Capturing Graded Knowledge and Uncertainty in a Modalized
Fragment of OWL
Hans-Ulrich Krieger
German Reasearch Center for Artificial Intelligence (DFKI), Saarbr
ucken, Germany
Knowledge Representation and Reasoning, Uncertainty in AI, Description Logics & OWL, Ontologies,
Graded Natural Language Statements, Representation of Controlled Natural Language.
Natural language statements uttered in diagnosis (e.g., in medicine), but more general in daily life are usually
graded, i.e., are associated with a degree of uncertainty about the validity of an assessment and is often
expressed through specific verbs, adverbs, or adjectives in natural language. In this paper, we look into a
representation of such graded statements by presenting a simple non-standard modal logic which comes with
a set of modal operators, directly associated with the words indicating the uncertainty and interpreted through
confidence intervals in the model theory. We complement the model theory by a set of RDFS-/OWL 2 RL-like
entailment (if-then) rules, acting on the syntactic representation of modalized statements. Our interest in such
a formalization is related to the use of OWL as the de facto language in today’s ontologies and its weakness to
represent and reason about assertional knowledge that is uncertain or that changes over time.
Medical natural language statements uttered by physi-
cians or other health professionals and found in med-
ical examination letters are usually graded, i.e., are
associated with a degree of uncertainty about the va-
lidity of a medical assessment. This uncertainty is
often expressed through specific verbs, adverbs, ad-
jectives, or even phrases in natural language which
we will call gradation words ( linguistic hedges);
e.g., Dr. X suspects that Y suffers from Hepatitis or
The patient probably has Hepatitis or (The diagnosis
of) Hepatitis is confirmed.
In this paper, we look into a representation of
such graded statements by presenting a simple non-
standard modal logic which comes with a small set
of partially-ordered modal operators, directly asso-
ciated with the words indicating the uncertainty and
interpreted through confidence intervals in the model
theory. The work presented here addresses modal-
ized propositional formulae in negation normal form
which can be seen as a canonical representation of
natural language sentences of the above form (a kind
of a controlled natural language).
Our interest in such a formalization is related
to the use of OWL in our projects as the de facto
standard for (medical) ontologies today (to represent
structural/terminological knowledge) and its weak-
ness to represent and reason about assertional knowl-
edge that is uncertain (Schulz et al., 2014) or that
changes over time (Krieger, 2012). There are two
principled ways to address such a restriction: either
by sticking with the existing formalism (viz., OWL)
and trying to find an encoding that still enables some
useful forms of reasoning (Schulz et al., 2014); or by
deviating from a defined standard in order to arrive, at
best, at an easier, intuitive, and less error-prone repre-
sentation (Krieger, 2012).
Here, we follow the latter avenue, but employ
and extend the standard entailment rules from (Hayes,
2004; ter Horst, 2005; Motik et al., 2012) for pos-
itive binary relation instances in RDFS and OWL
towards modalized n-ary relation instances, includ-
ing negation. These entailment rules talk about,
e.g., subsumption, class membership, or transitiv-
ity, and have been found useful in many applica-
tions. The proposed solution has been implemented
for the binary relation case (extended triples, quads)
in HFC (Krieger, 2013), a forward chaining engine
that builds Herbrand models which are compatible
with the open-world view underlying OWL.
Our approach is clearly not restricted to medi-
cal statements, but is applicable to graded statements
in general, e.g., in technical diagnosis (the engine
is probably overheated) or in everyday conversation
(I’m pretty sure that Joe has signed a contract with
Krieger, H-U.
Capturing Graded Knowledge and Uncertainty in a Modalized Fragment of OWL.
DOI: 10.5220/0005628100190030
In Proceedings of the 8th International Conference on Agents and Artificial Intelligence (ICAART 2016) - Volume 2, pages 19-30
ISBN: 978-989-758-172-4
2016 by SCITEPRESS Science and Technology Publications, Lda. All r ights reserved
Foo Inc.), involving trust (I’m not an expert, but ...)
which can be seen as the common case (contrary to
true universal statements).
We note here that the names of our initial modal op-
erators were inspired by the qualitative information
parts of diagnostic statements from (Schulz et al.,
2014) as shown in Figure 1.
Figure 1: Schematic mappings of the qualitative infor-
mation parts excluded (E), unlikely (U), not excluded (N),
likely (L), and confirmed (C) to confidence intervals. Picture
taken from (Schulz et al., 2014).
These qualitative parts were used in medical state-
ments about, e.g., liver inflammation with varying
levels of detail (Schulz et al., 2014) in order to infer,
e.g., if Hepatitis is confirmed then Hepatitis is likely
but not Hepatitis is unlikely. And if Viral Hepatitis B
is confirmed, then both Viral Hepatitis is confirmed
and Hepatitis is confirmed (generalization). Things
“turn around” when we look at the adjectival modi-
fiers excluded and unlikely: if Hepatitis is excluded
then Hepatitis is unlikely, but not Hepatitis is not ex-
cluded. Furthermore, if Hepatitis is excluded, then
both Viral Hepatitis is excluded and Viral Hepatitis B
is excluded (specialization).
(Schulz et al., 2014) consider five OWL encod-
ings, from which only two were able to fully repro-
duce the plausible inferences for the above Hepati-
tis use case. The encodings in (Schulz et al., 2014)
were quite cumbersome as the primary interest was to
stay within the limits of the underlying calculus. Be-
sides coming up with complex encodings, only minor
forms of reasoning were possible, viz., subsumption
reasoning. Furthermore, each combination of disease
and qualitative information part required a new OWL
class definition/new class name, and there exist a lot
of them!
These disadvantages are a result of two conscious
decisions: OWL only provides unary and binary re-
lations (concepts and roles) and comes up with a
(mostly) fixed set of entailment/tableaux rules.
In our approach, however, the qualitative informa-
tion parts from Figure 1 are first class citizens of the
object language (the modal operators) and diagnostic
statements from the Hepatitis use case are expressed
through the binary property suffersForm between p
(patients, people) and d (diseases, diagnoses). The
plausible inferences are then simply a byproduct of
the instantiation of the entailment rule schemas (G)
from Section 5.1, and (S1) and (S0) from Section 5.2
for property suffersForm (the rule variables are uni-
versally quantified; > = universal truth; C = con-
firmed; L = likely), e.g.,
(S1) ViralHepatitisB v ViralHepatitis
>ViralHepatitisB(d) >ViralHepatitis(d)
(G) CsuffersFrom(p, d) LsuffersFrom(p,d)
Two things are worth mentioning here. Firstly,
not only OWL properties can be graded, such as
CsuffersFrom(p, d) (= it is confirmed that p suf-
fers from d), but also class membership, e.g.,
CViralHepatitisB(d) (= it is confirmed that d is of
type Viral Hepatitis B). As the original OWL exam-
ple from (Schulz et al., 2014) can not make use of
any modals, we employ the special modal > here:
>ViralHepatitisB(d). Secondly, modal operators are
only applied to assertional knowledge (the ABox in
OWL)—neither TBox nor RBox axioms are being af-
fected by modals in our approach, as they are sup-
posed to express universal truth.
We address the confidence of an asserted (medical)
statement (Schulz et al., 2014) through graded modal-
ities applied to propositional formulae: E (excluded),
U (unlikely), N (not excluded), L (likely), and C (con-
firmed). For various (technical) reasons, we add a
wildcard modality ? (unknown), a complementary
failure modality ! (error), plus two further modali-
ties to syntactically state definite truth and falsity: >
(true) and (false).
Let 4 now denotes the set of all modalities:
4 := {?,!,>,, E,U,N,L,C}
A measure function
µ : 4 7→ [0,1] × [0, 1]
We also call > and propositional modals as they lift
propositional statements to the modal domain. We refer to ?
and ! as completion modals since they complete the modal
hierarchy by adding unique most general and most specific
elements (see Section 4.3).
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
is a mapping which returns the associated confidence
interval µ(δ) = [l,h] for a modality from δ 4 (l
h). We presuppose that
µ(?) = [0,1]
µ(>) = [1,1]
µ() = [0,0]
µ(!) =
In addition, we define two disjoint subsets of 4,
1 := {>,C,L, N}
0 := {⊥,E,U }
and again make a presupposition: the confidence in-
tervals for modals from 1 end in 1, whereas the confi-
dence intervals for 0 modals always start with 0. It is
worth noting that we do not make use of µ in the syn-
tax of the modal language (for which we employ the
modalities from 4), but in the semantics when deal-
ing with the satisfaction relation of the model theory
(see Section 4).
We have talked about confidence intervals now
several times without saying what we actually mean
by this. Suppose that a physician says that it is con-
firmed (= C) that patient p suffers from disease d,
for a set of observed symptoms (or evidence) S =
,. ..,S
}: CsuffersFrom(p,d).
Assuming that a different patient p
shows the
same symptoms S (and only S, and perhaps further
symptoms which are, however, independent from S),
we would assume that the same doctor would diag-
nose CsuffersFrom(p
Even an other, but similar trained physician is
supposed to grade the two patients similarly. This
similarity which originates from patients showing the
same symptoms and from physicians being taught at
the same medical school is addressed by confidence
intervals and not through a single (posterior) prob-
ability, as there are still variations in diagnostic ca-
pacity and daily mental state of the physician. By us-
ing intervals (instead of single values), we can usually
reach a consensus among people upon the meaning of
gradation words, even though the low/high values of
the confidence interval for, e.g., confirmed might de-
pend on the context.
Being a bit more theoretic, we define a confidence
interval as follows. Assume a Bernoulli experiment
(Krengel, 2003) that involves a large set of n patients
Recall that intervals are (usually infinite) sets of real
numbers, together with an ordering relations (e.g., < or )
over the elements, thus
0 is a perfect, although degraded
P, sharing the same symptoms S. W.r.t. our exam-
ple, we would like to know whether suffersFrom(p,d)
or ¬suffersFrom(p,d) is the case for every patient
p P, sharing S. Given a Bernoulli trials sequence
X = (X
,. ..,X
) with indicator random variables X
{0,1} for a patient sequence (p
,. .., p
), we can ap-
proximate the expected value E for suffersFrom being
true, given disease d and background symptoms S by
the arithmetic mean A:
X] A[
X] =
Due to the law of large numbers, we expect that
if the number of elements in a trials sequence goes
to infinity, the arithmetic mean will coincide with the
expected value:
X] = lim
Clearly, the arithmetic mean for each new finite
trials sequence is different, but we can try to locate
the expected value within an interval around the arith-
metic mean:
X] [A[
X] ε
X] + ε
For the moment, we assume ε
= ε
, so that A[
is in the center of this interval which we will call from
now on confidence interval.
Coming back to our example and assuming
µ(C) = [0.9,1], CsuffersFrom(p,d) can be read as
being true in 95% of all cases known to the physi-
cian, involving patients p potentially having disease
d and sharing the same prior symptoms (evidence)
,. ..,S
The variance of ±5% is related to varying diag-
nostic capabilities between (comparative) physicians,
daily mental form, undiscovered important symptoms
or examinations which have not been carried out (e.g.,
lab values), or perhaps even by the physical stature of
the patient (crooked vs. upright) which unconsciously
affects the final diagnosis, etc, as elaborated above.
Thus the individual modals from 4 express (via µ)
different forms of the physician’s confidence, depend-
ing on the set of already acquired symptoms as (po-
tential) explanations for a specific disease.
Let C denote the set of constants that serve as the
arguments of a relation instance. For instance, in
Capturing Graded Knowledge and Uncertainty in a Modalized Fragment of OWL
an RDF/OWL setting, C would exclusively consist
of XSD atoms, blank nodes, and URIs/IRIs. In
order to define basic n-ary propositional formulae
(ground atoms), let p(~c) abbreviates p(c
,. ..,c
), for
,. ..,c
C, given length(~c) = n. In case the num-
ber of arguments does not matter, we sometimes sim-
ply write p, instead of, e.g., p(c,d) or p(~c). As before,
we assume 4 = {?,!,>,,E,U,N,L,C}. We induc-
tively define the set of well-formed formulae φ of our
modal language as follows:
φ ::= p(~c) | ¬φ | φ φ
| φ φ
| 4φ
4.1 Simplification and Normal Form
We now syntactically simplify the set of well-formed
formulae φ by restricting the uses of negation and
modalities to the level of propositional letters π:
π ::= p(~c) | ¬p(~c)
φ ::= π | 4π | φ φ
| φ φ
The design of this language is driven by two main
reasons: firstly, we want to effectively implement the
logic (in our case, in HFC), and secondly, the applica-
tion of the below semantic-preserving simplification
rules in an offline pre-processing step makes the im-
plementation easier and guarantees a more efficient
runtime system. To address negation, we first need
the notion of a complement modal δ
for every δ 4,
) := µ(δ)
= µ(?) \ µ(δ) = [0, 1] \ µ(δ)
I.e., µ(δ
) is defined as the complementary inter-
val of µ(δ) (within the bounds of [0,1], of course). For
example, E and N (excluded, not excluded) or ? and !
(unknown, error) are already existing complementary
We also require mirror modals δ
for every δ 4
whose confidence interval µ(δ
) is derived by “mir-
roring” µ(δ) to the opposite side of the confidence in-
terval, either to the left or to the right:
if µ(δ) = [l, 1] then µ(δ
) := [0, 1 l]
if µ(δ) = [0,h] then µ(δ
) := [1 h,1]
It is easy to see that these two equations can be
unified and generalized
if µ(δ) = [l, h] then µ(δ
) := [1 h,1 l]
For example, E and C (excluded, confirmed) or >
and (top, bottom) are mirror modals. In order to
This construction procedure comes in handy when
dealing with in-the-middle modals, such as fifty-fifty or per-
haps, whose confidence intervals neither touch 0 nor 1.
Such modals have a real background in (medical) diagnosis.
transform φ into its negation normal form, we need
to apply simplification rules a finite number of times
(until rules are no longer applicable). We depict those
rules by using the ` relation, read as formula ` sim-
plified formula (ε = empty word):
1. ?φ ` ε % ?φ is not informative at all
2. ¬¬φ ` φ
3. ¬(φ φ
) ` ¬φ ¬φ
4. ¬(φ φ
) ` ¬φ ¬φ
5. ¬4φ ` 4
φ (example: ¬Eφ = E
φ = Nφ)
6. φ ` 4
φ (example: E¬φ = E
φ = Cφ)
Clearly, the mirror modals δ
(δ 4) are not nec-
essary as long as we explicitly allow for negated state-
ments (which we do), and thus case 6 can, in princi-
ple, be dropped.
What is the result of simplifying 4(φ φ
) and
4(φ φ
)? Let us start with the former case and con-
sider as an example the statement about an engine that
a mechanical failure m and an electrical failure e is
confirmed: C(m e). It seems plausible to simplify
this expression to Cm Ce. Commonsense tells us
furthermore that neither Em nor Ee is compatible with
this description (we should be alarmed if, e.g., both
Cm and Em happen to be the case).
Now consider the “opposite” statement E(m e)
which must not be rewritten to Em Ee, as either Cm
or Ce is well compatible with E(m e). Instead, we
rewrite this kind of “negated” statement as Em Ee,
and this works fine with either Cm or Ce.
In order to address the other modal operators, we
generalize these plausible inferences by making a dis-
tinction between 0 and 1 modals (cf. Section 3):
7a. 0(φ φ
) ` 0φ 0φ
7b. 1(φ φ
) ` 1φ 1φ
Let us now focus on disjunction inside the scope
of a modal operator. As we do allow for the full set of
Boolean operators, we are allowed to deduce
8. 4(φφ
) ` 4(¬(¬(φφ
))) ` 4(¬(¬φ¬φ
)) `
(¬φ ¬φ
This is, again, a conjunction, so we apply schemas
7a and 7b, giving us
8a. 0
(φ φ
) ` 0
(¬φ ¬φ
) ` 1(¬φ ¬φ
) ` 1¬φ
` 1
φ 1
` 0φ 0φ
8b. 1(φ φ
) ` 1
(¬φ ¬φ
) ` 0(¬φ ¬φ
) ` 0¬φ
` 0
φ 0
` 1φ 1φ
Note how the modals from 0 in 7a and 8a act as a
kind of negation operator to turn the logical operators
into their counterparts, similar to de Morgan’s law.
The final case considers two consecutive modals:
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
9. δ
φ ` (δ
We interpret the operator as a kind of function
composition, leading to a new modal δ which is the
result of δ
. We take a liberal stance here of what
the result is, but indicate that it depends on the do-
main and, again, plausible inferences we like to cap-
ture. The operator will probably be different from
the related operation which is used in Section 5.3.4.
4.2 Model Theory
In the following, we extend the standard definition of
modal (Kripke) frames and models (Blackburn et al.,
2001) for the graded modal operators from 4 by em-
ploying the confidence function µ and focussing on
the minimal definition for φ. A frame F for the prob-
abilistic modal language is a pair
F = hW ,4i
where W is a non-empty set of worlds (or situations,
states, points, vertices, etc.) and 4 a family of binary
relations over W × W , called accessibility relations.
In the following, we overload δ 4 below in that we
let δ both refer to the modal in the syntax as well as
to the accessibility relation R
in the semantics.
A model M for the probabilistic modal language
is a triple
M = hF , V ,µi
such that F is a frame, V is a valuation, assigning
each proposition φ a subset of W , viz., the set of
worlds in which φ holds, and µ is a mapping, returning
the confidence interval for a given modality from 4.
Note that we only require a definition for µ in M (the
model, but not in the frame), as F represents the rela-
tional structure without interpreting the edge labeling
(the modal names) of the graph.
The satisfaction relation |=, given a model M and
a specific world w is inductively defined over the set
of well-formed formulae in negation normal form (re-
member π ::= p(~c) | ¬p(~c)):
1. M ,w |= p(~c) iff w V (p(~c)) and w 6∈ V (¬p(~c))
2. M ,w |= ¬p(~c) iff w V (¬p(~c)) and
w 6∈ V (p(~c))
3. M ,w |= φ φ
iff M , w |= φ and M ,w |= φ
4. M ,w |= φ φ
iff M , w |= φ or M ,w |= φ
5. for all δ 4: M ,w |= δπ iff
#{u | (w, u) δ and M , u |= π}
{u | (w,u) δ
The last case of the satisfaction relation addresses
the modals: for a world w, we look for the successor
states u that are directly reachable via δ and in which
π holds, and divide the number of such states (#·) by
the number of all worlds that are reachable from w in
the denominator. This number, lying between 0 and
1, is then required to be an element of the confidence
interval µ(δ) of δ in order to satisfy δπ, given M ,w.
It is worth noting that the satisfaction relation
above differs from the standard definition in its han-
dling of M ,w |= ¬p(~c), as negation is not inter-
preted through the absence of p(~c) (M ,w 6|= p(~c)),
but through the existence of ¬p(~c). This treatment
addresses the open-world nature in OWL and the
evolvement of a (medical) domain over time.
We also note that the definition of the satisfaction
relation for modalities (last clause) is related to the
possibility operators M
· (=
·; k N) introduced
by (Fine, 1972) and counting modalities · n (Areces
et al., 2010), used in modal logic characterizations of
description logics with cardinality restrictions.
4.3 Well-behaved Frames
As we will see later, it is handy to assume that the
graded modals are arranged in a kind of hierarchy—
the more we move along the arrows in the hierar-
chy, the more a statement φ in the scope of a modal
δ 4 becomes uncertain. In order to address this, we
slightly extend the notion of a frame by a third com-
ponent 4 × 4, a partial order (i.e., a reflexive,
antisymmetric, and transitive binary relation) between
F = hW ,4, i
Let us consider the following modal hierarchy that
we build from the set 4 of already introduced modals
(cf. Figure 1):
This graphical representation is just a compact
way to specify a set of 33 binary relation instances
over 4 × 4, such as > >, > N, C N, ?,
or ! ?. The above mentioned form of uncertainty is
expressed by the measure function µ in that the asso-
ciated confidence intervals become larger:
if δ δ
then µ(δ) µ(δ
In order to arrive at a proper and intuitive model-
theoretic semantics which mirrors intuitions such as
if φ is confirmed (Cφ) then φ is likely (Lφ), we will
Capturing Graded Knowledge and Uncertainty in a Modalized Fragment of OWL
focus here on well-behaved frames F which enforce
the existence of edges in W , given and δ,δ
if (w, u) δ and δ δ
then (w, u) δ
However, by imposing this constraint, we also
need to adapt the last case of the satisfiability relation
from Section 4.2 above:
5. for all δ 4: M ,w |= δπ iff
{u | (w,u) δ
and M , u |= π}
{u | (w,u) δ
Not only are we scanning for edges (w,u) labeled
with δ and for successor states u of w in which π holds
in the numerator (original definition), but also take
into account edges marked with more general modals
: δ
δ. This mechanism implements a kind of
built-in model completion that is not necessary in or-
dinary modal logics as they deal with only a single
relation (viz., unlabeled arcs).
We now turn our attention, again, to the syntax of our
language and to the syntactic consequence relation.
This section addresses a restricted subset of entail-
ment rules which will unveil new (or implicit) knowl-
edge from already existing graded statements. Re-
call that these kind of statements (in negation normal
form) are a consequence of the application of simpli-
fication rules as depicted in Section 4.1. Thus, we
assume a pre-processing step here that “massages”
more complex statements that arise from a represen-
tation of graded (medical) statements in natural lan-
guage. The entailments which we will present in a
moment can either be directly implemented in a tuple-
based reasoner, such as HFC (Krieger, 2013), or in
triple-based engines (e.g., Jena (Carroll et al., 2004)
or OWLIM (Bishop et al., 2011)) which need to reify
the medical statements in order to be compliant with
the RDF triple model.
5.1 Modal Entailments
The entailments presented in this section deal with
plausible inference centered around modals δ, δ
which are, in part, also addressed in (Schulz et al.,
2014) in a pure OWL setting. We use the implication
sign to depict the entailment rules
lhs rhs
which act as completion (or materialization) rules
the way as described in, e.g., (Hayes, 2004) and (ter
Horst, 2005), and used in today’s semantic reposito-
ries (e.g., OWLIM ). We sometimes even use the bi-
conditional to address that the LHS and the RHS
are semantically equivalent, but will indicate the di-
rection that should be used in a practical setting. As
before, we define
π ::= p(~c) | ¬p(~c)
We furthermore assume that for every modal δ
4, a complement modal δ
and a mirror modal δ
exist (cf. Section 4.1).
5.1.1 Lift
(L) π >π
This rule interprets propositional statements as spe-
cial modal formulae. It might be dropped and can be
seen as a pre-processing step. We have used it in the
Hepatitis example above. Usage: left-to-right direc-
5.1.2 Generalize
(G) δπ δ δ
This rule schema can be instantiated in various ways,
using the modal hierarchy from Section 4.3, e.g.,
>π Cπ, Cπ Lπ, or Eπ U π. It has been used
in the Hepatitis example.
5.1.3 Complement
(C) ¬δπ δ
In principle, (C) is not needed in case the statement is
already in negation normal form. This schema might
be useful for natural language paraphrasing (explana-
tion). Given 4, there are four possible instantiations:
Eπ ¬Nπ, Nπ ¬Eπ, ?π ¬!π, and !π ¬?π.
5.1.4 Mirror
(M) δ¬π δ
Again, (D) is in principle not needed as long as
the modal proposition is in negation normal form,
since we do allow for negated propositional state-
ments ¬p(~c). This schema might be useful for nat-
ural language paraphrasing (explanation). For 4,
there are six possible instantiations, viz., Eπ C¬π,
Cπ E¬π, Lπ U¬π, Uπ L¬π, >π ⊥¬π, and
π π.
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
5.1.5 Uncertainty
(U) δπ ¬δπ δπ δ
π ?π
The co-occurrence of δπ and ¬δπ does not imply log-
ical inconsistency (propositional case: π ¬π), but
leads to complete uncertainty about the validity of π.
Remember that µ(?) = µ(δ) ] µ(δ
) = [0, 1]:
0 1
µ : |δ
π π
Usage: left-to-right direction.
5.1.6 Negation
(N) δ(π ¬π) δπ δ¬π δπ δ
¬π δ
π δ
(π ¬π)
(N) shows that δ(π ¬π) can be formulated equiva-
lently by using the mirror modal δ
0 1
µ : |δ
|——|δ |
π ¬π π ¬π
In general, (N) is not the modal counterpart of the law
of non-contradiction, as π ¬π is usually afflicted by
uncertainty, meaning that from δ(π ¬π), we can not
infer that π ¬π is the case for the concrete example
in question (recall the intention behind the confidence
intervals; cf. Section 3). There is one notable excep-
tion, involving the > and modals. This is formu-
lated by the next entailment rule.
5.1.7 Error
(E) >(π¬π) (π ¬π) !(π ¬π) !π
(E) is the modal counterpart of the law of non-
contradiction (note:
= >, >
= , !
= !). For
this reason and by definition, the error (or failure)
modal ! from Section 3 comes into play here. The
modal ! can serve as a hint to either stop a compu-
tation the first time it occurs, or to continue reason-
ing and to syntactically memorize the ground literal
π. Usage: left-to-right direction.
5.2 Subsumption Entailments
As before, we define two subsets of 4, called 1 =
{>,C,L,N} and 0 = {⊥,E,U }, thus 1 and 0 effec-
tively become
1 = {>,C,L,N,U
} 0 = {⊥,U, E,C
due to the use of complement modals δ
and mirror
modals δ
for every base modal δ 4 and by as-
suming that E = N
, E = C
, U = L
, and = >
together with the four “opposite” cases.
Now, let v abbreviate relation subsumption as
known from description logics and realized through
rdfs:subClassOf and rdfs:subPropertyOf.
Given this, we define two further very practical and
plausible modal entailments which can be seen as the
modal extension of the entailment rules (rdfs9) and
(rdfs7) for classes and properties in RDFS (Hayes,
(S1) 1p(~c) p v q 1q(~c)
(S0) 0q(~c) p v q 0p(~c)
Note how the use of p and q switches in the an-
tecedent and the consequent, even though p v q holds
in both cases. Note further that propositional state-
ments π are restricted to the positive case p(~c) and
q(~c), as their negation in the antecedent will not lead
to any valid entailments.
Here are four instantiations of (S0) and (S1) for
the unary and binary case (remember, C 1 and E
ViralHepatitisB v ViralHepatitis
CViralHepatitisB(x) CViralHepatitis(x)
ViralHepatitis v Hepatitis
EHepatitis(x) EViralHepatitis(x)
deeplyEnclosedIn v containedIn
CdeeplyEnclosedIn(x, y) CcontainedIn(x, y)
superficiallyLocatedIn v containedIn
5.3 Extended RDFS & OWL
In this section, we will consider further entailment
rules for RDFS (Hayes, 2004) and a restricted subset
of OWL (ter Horst, 2005; Motik et al., 2012). Re-
member that modals only head positive and negative
propositional letters π, not TBox or RBox axioms.
Concerning the original entailment rules, we will dis-
tinguish four principal cases to which the extended
rules belong (we will only consider the unary and bi-
nary case here as used in description logics/OWL):
1. TBox and RBox axiom schemas will not undergo
a modal extension;
2. rules get extended in the antecedent;
3. rules take over modals from the antecedent to the
4. rules aggregate several modals from the an-
tecedent in the consequent.
We will illustrate the individual cases in the fol-
lowing subsections with examples by using a kind of
description logic rule syntax. Clearly, the set of ex-
tended entailments depicted here is not complete.
Capturing Graded Knowledge and Uncertainty in a Modalized Fragment of OWL
5.3.1 Case-1: No Modals
Entailment rule (rdfs11) from (Hayes, 2004) deals
with class subsumption: C v D D v E C v E.
As this is a terminological axiom schema, the rule
stays constant in the modal domain. Example rule
ViralHepatitisB v ViralHepatitis
ViralHepatitis v Hepatitis
ViralHepatitisB v Hepatitis
5.3.2 Case-2: Modals on LHS, No Modals on
The following original rule (rdfs3) from (Hayes,
2004) imposes a range restriction on objects of binary
ABox relation instances: P.C P(x,y) C(y). The
extended version needs to address the ABox proposi-
tion in the antecedent (don’t care modal δ), but must
not change the consequent (even though we always
use the > modality here—the range restriction C(y) is
always true, independent of the uncertainty of P(x, y);
cf. Section 2 example):
(Mrdfs3) P.C δP(x,y) >C(y)
Example rule instantiation:
suffersFrom.Disease LsuffersFrom(x,y)
5.3.3 Case-3: Keeping LHS Modals on RHS
Inverse properties switch their arguments
(ter Horst, 2005) as described by (rdfp8):
P(x,y) Q(y,x). The extended ver-
sion simply keeps the modal operator:
(Mrdfp8) P Q
δP(x,y) δQ(y,x)
Example rule instantiation:
containedIn contains
CcontainedIn(x, y)
5.3.4 Case-4: Aggregating LHS Modals on RHS
Now comes the most interesting case of modalized
RDFS & OWL entailment rules, that offers several
possibilities on a varying scale between skeptical and
credulous entailments, depending on the degree of un-
certainty, as expressed by the measuring function µ of
the modal operator. Consider the original rule (rdfp4)
from (ter Horst, 2005) for transitive properties:
v P P(x,y) P(y,z) P(x,z).
Now, how does the modal on the RHS of the ex-
tended rule look like, depending on the two LHS
modals? There are several possibilities. By operat-
ing directly on the modal hierarchy, we are allowed to
talk about, e.g., the least upper bound or the greatest
lower bound of δ
and δ
. When taking the associ-
ated confidence intervals into account, we might play
with the low and high numbers of the intervals, say,
by applying min/max, the arithmetic mean or even by
multiplying the corresponding numbers.
Let us first consider the general rule from which
more specialized versions can be derived, simply by
instantiating the combination operator :
(Mrdfp4) P
v P δ
P(x,y) δ
Here is an instantiation of Mrdfp4, dealing with the
transitive relation contains from above, assuming that
reduces to the least upper bound (i.e., C L = L):
Ccontains(x, y) Lcontains(y,z)
What is the general result of δ
? It depends,
probably both on the application domain and the epis-
temic commitment one is willing to accept about the
“meaning” of gradation words/modal operators. To
enforce that is at least both commutative and as-
sociative (as is the least upper bound) is probably a
good idea, making the sequence of modal clauses or-
der independent. And to work on the modal hierarchy
instead of combining low/high numbers of the cor-
responding intervals is probably a good decision for
forward chaining engines, as the latter strategy might
introduce new individuals through operations such as
multiplication, thus posing a problem for the imple-
mentation of the generalization schema (G) (see Sec-
tion 5.1.2).
5.4 Custom Entailments: An Example
from the Medical Domain
Consider that Hepatitis B is an infectious disease
ViralHepatitisB v InfectiousDisease v Disease
and note that there exist vaccines against it. Assume
that the liver l of patient p quite hurts
ChasPain(p, l),
but p has been definitely vaccinated against Hepatitis
B before:
>vaccinatedAgainst(p, ViralHepatitisB).
We apply OWL2-like punning here when using the
class ViralHepatitisB (not an instance), as the second
argument of vaccinatedAgainst; cf. (Golbreich and
Wallace, 2012).
Given that p received a vaccination, the follow-
ing custom rule will not fire (x, y below are now
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
universally-quantified variables; z an existentially-
quantified RHS-only variable):
>Patient(x) >Liver(y) ChasPain(x,y)
NViralHepatitisB(z) NsuffersFrom(x,z)
Now assume another person p
that is pretty sure (s)he
was never vaccinated:
Given the above custom rule, we are allowed to infer
that (h instantiation of z)
NViralHepatitisB(h) NsuffersFrom(p
The subclass axiom from above thus assigns
so that we can query for patients for whom an infec-
tious disease is not excluded (= N), in order to initiate
appropriate methods (e.g., further medical investiga-
5.5 Implementing Modal Entailments
The negation normal form from Section 4.1 makes
it relatively easy to implement entailment rules in-
volving modalized propositional letters of the form
δ ± p(~c). ± is a polarity value as known from situ-
ation theory (Devlin, 2006) in order to make negative
property assertions available in the object language.
We have implemented a modalized extension of
the RDFS and OWL rule sets (Hayes, 2004; ter Horst,
2005) by employing the tuple-based rule engine HFC
(Krieger, 2012; Krieger, 2013). Without loss of gen-
erality, let us focus here on the positive case for the
three binary entailment schemas from Section 5.3.2,
5.3.3, and 5.3.4 and their HFC rule representation, as
negation inside the scope of a modal can be rewrit-
ten using the mirror modal, thus turning the quintuple
into a quad (rule variables start with a ?):
(Mrdfs3) P.C δP(x,y) >C(y)
?p rdfs:range ?c
?modal ?x ?p ?y
mod:T ?y rdf:type ?c
(Mrdfp8) P Q
δP(x,y) δQ(y,x)
?p owl:inverseOf ?q
?modal ?x ?p ?y
?modal ?y ?q ?x
(Mrdfp4) P
v P δP(x,y) δ
(δ δ
?p rdf:type owl:TransitiveProperty
?modal1 ?x ?p ?y
?modal2 ?y ?p ?z
?modal ?x ?p ?z
?modal = CombineModals ?modal1 ?modal2
Triple-based engines, such as OWLIM clearly
need to reify such extended descriptions (expensive;
no termination guarantee). Even more important, ad-
ditional tests going beyond simple symbol match-
ing and function calls, such as CombineModals (the
equivalent to in the abstract syntax) in the HFC
version of (Mrdfp4) above, are rarely available in to-
day’s RDFS/OWL reasoning engines, thus making it
impossible for them to implement such modal entail-
We finally describe how the implementation of the
generalization schema (G) (Section 5.1.2) works. As
explained in Section 4.3, the modal operators δ are
arranged in a modal hierarchy that is based on the in-
clusion of their confidence intervals µ(δ). This hierar-
chy is realized in OWL through a subclass hierarchy,
using rdfs:subClassOf to implement :
(G) δP(x,y) δ δ
?modal1 ?x ?p ?y
?modal1 rdfs:subClassOf ?modal2
?modal2 ?x ?p ?z
The two modalities and from standard modal
logic are often called dual as they can be defined in
terms of each other: φ ¬¬φ and φ ¬¬φ,
resp. At first sight, it seems that our non-standard
modal logic is missing a similar property, as we origi-
nally dealt with five modal operators, extended by the
propositional modals > and , and the completion
modals ? and !. For every such modal δ, we can fur-
thermore think of additional complement modals δ
and additional mirror modals δ
whose confidence
intervals µ(δ
) and µ(δ
) can be derived from µ(δ)
(cf. Section 4.1). Some of these modals coincide with
original modals from 4, others do not have a direct
counterpart. However, the confidence intervals for
the “anonymous” modals can be trivially computed
by applying the two equations from Section 4.1.
Coming back to the question of whether dual
modals exist for every δ 4, we need to simplify
¬δ¬φ by applying the schemas from Section 4.1. We
can either start with the inner or with the outer nega-
tion, resulting in either mirror modals or complement
Capturing Graded Knowledge and Uncertainty in a Modalized Fragment of OWL
modals. Interestingly, the resulting confidence inter-
vals at which we reach in the end are the same, and
this is clearly a good point and desirable, as simpli-
fication is supposed to be an order-independent pro-
/ \
¬φ ¬δ
| |
φ δ
Thus, δ
, for every δ 4 which can be
shown by applying the definitions for complement
and mirror modals from Section 4.1. The deeper rea-
son why this is so is related to the inherent properties
of the two operations complementation and mirror-
ing. Contrary to complement and mirror modals, dual
modals δ
are either supersets or subsets of µ(δ), i.e.,
if δ is a 1- or 0-modal, so is δ
It is worth noting to state that this paper is interested
in the representation of and reasoning with uncertain
assertional knowledge, and neither in dealing with
vagueness/fuzziness found in natural language (very
small, hot), nor in handling defaults and exceptions in
terminological knowledge (penguins can’t fly).
To the best of our knowledge, the modal logic pre-
sented in this paper uses for the first time modal op-
erators for expressing the degree of (un)certainty of
propositions. These modal operators are interpreted
in the model theory through confidence intervals via
measure function µ. From a model point of view,
our modal operators are related to counting modali-
(Fine, 1972; Areces et al., 2010). However,
for M ,w |= δπ to be the case, we do not require a fixed
number k N of reachable successor states (absolute
frequency), but instead divide the number of worlds
reached through label δ 4 and in which π holds by
the number of all directly reachable worlds, yielding
fraction 0 p 1. This number then is further con-
strained by requiring p µ(δ) (relative frequency), as
defined in case 5 of the satisfaction relation in Section
4.2 and extended in Section 4.3.
As (Wikipedia, 2015) precisely put it: ... what
axioms and rules must be added to the propositional
calculus to create a usable system of modal logic is
a matter of philosophical opinion, often driven by
the theorems one wishes to prove ...”. Clearly, the
logic presented here is no exception and its design is
driven by commonsense knowledge and plausible in-
ferences, we try to capture and generalize. In a strict
sense, it is a non-standard modal logic in that it is not
an instance of the normal modal logic K = (N) + (K)
(N) p p
(K) (p q) (p q)
as the necessitation rule (N) and the distribution ax-
iom (K) does not hold for every δ 4. However, we
can show that restricted generalized forms of these
axioms are in fact the case for our logic (1
1-modals whose low value is 0.5 and 0
are 0-
modals whose high value is 0.5):
(N1) p 1p
(N0) ¬p 0p
) 1
(p q) (1
p 1
) 0
(p q) (0
p 0
In addition, the well-behaved frames condition
(Section 4.3) generalizes the seriality condition (D)
on frames and a kind of forward monotonicity, we
would like to keep for an evolving domain, is directly
related to transitivity (4) of the accessibility relations
from 4 in F :
(D) δp δ δ
(4) δp δδp
Several approaches to representing and reason-
ing with uncertainty have been investigated in Ar-
tificial Intelligence; see (Halpern, 2003) for a (bi-
ased) overview. (Halpern, 1990) was probably the
first attempt of a first-order logic which unifies prob-
ability distributions over classes and individuals.
Weaker decidable propositional formalisms such as
Bayesian Networks (Pearl, 1988) and related prob-
abilistic graphical models (Koller and Friedmann,
2009) have found their way into causal (medical)
reasoning (Lucas et al., 2004). Programming lan-
guages for these kind of models exist; e.g., Alchemy
for Markov Logic Networks (Richardson and Domin-
gos, 2006). In Markov Logic, first-order formulae
are associated with a numerical value which softens
hard first-order constraints and a violation makes a
possible world not impossible, but less probable (the
higher the weight, the stronger the rule). For example,
the Markov Logic rule smoking causes cancer with
weight 1.5 (Richardson and Domingos, 2006, p. 111)
1.5 : x. smokes(x) hasCancer(x)
might be approximated in our approach through the
use of modals:
>smokes(x) LhasCancer(x)
Very less so has been researched in the Descrip-
tion Logic community (as it is smaller) and little or
nothing of this research has find its way into imple-
mented description logic systems. As we focus in
this paper on a modalized extension of OWL, let us
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
4 meaning confidence belief disbelief uncertainty
! error
0 0.5 0.5 0
false [0,0] 0 1 0
E excluded [0,0.1] 0 0.9 0.1
U unlikely [0,0.3] 0 0.7 0.3
PN perhaps not [0.4,0.5] 0.4 0.5 0.1
FF fifty-fifty [0.45,0.55] 0.45 0.45 0.1
P perhaps [0.5,0.6] 0.5 0.4 0.1
N not excluded [0.1, 1] 0.3 0 0.7
L likely [0.7,1] 0.7 0 0.3
C confirmed [0.9,1] 0.9 0 0.1
> true [1,1] 1 0 0
? unknown [0,1] 0 0 1
Figure 2: Representation of modal operators from 4 (incl. three in-the-middle modals) in terms of opinions in Subjective
Logic. The confidence intervals for the five initial modals roughly coincide with the numbers depicted in Figure 1.
review here some of the work carried out in descrip-
tion logics. (Heinsohn, 1993) and (Jaeger, 1994) con-
sider uncertainty in ALC concept hierarchies, plus
concept typing of individuals (unary relations) in dif-
ferent ways (probability values vs. intervals; condi-
tional probabilities in TBox vs. TBox+ABox). They
do not address uncertain binary (or even n-ary) rela-
tions. (Tresp and Molitor, 1998) investigates vague-
ness in ALC concept descriptions to address state-
ments, such as the patient’s temperature is high, but
also for determining membership degree (38.5 °C).
This is achieved through membership manipulators
which are functions, returning a truth value between 0
and 1, thus deviating from a two-valued logic. (Strac-
cia, 2001) defines a fuzzy extension of ALC , based
on Zadeh’s Fuzzy Logic. As in (Tresp and Moli-
tor, 1998), the truth value of an assertion is replaced
by a membership value from [0, 1]. ALC assertions
α in (Straccia, 2001) are made fuzzy by writing,
e.g., hα ni, thus taking a single truth value from
[0,1]. An even more expressive theoretical descrip-
tion logic, Fuzzy OWL, based on OWL DL, is inves-
tigated in (Stoilos et al., 2005).
Our work might also be viewed as a modalized
version of a restricted fragment of Subjective Logic
(Jøsang, 1997; Jøsang, 2001), a probabilistic logic
that can be seen as an extension of Dempster-Shafer
belief theory (Wilson, 2000). Subjective Logic ad-
dresses subjective believes by requiring numerical
values for believe b, disbelieve d, and uncertainty u,
called (subjective) opinions. For each proposition, it
is required that b + d + u = 1.
The translation from modals δ to hb,d,ui is deter-
mined by the length of the confidence interval µ(δ) =
[l,h] and its starting/ending numbers, viz., u := h l,
b := l, and d := 1 h (cf. Figure 2).
These definitions also address in-the-middle
modals (cf. footnote 3). Such modals even do not
need to be symmetrical, i.e., being around the center
of the confidence interval. The definitions are clearly
not applicable to the error modal ! (cf. Section 5.1.7)
and it makes perfect sense to assume u = 0 here (re-
member, µ(!) =
0), and thus bisecting the belief mass
for this corner case, i.e., b = 0.5 and d = 0.5.
The simplification and entailment rules of the for-
malism (Sections 4.1 and 5) allow rule-based (for-
ward) engines to easily implement this conservative
extension of OWL. Through these rules, the formal-
ism is compositional by nature and thus afflicted with
all the problems, reviewers have already noted on the
interplay between logic and uncertainty (Dubois and
Prade, 1994). Due to the finite number of modal oper-
ators, the approach is only able to approximately com-
pute the degree of uncertainty of new knowledge in-
stead of giving more precise estimations, by combin-
ing the low/high numbers of the confidence intervals
through min/max, multiplication, addition, etc. Con-
trary to other approaches, we do not talk about the un-
certainty of complex propositions (conjunction, dis-
junction) or sets of beliefs, but instead focus merely
on the uncertainty of atomic ABox propositions.
The research described in this paper has been funded
by the German Federal Ministry of Education and
Research (BMBF) through the project HySocia-
Tea (Hybrid Social Teams for Long-Term Collab-
oration in Cyber-Physical Environments, grant no.
01IW14001). I have profited from discussions with
my colleagues Miroslav Jan
cek, Bernd Kiefer, and
Stefan Schulz and would like to thank the ICAART
reviewers for their detailed and useful suggestions—
thank you guys!
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