It is widely accepted that most natural language terms do not have precise universally agreed de nitions that x their meanings. Instead, humans use terms in a variety of ways that adapt to di erent contexts and points of view. In this paper we present a framework based on Supervaluation Semantics for interpreting languages in the presence of semantic variability. This work builds on supervaluationist accounts, which explain linguistic indeterminacy in terms of a collection of possible precise interpretations of the terms of the language. We extend the basic supervaluation semantics by adding the notion of standpoint. A multi-modal logical language for describing standpoints is presented. The language includes a modal operator s for each standpoint s, such that s means that proposition is unequivocally true according to standpoint s | i.e. is true at all precisi cations compatible with s. We show how it can be used to represent logical properties and connections between alternative ways of describing a domain and di erent accounts of the semantics of terms.
It is widely accepted that most natural language terms do not have precise
universally agreed de nitions that x 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 di er greatly in the speci c 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 [
This phenomenon has been approached in many ways within a highly
interdisciplinary area of research, from philosophy to linguistics, cognitive science
and arti cial intelligence. Philosophical interest on this issue dates well back
to Ancient Greek philosophy [
The current research is mainly concerned with the problem of reasoning within a scenario of semantic heterogeneity. Instead of aiming at representing the full variation of certain natural language terms (e.g. via a fuzzy set), we focus on providing a means to model scenarios in which certain interpretations or standpoints coexist and investigate reasoning within such settings. In this way, our framework can be seen as complementary rather than rival to others.
Furthermore, ontology matching techniques are mainly conceived to `solve' heterogeneous situations instead of `representing' them. Our intention is to supplement the existing work on the topic of conceptual indeterminacy by providing explicit support for semantic variation within the representation framework. Nevertheless, as discussed in the future work section of our paper, we consider that ontology matching techniques could have a huge potential to populate supervaluated ontologies as well as to guide both logic and heuristic reasoning within them.
The rest of the paper is organised as follows. It rst presents a motivation example in chapter 2. We then introduce the supervaluationist account vagueness in chapter 3, overview the core ideas of standpoint semantics in chapter 4 and give a formal speci cation of the standpoint logic in chapter 5. Following this, we illustrate the expressive capabilities of the logic (chapter 6) and discuss its application to combining ontologies (chapter 7). We revisit the use case in chapter 8 and conclude with some nal remarks and indication of future work. 2
How much forest is there in the world? is a surprisingly di cult question to
answer [
We consider the situation of a decision maker needing to understand a
complex scenario in which data and conceptual models coming from di erent
contexts need to be critically analysed. We consider ve examples of inferences
relevant to the aspects in which the diversity of forest de nitions has been
reported to pose challenges for the acquisition of global knowledge [
The scenario is not dynamic or informal. On the contrary, the agents that set their standpoints are institutions and/or scienti c communities that produce and distribute data relevant to speci c contexts. E.g. An area in Iran may have been considered a forest by a national agency while according to a global de nition it may not be. We shall return to this example towards the end of the paper, once we have presented our logic of standpoints, and consider how our representation can be used to express the requirements of these queries. 3
In supervaluationist accounts of linguistic indeterminacy the meanings of vague
terms are explained in terms of a collection (a set or some more structured
ensemble) of possible precise interpretations (often called precisi cations). An
early proposal that vagueness can be analysed in terms of multiple precise senses
was made by Mehlberg [
Apart from providing a general framework for specifying a semantics of
vagueness, the supervaluationist idea is also attractive in that it can account
for penumbral connection [
Moreover, such frameworks o er a natural treatment of what we call conceptual vagueness 2, which arises when there is a lack of clarity on which attributes or conditions are essential to the meaning of a given term, so that it is controversial how it should be de ned. This kind of conceptual indeterminacy underlies the controversy about whether to de ne forest in terms of land cover or land use, and this case is a good illustration of the overlapping of the applicability of the two interpretations, since the use to which land can be put depends to a large extent on the material and ecological properties of its land cover. 4
We now overview the ideas of standpoint semantics, which is essentially an
elaboration of supervaluation semantics introduced in [
A precisi cation in standpoint semantics is identi ed with a precise interpretation of the language, such that for any state of the world there is a unique extension for every predicate. This notion di ers from Fine's terminology, where a precisi cation need not be completely precise, but leave certain propositions indeterminate. Fine's precisi cations form a partial order where p2 is more precise than p1 if all propositions with a de nite truth value in p1 have the same truth value in p2 but some propositions that are indeterminate in p1 have a determinate truth value in p2.
The standpoint semantics [
Standpoints are collections of precisi cations. They are typically organised by constraints (axioms and threshold limitations) that pick out a corresponding set of admissible precisi cations that satisfy these constraints. For example, a standpoint might be constrained by the de nition E1, but the value of the threshold t tall might not be xed although constrained to lie between certain values E2.
2 The distinction between sorites vagueness (the applicability of a predicate depends
on measurable parameters but their thresholds are undetermined) and conceptual
vagueness is introduced in [
The formalism that we present here goes signi cantly beyond [
Standpoint logic is a multimodal logic with partially ordered modalities. Standpoints are modelled as a set of modal operators: If s is a standpoint and s then is the case for all precisi cations which are compatible with the standpoint s. The partial ordering encodes the subset relation which may hold between two standpoints s and s0. 5.1
Our formal language LS is an extension of classical rst-order calculus including numerical symbols and comparison relations (= and <) as well as the usual boolean operators and quanti ers.
Vocabulary. The non-logical symbols of the language are speci ed by a vocabulary, which is a tuple of the form:
V = hN ; X ; R; F i ; where N is a set of nominal constants, X is a set of nominal variables, R = (R1; [ : : : [ Rn [ : : :) is the set of predicate/relation symbols, whose subsets Rn are the sets of n-ary predicate symbols and F = (F1; [ : : : [ Fn [ : : :) is the set of function symbols, subsets Fn being the sets of n-ary function symbols. Terms. The language has two types of terms: one type refer to individual entities and the other refer to numerical magnitudes: { Tn = N [ X is the set of nominal terms of the language. { Tm = TD [ ff ( 1; : : : ; n) j f 2 Fn ^ 1; : : : n 2 Tng is the set of magnitude terms.
The set Tm includes the set TD of decimal numerals, as well as terms formed by applying function symbols to nominal terms, which give the value of some scalar property of an entity (e.g. height) or tuple of entities (e.g. the distance between two entities).
Atomic Propositions. The language has the following forms of atomic proposition: { R( 1; : : : n), { 1 = 2, { 1 2, where 1; : : : n 2 Tn, where 1; : : : 2 2 (Tn [ Tm), where 1; : : : 2 2 Tm
R( 1; : : : n) asserts that relation R holds of the nominal terms 1; : : : n (which are named entities and/or quanti ed variables) and 1 = 2 is the usual equality relation, that can hold either between named entities and/or variables. Complex Propositions. For any ; propositions are also in LS : 2 LS , and x 2 X the following complex { : , ( ^ ), ( _ ), ( ! ), ( $
operators and rst-order quanti ers, { s meaning is true in standpoint s, i.e. in all precisi cations compatible with standpoint s. { | meaning is true in all precisi cations. is a special case of s where is the standpoint containing the set containing all precisi cations.
), 8x[ ], 9x[ ] | the standard boolean
LS is the smallest set containing all atomic propositions and all complex propositions formed by these constructions.
We can easily de ne other useful operators such as s def : s : , i.e. according to standpoint s may be considered true; Is def ( s ^ s : ), i.e. the truth of is indeterminate with respect to standpoint s, or Ds def ( s _ s : ), i.e. according to standpoint s has a determinate truth value. 5.2
A standpoint semantics interpretation structure is a tuple
S = hP; D; V; S; ; ; i ; where: { P is the set of precisi cations, { D is a non-empty set, the domain of individuals, { V = hN ; X ; R; F i is a vocabulary, { S is the set of standpoint symbols. { : S ! (2P =;) is a function mapping each standpoint symbol to a nonempty set of precisi cations, { = n [ m, where n : Tn ! D maps each nominal term to an element of the domain of individuals and m : Tm ! Q maps each magnitude term to a real number. { = ( 1 [ : : : [ n : : :), where n : Rn P Dn ! ft; f g. So maps, each n-ary predicate, precisi cation and n-tuple of individuals to a truth value.
With respect to the interpretation structure, formulae are interpreted as follows: {{ [[[[(r(11=;: : 2:)]n]Sp)]]Sp = (r; p; h ( 1); : : : ( ni)), = t if ( 1) = ( 2), else = f , { [[( 1 2)]]Sp = t if ( 1) ( 2), else = f , {{ [[[[: ^]]Sp ]]Sp == tt iiff [[[[ ]]]]SpSp == ft,aneldse[[ =]]Spf ,= t, else = f , { [[8x[ ]]]Sp = t if ( [[ ]]Sp0w = t, for every interpretation structure S0 = hP; D; V; S; ; 0; i, such that 0 is identical to , except that 0(x) may have a di erent value from (x) ), else = f , { [[ s ]]Sp = t if [[ ]]p0w = t for all p0 2 (s), else = f . Note that this includes the case there s = . 5.3
We now give a set of axioms for the logic, which capture signi cant aspects of the semantics. We believe all our axioms are sound with respect to the semantics, but we do not claim they are complete. Establishing a proof system that we can prove sound and complete is the subject of ongoing work.
Being built upon an underlying classical logic we have that is a theorem of Ls if it is an axiom of rst-order logic (where we treat all modal sub-formulae as atomic propositions).
Modal Axioms. In the following axioms hold for any of the operators the special case . si and AN AK.
AT.
AD.
A4.
s s( s s for any theorem ! ) ! ( s ! ! s ! s s ! s )
A5.
A1.
A2.
A3.
s s s ! ! $ ! s s s0 s s for (s0 s) 3
Thus, the operators is an S5 modality, whereas the rest of s operators satisfy the axioms of the modal logic KD45.
Axiom A1 can also be expressed in the form s0 ! s for (s0 s) and it captures the partial order between standpoints, by ensuring that any proposition considered de nite in a given standpoint, is also considered de nite in any sharper standpoint. A3 is easily derivable from A1. A2 captures the property of the semantics whereby once we apply a standpoint operator to a propositional formula, we will get a proposition that is either true at all precisi cations or false at all precisi cations. This is like in the well-known modal logic S5, where ! and ! . By combining A2 and A3, we get: T1.
s0 s $
s 3 s0 s asserts that s0 is at least as sharp as s i.e. (s1) that is unequivocally true in s is also true in s0. (s2); every proposition Quanti er Axioms. The universal quanti er satis es its classical axioms, which are covered by axiom C above. What about axioms specifying the interaction between the modal operators and quanti cation?
Since our semantics is based on a single domain of individuals, its models will satisfy the Barcan formula: A7. 8x[ s (x)] ! Issues Regarding the Domain of Quanti cation. One can argue that if we change the precisi cation according to which the world is classi ed, the set of entities is likely to change. For example, under one precisi cation, a particular tree-covered area might be determined to form a single forest under one precisi cation, whereas under another it might be determined as consisting of two forests separated by a band of heathland. However, one could take the contrary view that, the set of entities can be regarded as the same even though their classi cation changes. This is consistent with a de dicto view of vagueness, in which it is only linguistic descriptions that are vague, not the objects that they describe. In fact, we believe that our semantics is inadequate to adequately express correspondencies between entities at di erent worlds and/or precisi cations, e.g. we cannot say that di erent tree covered areas correspond to the same forest seen from di erent standpoints, which agree that there is one forest within a given region, but disagree about its extent. 6
Expressive Capabilities of LS We now examine the expressive capabilities of our standpoint logic and illustrate them with a variety of examples.
soning with vague predicates are the subject of a huge amount of debate in the philosophical literature. Here we do not go into detail about the issues involved, but just give a simple example of how our formalism can deal with some aspects of this problem in relation to attributions of the typical graded adjectives `tall' and `short' for two standpoints s1 and s2 where s2 s1.
E3.
E4.
E5.
E6.
:9x[Tall(x) ^ Short(x)] [height (tara) = 186cm] [height (simon) = 160cm] s1 [8x[Tall(x) $ height (x) > t tall]
E7.
E8.
E9.
s1 [t tall < 185cm] s1 [Short(simon)] s2 [t tall = 180cm]
All precisi cations must satisfy the given penumbral connection axiom E3: nothing can be both Tall and Short. Thus this formula will be true in all standpoints. Formulae E4 and E5 express objective, non-vague facts that are taken to be true for all precisi cations. Given E3, E4 and E7 we know that s1 [Tall(tara)]. Because we know that s2 is at least as sharp as s1, by using axiom A1 we can infer that s2 [Tall(tara)]. Note that s2 s1 implies that at least those de nite tall cases in s1 are also de nite in s2, so the threshold for tallness in s2 must be equal or lower than that of s1. Also, with E8, we know s1 [t tall > 160cm]. Conceptual Variation. In LS it is easy to stipulate that a particular de nitional axiom holds for a particular standpoint. In the domain of forests we might have: E10.
E11.
s1 8x[Plant(x) $ Embryophyta(x)] s1 8x[Tree(x) $ (Plant(x) ^ Woody(x) ^ Tall(x))] In another standpoint s2 we might have: E12.
s2 8x[Plant(x) $ Organism(x) ^ :Animal(x)]
E10 and E12 express inconsistent de nitions (a mushroom would be a Plant
accordint to s2 but not according to s1) but the formulae can coexist consistently
within LS because the de nitions are within di erent standpoint operators.
Penumbral Connections. Supervaluation semantics regards precisi cations
as applying to the whole language. This is because the meaning of related vague
concepts cannot always vary independently. We do not allow precisi cation in
which related concepts are interpreted inconsistently (e.g. someone can be both
`tall' and `short'). Modelling this kind of penumbral connection is one of the main
motivations of supervaluation semantics [
It may also be useful to specify that concepts are semantically independent within a standpoint (or globally). This may allow for the modularisation of a vocabulary into sets of terms that do not impinge on each other (e.g. concepts relating to forests can be modelled without worrying about mountains or buildings). We say that two vague propositions and are penumbrally independent when every sense in which can be interpreted is compatible with every sense which can be interpreted. In LS logic we can express this by: ( ( s ( s : ^ ^ s ) ! s ) ! s( ^ ) s(: ^ ) ^ ^ ( s ( s : ^ s : ) ! s(
^ : ) ^ ^ : s ) ! s(: ^ : )) Concept sharpness. We can say that a particular predicate R is extensionally sharp in a given standpoint s with a formula such as 8xDs[R(x)], where x stands for n di erent universally quanti ed variables and where n is the arity of R. (s0 7 Moreover, we can express the condition that `standpoint s0 has an extensionally sharper interpretation of concept C than standpoint s', which we represent by the form (s0 s) : x[C(x)]. This is just syntactic sugar for an actual formula of LS that can be de ned as follows: s) : ( x[C(x)]) ( 8x[ s C(x) ! s0 C(x)] ^ 8x[ s :C(x) ! s0 :C(x)] )
In the process of de ning an ontology, there may be a number of di erent
meanings attached to certain terms. Moreover, users may hold that having all these
meanings is useful, as they are relevant for di erent contexts [
We thus see the potential for supervaluated ontologies by design, that are not the fruit of the integration of di erent models but a direct formalisation of a domain in which the semantic variablility of its terms is su ciently meaningful and/or relevant to be represented in the ontology.
Embedding Ontologies within LS. Moreover, our language LS provides many ways in which the relationships between terms in di erent ontologies can be described and we are still investigating which of these are most practical. However, we now outline some possibilities. Any given ontology Oi is identi ed with a set of formulae incorporating both semantic constraints (its T-Box ) and facts (its A-Box ). We also associate Oi with a standpoint si. Our aim is to construct a theory in LS in which we embed two or more unaligned ontologies.
We assume that every formula of the ontology is unequivocally true with respect to that standpoint, so an ontology's standpoint can only be inde nite about formulae that are not explicitly stated or implied by its explicit statements.
We suggest that when interpreting and combining ontologies it will be useful to also create a general umbrella ontology O . This will contain a set of constraining formulae, which we would expect to hold in any reasonable interpretation of the vocabulary under consideration. Thus, for each 2 O we have 2 . This means that the standpoint of each ontology will inherit any penumbral connections that we consider to be essential, but may have been omitted from the ontology itself.
Inter-Ontology Concept Alignment and Interaction. We address the general and common case where the ontologies are poorly aligned both in that: (1) only some of the terms in each ontology have a closely corresponding counterpart in the other ontology; and (2) even where a concept in one ontology has a `matching' counterpart in the other, these counterparts are not exactly equivalent. Such counterparts may di er either in terms of explicit de nitions or axioms within the ontology itself, or in terms the criteria that have been used to determine the instances of a predicate (for instance a particular settlement might be classi ed as a Forest in one ontology but not in the other (e.g. because that ontology requires that a forest must have a certain minimal area), where it is instead considered to be an instance of Woodland. We assume that some form of alignment and matching of conceptual terms has already been carried out. E13. 8x[ s1 C(x) $ s2 D(x)] E14. (s0 s) : ( x[C(x)]) E15. 8x[:( s1 C(x) ^ E16. 8x[ s1 C(x) ! E17. 9x[ s1 C(x) ^ E18. 9x[ s1 C(x) ^
s2 D(x))] s2 D(x)] s2 D(x)] s2 D(x)] (equivalence) (sharpening)
(exclusion) (subsumption) (strong overlap) (weak overlap)
For present purposes we simply assume that certain entities can be identi ed between ontologies. In practice, identifying objects between di erent ontologies may be non-trivial and involve complex issues. These issues are related to the problems of establishing correspondences between entities at di erent possible worlds and/or precisi cations. 8
As described in the Motivation Example, for the current purpose we set a scenario in which a stakeholder is using the proposed framework to analyse the conceptual variation between the forest concepts of the di erent data sources of an online platform (e.g. Global Forest Watch). In this section we show how the answers to the questions set in the Motivating Example section may be inferred. Q1. Is area a consistently forest for all forest interpretations? Otherwise, is it arguably a forest?:
[Forest(a)]; [Forest(a)] ^ : [Forest(a)] Q2. A degraded area a stopped being classed as Intact Forest4 and is now only arguably a forest. What is it according to the other views? 8s[ s :Forest(a) ! s Landtype(a)] Q3. What can we infer from an area that satis es some but not all the interpretations? Consider an example where we have three standpoints, s1, s2 and s3 where s3 s2, two potential forests p and q and some facts:
4 IFL are unfragmented areas with at least 50,000ha and 10Km width. These were
then mapped from Landsat satellite imagery for the year 2000. [
[8xForest(x) $ :Savannah(x)] s3 Savannah(q), canopy (q) = 35 We can infer the following: E29. av tree height (p) < 2m E30. s2 :Forest(q) E31. : s1 Forest(q)
E32. s3 [t can > 35] Q4. Are there inconsistencies among the data or only di erent data classi cations? e.g. Are there areas reported to satisfy some criteria that don't t the classi cations? Hypothesis: temporal clearings? E33. canopy (p) = 25 Consistency is ensured within precisi cations and thus the system would show any inconsistent standpoint s to be ;. In this case E24, E25 and E33 imply that s1 [t can < 25], which is inconsistent with E25 in s1. 9
We have presented a formal language which we believe is well-suited for describing the variability of vague concepts and also for expressing relationships between di erent points of view regarding concept meaning in di erent contexts. Development of our system is ongoing, and many questions remain regarding its generality and practical applicability.
We have given both a semantics and a set of axioms for our formal language.
In future work we will provide alternative Kripke style semantics with partially
order operators, along the lines of [
With regard to applications of the theory we plan to develop its use for combining non-aligned ontologies and to test this on speci c examples, particularly concentrating on the forestry domain, which provides many interesting scenarios and challenges. Regarding developing the underlying semantics, there are some very interesting and tricky issues concerning correspondences between individuals identi ed with respect to di erent standpoints and/or possible worlds.