were reported in [
This scenario poses challenges both for the acquisition of global forest knowledge,
particularly its extent and spatial distribution [
Despite the call for addressing the variability of interpretations and definitions of
‘forest’ in the literature [
Our intent is to provide a novel way of tackling the prior issues by approaching them from the perspective of vagueness in ontology, more specifically conceptual vagueness. In this section, we briefly review some approaches to vagueness and concept creation and negotiation, and how they relate to our research on forest definitions.
The most broadly used logic-based techniques to model vagueness in information
systems are multivalued logics [
3The Food and Agriculture Organization of the United Nations.
less to more likely. However, fuzzy sets do not fully characterise the different precise
overlapping meanings that a term can adopt, which can be sharp but diverse, and fails to
incorporate penumbral connections [
Conceptual spaces (Gardenfors, [
Semantic heterogeneity and concept negotiation in information systems have been
mainly investigated as a separate phenomenon from that of concept creation and
vagueness. Instead, they are approached as a phenomenon that arises in the context of the need
for interoperability of two systems, ontologies or agents, and the necessary conciliations
for their successful interoperation. In the area of ontology matching, a wide literature
review is provided in [
Standpoint Semantics is our theoretical framework for representing, interpreting and reasoning about information expressed using vague terminology. This theory is an elaboration of the Supervaluation approach, in which the semantics of a vague language is modelled by a set of precise interpretations called precisifications, where each precisification corresponds to a precise and coherent interpretation of all vocabulary of the language.
Consider a formal language based on classical first order logic, with a vocabulary V = N [ C [ R, where N is a set of name symbols (these may be used as constants or as variables when used with a quantifier), C a set of unary concept terms and R a set of binary relation symbols. The set L of formulae of the language is the smallest set that contains Ca and Rab for every C 2 C every R 2 R and every a; b 2 N , and, for every j and y in L , formulae :j, j ^ y and j _ y are also in L .4
We define an interpretation structure I = hP; D; V ; d i, where:
D is a set of individuals (the domain of discourse), 4Here we omit the implication symbol ‘!’ from the syntax. But it can easily be defined by j ! y def :j _ y.
d is a denotation function, which, for each precisification and each vocabulary symbol in V , gives the semantic value of that symbol. It is a compound of the following sub-functions: — dN : (P N ) ! D — dC : (P C ) ! 2D — dR : (P R) ! 2D D maps name symbols to elements of the domain, maps concept terms to subsets of the domain, maps relation symbols to sets of ordered pairs of elements of the domain.
The only difference from the usual classical logic semantics is that the denotation of each symbol is relative to a precisification p 2 P, which determines a precise interpretation of that predicate. We write p I j to mean that the formula j is true for precisification p, in interpretation structure I . This determines the truth of propositions as follows: p p
I Ca I Rab iff d (p; a) 2 d (p;C) iff hd (p; a); d (p; b)i 2 d (p; R) p I :j iff it is not the case that p I j p I j ^ y iff p I j and p I y p I j _ y iff p I j or p I y p I 8x[j] iff p I 0 j for every interpretation structure I 0 = hP; D; V ; d 0i that is identical to I , except that the value of d 0(p; x) may be any element of D.5 Notice that this semantics not only accounts for variability in the meaning of vocabulary terms but also allows one to enforce dependencies between the terms occurring in a formula. For example, precisifications might vary in the set of people considered tall. If precisification p1 sets a greater height threshold than p2 for applicability of the vague property Tall, we might have d (p1; Tall) = fsally; tomg, d (p2; Tall) = fsally; tom; ulig. Hence, we would have p1 I Tall(uli) and p2 I :Tall(uli). But no precisification can make the formula Tall(uli) ^ :Tall(uli) true. Moreover, we can also enforce constraints between different terms in a formula. For instance, we might require that the denotations of Tall and Short are disjoint for all precisifications. Thus, no formula of the form Tall(x) ^ :Short(x) would be true in any precisification.
The supervaluation semantics allows us to to augment the logical language with operators that are interpreted relative to the set of precisifications. In particular, we can specify a semantics for Uj, meaning that j is unequivocally true, and Sj, meaning that j is in some sense true: p p
I Uj I Sj iff iff q q
I j for every precisification q 2 P,
I j for some precisification q 2 P.
Standpoint semantics adds detail to the basic supervaluation approach in several ways. The semantic choices that determine each particular precisification are modelled explicitly. These consist of choices of (a) threshold values that determine the applicability of ‘sorites vague’ predicates such as ‘tall’ and ‘bald’; and (b) choices of definitions that resolve conceptual ambiguities, such as which kinds of vegetation species can be considered as constituting a forest.
5Here, for simplicity, we are assuming that the domain of entities will be the same for every precisification. This is not plausible in general.
Standpoint semantics explicitly models the variability of vague terminology in relation to precise observable measurements and properties. Such measurements could be heights, weights, distances and the properties might relate to physical composition, topological relationships. This is the kind of information one might store in a database (or compute directly from the information in a database). Of course, in practice such information might be inaccurate for various reasons such as limitations of measuring equipment or errors in data capture. But these issues are not due to vagueness of terminology, so in formulating the standpoint theory we need not worry about the correctness of the measurement data: we take it as we find it.
We now specify a classical first order language Lo for describing entities in terms of objective measurements, properties and relations. Let its vocabulary be the set Vo = N [ D [ F [ G [ C [ R, where N is a set of names, D is a set of numerical expressions (e.g. standard decimals), F is a set of unary function symbols, G is a set of binary function symbols, C is a set of (precise) unary concepts and R is a set of (precise) binary relations. So in a forestry related domain a vocabulary might be something like: hftree1; :::; boris; :::g; fheight; radiusg; fdistanceg; fOak; Beech; :::g; fOwns; :::gi : Here we have identifiers for individual trees, names of people, unary functions giving height and (canopy) radius measurements of the trees, a binary function giving the distance between any two trees, predicates specifying the species of trees and an owenership relation between trees and people.
In addition to the predications, Ca and Rab, of L , the vocabulary of Vo enables us to include in Lo atomic formulae of the forms t1 = t2 and t1 t2, where each of t1 and t2 can be either a functional term (of the form f (a) or g(a; b)) or a numerical expression (e.g. in standard decimal notation). So we can write formulae such as:
distance(tree1; tree2) = 23:5 or height(tree6) height(tree57).
Like L , the language Lo is also closed under combination by Boolean connectives and the quantification operators. An interpretation of Lo is determined by a structure Io = hD; Vo; doi, where
D is a domain of entities, do maps the vocabulary in Vo to their semantic denotations and is a compound of the following sub-functions: — dN : N ! D — dD : D ! Q — dF : F ! (D ! Q) — — — dG : G ! ((D
D) ! Q) dC : C ! 2D dR : R ! 2D D maps names to elements of the domain, maps numerical expressions to rationals, maps unary function symbols to functions from elements of the domain to rationals, maps binary function symbols to functions from pairs of elements of the domain to rationals, maps concept terms to subsets of the domain, maps relation symbols to sets of ordered pairs of elements of the domain.
The truth conditions of the language Lo relative to interpretation Io are as follows: Io Ca Io Rab iff d (a) 2 d (C) iff hd (a); d (b)i 2 d (R) iff d (t1) = d (t2) iff d (t1) d (t2)
The interpretation of formulae formed by the Boolean connectives and quantifiers will be exactly the same as for standard first order logic, so we do not specify it here.
We now specify a language Lv in which we can specify possible precise interpretations of vague predicates. In adition to Vo, we also have the vocabulary Vv = Cv [ Rv [ Tv. where Cv is a set of vague unary predicates, R a set of vague binary relations Tv a set of threshold variables. The specification is technically complex, although it has a relatively simple informal explanation. Each precisification determines a mapping of vague conceptual terms and relations to possible definitions and also determiens a valuation of threshold variables that fix the limits of applicability of vague graded predicates. Thus each precisification provides a translation from formulae containing vague terms of Vv into formulae containing only the precise objective vocabulary of Vo.
Let Ld be the set of possible defining formulae. These are any well formed formulae over the vocabulary Vo [ Tv, where as well as measurement functions and decimals, the terms that can occur in atomic formulae of Ld formed with the = and relations also include threshold variables. So for example, we could have height(x) thresh1. We write Ldn to refer to the subset of Ldn containing those formulae that have exactly n distinct free variables — i.e. formulae in which exactly n symbols in N occur outside the scope of any quantifier.
We now give an explicit specification of the particular interpretation associated with a precisification. Let each precisification p be associated with a tuple hpT ; pC; pRi, where: pT : Tv ! Q maps each threshold variable to a number, pC : Cv ! Ld1 maps each vague unary predicate to a precise unary definition, pR : Rv ! Ld2 maps each vague binary predicate to a precise binary definition.
pC(Tree) = Plant(x) ^ Woody(x) ^ min tree height thresh height(x) and pT (min tree height) = 10, so precisification p defines Tree to be a woody plant of height greater than or equal to 10 (metres).
We can now give an interpretation function for the full language Lv in which formulae containing vague terms are interpreted relative to a precisification, which maps them into precise formulae of Lo. The interpretation is specified by Iv = hD; Vo; do; P; Vv; dvi, where the interpretation of symbols in Vo is given as for the structure Vo given above, and the interpretation of formulae in Lv is given by: p Iv j is evaluated as Io j iff j does not contain any symbol in Vv, p Iv t1 = t2 iff d 0(p; t1) = d 0(p; t2), where d 0(p; t) = do(t) for all terms in the precise language Lo (for whose interpretation p is not relevant) and d 0(p; t) = pT (p; t), where t is a threshold variable (i.e. t 2 Tv), p Iv t1 t2 iff d 0(p; t1) d 0(p; t2), where d 0(t) is defined the same way as in the previous clause, p Iv Ca iff Io pC(C)(a=x), where pC(C)(a=x) is the result of substituting a for the free variable x occurring in the precise definition of C determined by the function pC of precisification p, p Iv Rab iff Io pR(R)(a=x; b=y), where pR(R)(a=x; b=y) is the result of substituting a and b for the free variables x and y, in order of their occurrence in the definition of R determined by the function pR of precisification p,
In this section we explore the potential of the proposed system to enable communication between agents holding diverse standpoints, by providing a shared ontology that supports vague semantics. Our framework enables agents to both link data to the ontology according to their precise interpretation and also to query and reason within the ontology according to a specific standpoint, which can be modified during the interaction. In a context of sharing information, the proposed framework offers tools that enable both the analysis of the relations that hold between precisifications and the execution of modifying operations such as calculating the intersection or union of a pair of precisifications. This, together with information on the instances satisfying these precisifications, serves as a support to the agent for the specification of sound standpoints that guarantee integrity and enable interoperation with the ontology.
Assuming that a concept negotiation is necessary, we support the analysis of the relations that hold between precisifications. We identify five main relations, analogous to RCC5, to be inferred by the system. These are: 1. Equivalence: p1 $ p2. Anything that is a ‘forest’ in p1 must be a ‘forest’ in p2 and vice versa. 2. Subsumption: p2 ! p1. Anything that is a ‘forest’ in p2 must be a ‘forest’ in p1. 3. Inverse subsumption: p1 ! p2. Anything that is a ‘forest’ in p1 must also be in p2. 4. Disjunction: p1 ! :p2. No entity can be a ‘forest’ in both p1 and p2 5. Overlap: None of the previous (NP) relations hold between p1 and p2. Thus there may be entities that are classified as ‘forest’ in both p1 and p2, although neither precisification has a stronger definition that the other.
Relations between precisifications show, to some degree, the connections among them. While overlap is the most common scenario, other relations such as subsumption and disjunction provide valuable information to the agent, to the point of interoperation becoming trivial (such as when information is linked to a definition that is subsumed by the agent’s definition) or not feasible because of explicitly conflicting commitments (in the case of disjoint definitions).
It must be noted that the relation between precisifications is purely logical, i.e. is a relationship that holds between the formal representation of two interpretations of the semantics of the ontology, p1 and p2, which is not dependent on the real world objects that may instantiate them. Consequently, the relation between the logical relations of the precisifications and that of the sets of the real instances (in this world) may not match up. However, the former do constrain the possibilities of the latter. This is shown in Table 2 and discussed in more detail in Section 5.4. 5.2. Operations In order to support both the analysis of the differences and variation among the existing instantiations of the concepts of the ontology and also to facilitate the specification of standpoints that comprise the appropriate subset of precisifications, we define four basic operations between precisifications.
1. Union: p1 [ p2. All the instances that satisfy either p1 or p2 (or both). E.g. Both p1 and p2 are considered admissible precisifications for a specific use of the ontology by a certain agent. 2. Intersection: p1 \ p2 . All the instances that satisfy both p1 and p2. E.g. An agent is interested on the intersection between their own interpretation p1 and a precisification in the ontology p2. 3. Complement: p2 n p1. All instances that satisfy p2 but not p1. E.g. An agent finds p1 conflicting with its interpretation and wants to prevent instances of p1 from his query on p2. 4. Symmetric difference: p1 D p2. All instances that satisfy either p1 or p2 but not both. E.g. An agent is interested in exploring the borderline scenarios in which p1 and p2 disagree.
In the context of knowledge sharing, operations between precisifications are expected to be used for two main purposes, namely concept negotiation and specification of the standpoints. In practice, concept negotiation not only involves the analysis of the objective meaning and ontological commitments formalised on the different precisifications, but also of the real world implications of such commitments. This analysis is possible by querying the ontology with the previous operations, as the agent can gain insight on which instances fall within the borderline scenarios, which instances comply with all the desired precisifications and so on. In the case of GIS, this is enhanced because a spatial projection is available, thus providing powerful means for understanding the way definitions perform in specific areas of interest.
Moreover, once the process of concept negotiation is complete for a particular use of the ontology, operations can be used to support the formalization of the agent’s standpoint s1. During further interactions of the agent with the ontology, queries will be associated with this standpoint. s1 acts by restricting the admissible precisifications of the ontology to those which are consistent with s1 and enabling reasoning and information retrieval within the subset.
In the case of geographical information, the spatial projection of the instances can be particularly relevant to complement the analysis of the relations that hold between precisifications. One might expect that the possible relations holding between precisifications (equivalence, subsumption, inverse subsumption, overlap and disjunction) would map to the analogous RCC5 relations (equal, proper part, inverse proper part, partial overlap and discrete) between the total spatial area covered by sets of instances satisfying each precisification. However, that does not necessarily need to be the case. Instead, relations between precisifications only restrict the space of possibilities but leave some variation open.
(a) (b)
In Table 2 we show the possible relations holding in the different scenarios. While the analogous to RCC5 is the expected relation to hold, other possibilities are allowed. These may be symptomatic of different circumstances. Take, for example, the case where the spatial projection of p1 is equal to that of p2. It may be the case that p1 and p2 are equivalent. p1 could also be weaker than p2 by allowing some more scenarios, but they never manifest in the available data about the state of the world. Even, it could be that p1 and p2 describe forest in a logically different way (e.g. one uses tree proximity and the other canopy cover) that is highly correlated, and therefore both map to the same objects. In other cases it may be due to mere exemplar clustering6. Finally, even when the expected spatial relation holds between the projections, it is expected to be useful for the agent to know how populated are the intersections or borderline scenarios of the definitions under consideration.
The diagrams in Figure 1 give some output examples from our prototype standpoint semantics based software, which enables visualisation of forest data in accordance with different standpoints. In Figure 1(a) we see forest extensions according to three different precisifications in different shades of green. The darker green corresponds to a stricter definition, which requires trees to be closer together to count as constituents of a forest. None of these precisifications consider the shrubs (small brown discs) to be forest constituents. In Figure 1(b) another precisification (demarcated by a black dashed line) is overlayed over the map. According to this precisification the shrubs are counted as forest constituents, so the forest takes on a very different shape, overlapping the extensions associated with the other precisifications.
6In ordinary situations, objects that exhibit one property, will very often also exhibit another property and
vice versa, even though there is no necessary connection between the properties. The cause may be because of
patterns and regularities that are essentially contingent [
It is widely agreed that, while domain specific ontologies can be used within a
community of users for achieving a consensus about conceptualizing, structuring and sharing
domain knowledge [
We present a novel approach, consisting on a framework based on ‘Standpoint Semantics’, that makes explicit the semantic variability of the terms of an ontology. We suggest that our framework can support agents with different interpretations and standpoints on their interoperation with a common ontology. Our aim is to provide a description of such a theory and to outline the particular features that can support the conceptual negotiation between agents.
Further work remains to be done to provide a fully operable standpoint semantics ontology, so, in a sense, this paper is only a preliminary step. However, we expect to open the possibility for a complementary approach to the research on semantic alignment in distributed systems. We consider that, in the current context, it is key to support ontologies that can both bring together the views of interdisciplinary domains and that are expressive enough to prevent reasoning and inference from what can be inconsistent interpretations. Moreover, we believe that the explicit acknowledgement of the semantic variability of the terms of the ontology is key for that purpose.