Datalog for Inconsistency-tolerant Knowledge Engineering
Hendrik Decker
Instituto Tecnol´ogico de Inform´atica, Universidad Polit´ecnica de Valencia, Valencia, Spain
Inconsistency Tolerance, Datalog, Integrity.
Inconsistency tolerance is widely discussed and accepted in the scientific community of knowledge engineer-
ing. From a principled, theoretical point of view, however, the fundamental conflict of sound reasoning with
unsound data has remained largely unresolved. The vast majority of applications that need inconsistency tol-
erance either does not care about a firm theoretical underpinning, or recurs on non-standard logics, or superfi-
cially refers to well-established classical foundations. We argue that hardly any of these paradigms will survive
in the long run. We defend the position that datalog (Abiteboul et al., 1995), including integrity constraints,
is a viable candidate for a sound and robust foundation of inconsistency-tolerant knowledge engineering. We
line our argument by a propaedeutic glance at the history of issues related to inconsistency.
In computing, the term “inconsistency tolerance” de-
nominates the capacity of software systems to pro-
cess data correctly even if the data are inconsistent.
In knowledge engineering (abbr. KE), which com-
prises variouskinds of automated reasoning with data,
inconsistency tolerance also means to produce valid
conclusions from inconsistent data, and the capability
to make guarantees about the correctness of results in
the presence of inconsistency.
KEsubsumes subfields of database management
such as query answering, updating, integrity manage-
ment, the evolution and integration of schemas and
ontologies, etc. These are the fields of interest in this
paper. Due to limitations of space and time, we do
not deal here with KE subfields such as requirements
engineering, knowledge acquisition and conceptual
In the KE literature, inconsistency tolerance is
widely discussed and accepted as a desirable feature
(Chopra and Parikh, 1999; Nuseibeh et al., 2000;
Koogan Breitman et al., 2003; Bertossi et al., 2005;
Calvanese et al., 2008; Imam and MacCaull, 2009;
Qi et al., 2009; Hinrichs et al., 2009; Dunnei et al.,
2009; Calvanese et al., 2012).
Unfortunately, however, solid formal foundations
are largely missing. That is deplorable, since a lack
of firm foundations of any technical approach always
tends to abet doubts in its validity and universality.
Occasionally, such foundations are considered an
unnecessary luxury or a practically irrelevant play-
ground for egghead theoreticians. Such an attitude
is frequently encountered with “application-oriented”
people (who might very well have an admirable tal-
ent of producing amazing special-purpose solutions
for intricate problems). However, solutions that are
not grounded on theoretical foundations that have
withstood the test of time tend to suffer the fate of
most ad-hoc solutions: they lack generalizability, are
hard to maintain, difficult to evolve and become old-
fashioned soon after their noveltyappeal has worn off.
Less lamentable, perhaps, are those proposals that
favour some non-standardlogic for capturing the prin-
ciples that underly automated reasoning in the pres-
ence of inconsistency. Calculi that are paraconsistent,
multi-valued, annotated, probabilistic or possibilistic
are among the most frequently used technical means
to provide a formal framework for consistent reason-
ing with inconsistency. Yet, those logics are largely
divergent, and none of them has ever attained a status
of acceptance that could be called a standard.
Several other approaches simply are content with
relying more or less explicitly on conventional first-
order predicate logic as a theoretical underpinning for
their ways to cope in a reasonable way with incon-
sistency. However, they usually ignore or pass by the
devastating effects that a deployment of full-fledged
classical logic can have, due to the principle known as
ex contradictione sequitur quodlibet (ECQ), i.e., that
everything, and thus nothing reasonable at all, can be
inferred from inconsistency.
In fact, the problem is not just to find workarounds
for avoiding the explosive effects of inconsistency.
The deep problem is that a foundation of inconsis-
tency tolerance must provide a meta-level for rea-
Decker H..
Datalog for Inconsistency-tolerant Knowledge Engineering.
DOI: 10.5220/0004172202960301
In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2012), pages 296-301
ISBN: 978-989-8565-30-3
2012 SCITEPRESS (Science and Technology Publications, Lda.)
soning about reasoning in the presence of inconsis-
tency on the object-level. Such meta-level reasoning
is bound to recur on irrefutable principles of rational
reflection and argumentation that do not tolerate in-
consistency, as long as it argues in its own defense.
In particular, meta-level reasoning may insist on fun-
damental principles of logic such as the law of non-
contradiction (LNC) (i.e., no statement can be both
true and false), and reductio ad absurdum (RaA) (i.e.,
to infer the negation of a hypothesis that would lead
to a contradiction), which, at the same time, are abdi-
cated on the object-level. In other words, reasoning
consistently with inconsistent data risks to be self-
In Section 2, we take a propaedeutic look on the
historical roots of LNC, RaA and ECQ, and their re-
lationship to inconsistency. In Section 3, we argue
why datalog, augmented with integrity constraints, is
not just a lucky compromise or a pragmatically con-
venient tool, but an appropriate candidate for a sound
and robust foundation of inconsistency-tolerant KE. It
respects and deploys LNC and RaA while tolerating
inconsistency and avoiding the application of ECQ. In
Section 4, we conclude.
In logic, consistency is understood as the absence of
contradiction, and inconsistency as the presence of
contradiction. LNC denunciates inconsistency as il-
logical. Consistency and LNC have played constitu-
tional roles in western philosophy, logic and compu-
tation. Aristotle and Kant took LNC to be a condi-
tio sine qua non of all reasoning. In mathematical
logic, LNC has had a solid standing since Leibniz
formalized it as the fundamental principle of human
comprehension. Frege attempted to base mathemat-
ics on LNC, defining the elementary number 0 as the
cardinality of the set of true contradictions. Hilbert
and G¨odel required proofs of consistency as the most
desirable property of any mathematical theory. The
semantic consistency of stored data, a.k.a. integrity,
is a key requirement in most database systems. The
vast majority of established proof procedures, includ-
ing abductive ones, implicitly or explicitly use LNC
for making valid inferences.
Also inconsistency is foundational in many philo-
sophical, logical and computational systems. Herak-
lit, Zenon, Plato, Hegel, C.G. Jung and others have
taken inconsistency (as embodied by contradictions,
paradoxes, dialectic aporias, thesis/antithesis or op-
posed polarities) as a constitutive element of human
cognition. Popper recalled the ancient wisdom that
universal sentences can never be proved by experi-
ence, but only be falsified by contradiction. More-
over, as already indicated, inconsistency is constitu-
tional in the principle of RaA, which has been a ven-
erable inference rule since the ancient Greeks. Many
automated theorem provers are based on RaA, i.e.,
proving a sentence by showing its negation to be in-
consistent. Nowadays, data mining may infer use-
ful information from detected contradictions, e.g., for
information integration (M¨uller et al., 200), decision
support (Padmanabhan and Tuzhilin, 2002), or identi-
fying fraudulent tax declarations (Bonchi et al., 1999;
Yu et al., 2003).
On the other hand, LNC has not always en-
joyed unanimous approval. Aristotle discussed his
own doubts about LNC. Medieval and modern-day
philosophers became aware of the potentially devas-
tating logical effects of any violation of LNC by the
ECQ principle, by which anything (thus, nothing use-
ful at all) can be derived from contradiction. Similar
to abandoningthe axiom of parallels in non-Euklidian
geometry, Peirce, Lukasiewiczand Post contemplated
to abandon LNC, and Vasiliev effectively did so, in
systems of non-Aristotelian logic.
Russell shattered Frege’s LNC-based attempt by
showing nıve set theory to be inconsistent. That,
and the inflationary effects of ECQ ignited Orlov
and Lewis to tame ECQ by introducing less powerful
forms of implication. The more radical approaches
proposed by Jakowski, da Costa and others rejected
the universality of LNC or, in some cases, the law of
excluded middle (LEM), a.k.a. tertium non datur, in
order to avoid ECQ.
According to (Diamond, 1976), Wittgenstein de-
nied the malignance of LNC and ECQ, by (1) qualify-
ing known and unknowninconsistencies as innocuous
(“When an inconsistency comes out into the open it
can do no harm” and “As long as its hidden an incon-
sistency is as good as gold”), (2) suggesting a kind of
exception handling for known inconsistency (“If an
inconsistency were to arise (. . . ), all we have to do
is to make a new stipulation to cover the case where
the rules conflict and the matters resolved””), and
(3) proposing to confine inconsistency by not draw-
ing any conclusions from it (“You might get p.p by
means of Freges system. If you can draw any conclu-
sion you like from it, then (. . . ) I would say, ‘Well,
then, just dont draw any conclusions from a contra-
By common standards, Wittgenstein’s attitude to-
ward inconsistency seems frivolous. In fact, (1)
does not consider that inconsistent data which are not
known to be inconsistent may be used bona fide to
derive possibly fatal consequences; (2) does not take
into account that exception handling easily gets out
of hand as the number of exceptions grows; finally,
(3) can only be qualified as a much too careless state-
ment, since wrong conclusions can be drawn from
data involved in contradiction “without actually go-
ing through the contradiction”, as remarked by Turing
(Diamond, 1976).
Nevertheless, a philosophical debate has been go-
ing on about the untenability or, resp., the justifiabil-
ity of Wittgenstein’s enunciations on contradictions
and inconsistency. For instance, see (Chihara, 1977;
Wright, 1980; Wrigley, 1980; Caruana, 2004; Bagni,
2008; Decker, 2010). Basically, the positions are that,
either Wittgenstein’s arguments are not cogent, un-
convincing, vague, besides the point, if not plainly
wrong, or that his utterances should be appreciated
in a pragmatic sense, or in the context of his own
mindset and certain tendencies of his times. Any-
way, Wittgensteins suggestion to refrain from infer-
ring arbitrary conclusions from contradictions remain
incomplete, since he did not hint at any systematic
approach of how to achieve that.
In this section, we are going to see that Wittgensteins
recommendation of how to confine inconsistency is
complied with easily, in the framework of datalog.
In datalog, each theory T is usually represented
by a set of Horn clauses of the form H B, where
H either is a positive literal called head or is empty,
and B is a possibly empty conjunction of positive lit-
erals called body. H can be inferred in T if B can be
inferred in T. Atoms that do not match any literal in
the head of any clause in T cannot be inferred in T.
Thus, as opposed to ECQ, there is no way to derive
any arbitrary conclusion from T.
Each such T in datalog is partitioned into a
database D and an integrity theory IC. For each clause
in D, its head is not empty. Each clause in IC, called
integrity constraint (or, in short, constraint), is repre-
sented as a denial, i.e., a clause with empty head, from
which nothing can be inferred. The body of each con-
straint expresses a condition that should not hold. If
it does, then the database is inconsistent. Thus, in-
consistency is syntactically hedged in datalog: incon-
sistency of T = D IC means that some I in IC is
violated, i.e., the body of I can be inferred in D. And
nothing more can be derived from that.
This way of representing inconsistency in data-
log is sometimes emphasized by rewriting each de-
nial B in IC as violated B, where violated is a
distinguished 0-ary predicate that does not occur in
the body of any clause in IC nor in any clause of D.
Thus, in each datalog theory T, inferences are im-
mune against any inconvenience associated with in-
consistency, since nothing (or, at most, violated) can
be derived from a contradiction such as A D and
Early on, Kowalski has pointed out that logic pro-
gramming (LP), and hence datalog (which is a some-
what restricted form of LP), has the potential of para-
consistency (Kowalski, 1979). In LP, no use is ever
made of the law of disjunctive weakening (LDW) by
which conclusions p q can be inferred from any
premise p for arbitrary q, so that ECQ cannot become
effective. That is, q cannot be inferred from contradic-
tory premises p, p by inferring p q and resolving
that with q.
Datalog has been characterized as a form of
resource-constrained first-order predicate logic. In
practice, the available computer memory and time
may always constrain the power of computation.
Apart from that, however, the resource-constrained
approachof inferencing in datalog is indeed beneficial
in terms of inconsistency tolerance, as observed in
(Kowalski, 1988), and also in terms of the oxymoron
of reasoning consistently with inconsistent knowl-
edge, as exposed in the introduction.
More precisely, datalog renounces on several re-
sources of inference mechanisms that are available in
classical logic, without sacrificing the computational
power and deductive capacities that are needed for
knowledge engineering. While Datalog involves RaA
as an essential inference principle, it never applies
LDW, and thereby avoids the effects of ECQ.
Moreover, the goal-orientedness of datalog can be
interpreted as another form of constraining resources,
since inference steps that evidently are not conductive
to reach the goal (which either is to deduce an answer
to a query or to test if a constraint is violated) simply
are not taken.
Yet, apart from the possibly explosive effect of
LDW, there is another possible cause of ECQ becom-
ing effective, as identified in (Hewitt, 2012). That
possible cause uses RaA, which, similar to LDW, is
an inference rule, i.e., a principle on the meta-level
of reasoning, and goes as follows. Let T = {p, p}
and q a hypothesis, where q is an arbitrary sen-
tence. Then, (T {∼q}) p p holds trivially.
From that, RaA infers q.
However, the only time RaA is applied in datalog
is the moment in which a goal clause (i.e., a query in
denial form or a constraint) is refuted by input clauses
from the database. The refutation of a goal of the form
B in a database D means that D {← B} is incon-
sistent, and hence the existential closure B is inferred
from D. (Moreover, the refutation of B in D also
computes a set of answer substitutions {θ
,. . . ,θ
(n > 0) of the variables that are free in B such that
D Bθ
(1 i n), but that is not of importance in
this context).
Now, if it happens that, e.g., A is a unit clause in a
database D and A is a constraint in an integrity the-
ory IC, then the refutation of the goal A in D cor-
rectly states that A is true in D (or, proof-theoretically
speaking, A can be inferred from D), and that A is
violated in D. But no other consequence is inferred
from that inconsistency, and in particular not in the
way arbitrary sentences can be inferred from RaA in
general, as indicated above.
Now, let us wrap up what we have seen up to this
point. By amalgamating object- and meta-level rea-
soning (Bowen and Kowalski, 1982), datalog is a self-
consistent problem solving paradigm that may consis-
tently reason about its own reasoning, even if the lat-
ter is done with inconsistent knowledge. Thus, data-
log appears to be an ideal candidate for inconsistency-
tolerant knowledge engineering.
In particular, datalog can be seen as a realization
of Wittgenstein’s advice to simply not draw any con-
clusion from inconsistent sentences. Similarly, each
paraconsistent logic can be seen that way. Yet, the
essential difference is that datalog is much closer to
classical logic than any other paraconsistent form of
However, as soon as other reasoning principles are
used in datalog applications, i.e., on top of datalog,
more caution has to be taken. For example, for check-
ing if updates would violate integrity, most integrity
checking methods assume the total integrity premise,
i.e., that the theory before the update is consistent.
For instance, let D be a database containing r(a, a)
and IC an integrity theory containing r(x, x) and
r(a, y) s(y), where r, s are predicates, x, y are
variables and a is a constant. Clearly, D IC is incon-
sistent. For checking if the updateU = insert s(a) in D
violates integrity, the instance r(a, a) s(a) of the
constraint r(a, y) s(y) is considered relevant by
most methods, since U may violate it, while r(x, x)
is considered irrelevant, since it cannot be violated by
U. By the total integrity premise, r(x, x) is not vi-
olated before the update. Hence, some methods (e.g.,
the one in (Gupta et al., 1994)) wrongly infer that
r(x, x) s(a) also cannot be violated by U, since
r(a, a) s(a) is subsumed by r(x, x). Thus, such
methods do not confine inconsistency, since they risk
to miss an increase of inconsistency across updates.
As opposed to that, it is shown in (Decker
and Martinenghi, 2011) how to confine inconsis-
tency in databases across updates, by methods for
inconsistency-tolerant integrity checking (ITIC) that
do not recur on the total integrity premise. A more
general approach to ITIC in database theories of the
form (D, IC) is discussed in (Decker, 2012). It
is based on a definition of inconsistency measures,
which size the amount of inconsistency in a database,
and allow to determine if an update increases or de-
creases the current amount of inconsistency. Hence,
an integrity checking method is re-defined to be
inconsistency-tolerant if it only accepts updates that
do not increase the amount of integrity violation in
the updated database.
Inconsistency measures also may serve for com-
puting consistency-preserving updates and partial re-
pairs of inconsistent databases that decrease the
amount of constraint violations, as shown in (Decker,
2012). Another application of inconsistencymeasures
is an inconsistency-tolerant approach to the evolution
of database schemas, as described for some specific
measures in (Decker, 2011b). Also, the consistency
preservation of concurrent transactions can be con-
trolled in an inconsistency-tolerant manner by incon-
sistency measures, as shown in (Decker and Mu˜noz-
Esco´ı, 2010) and (Cuzzocrea et al., 2012), where in-
consistency measures are also used for uncertainty
management. Moreover, specific inconsistency mea-
sures allow to determine if an answer to a given query
“has integrity” or not, by checking if the data involved
in computing answer substitutions are disjoint or not
from the data involved in any constraint violation, as
shown in (Decker, 2011a).
Even though a lot of datalog-based KE methods
had not been conceived for working in the presence
of inconsistency, many of them have turned out to be
inconsistency-tolerant, in the sense of confining ex-
tant integrity violations. Thus, the capacity of be-
ing inconsistency-tolerantcomes for free in most con-
ventional methods. This observation confirms the
main point of this paper, which is that the inconsis-
tency tolerance of datalog, as well as of many impor-
tant datalog-based KE applications, provides a reli-
able reasoning that guarantees consistency in the pres-
ence of inconsistency without further ado.
In this paper, we have argued that datalog is a viable
solution to the problem of pragmatic but theoretically
well-founded reasoning with data that are possibly in-
To facilitate our arguments, we have confined at-
tention to the definite case of datalog, i.e., we have
not considered its extension by non-monotonic nega-
tion, as obtained by an abductive or argumentation-
theoretic interpretation of negative literals in bodies
of clauses. Also, we have not considered extension to
a more general syntax and semantics of integrity con-
straints that allow disjunctions of atoms in the head
of clauses, as proposed, e.g., in a variety of papers by
Robert Kowalski his co-authors.
Future work of ours is concerned with defending
the claim that the so-extended datalog continues to
go out of the way of any inadvertent application of
ECQ, and thus is an even more powerful paradigm for
inconsistency-tolerant KE. Here, we already remark
that the abductive interpretation of negation involves
an active use of LNC. As opposed to that, abductive
datalog is careful with applying LEM, an unbridled
use of which may lead to inconsistent conclusions, as
shown in (Dung, 1995).
A more radical approach to embrace inconsistency
as an ubiquitous feature in computing and KE on
a foundational level has been proposed by (Hewitt,
2012). As opposed to datalog, which, by its avoidance
of LDW and its controlled, goal-oriented use of RaA,
is consistent on the meta-level, Hewitt’s Direct Logic
(which does not support RaA) is inherently inconsis-
tent, on purpose, and arguably is even more in line
with Wittgenstein’s thoughts on inconsistency. Per-
haps, time will tell if the conservative stance of data-
log (by which inconsistency on the object-levelcan be
kept at bay by a consistent, resource-constrained way
of reasoning on the meta-level) could prevail over an
approach that fully embraces inconsistency.
The work of the author for this publication has
been partially supported by FEDER (European Fund
for Regional Development) and the grants TIN2009-
14460-C03 and TIN2010-17139 from the Spanish
Ministry of Economy and Competitiveness.
Abiteboul, S., Hull, R., and Vianu, V. (1995). Foundations
of Databases. Addison-Wesley.
Bagni, G. (2008). Obeying a rule: Ludwig Wittgenstein
and the foundations of set theory. The Montana Math-
ematics Enthusiast, 5(2,3):215–222.
Bertossi, L., Hunter, A., and Schaub, T. (2005). Inconsis-
tency Tolerance, volume 3300 of LNCS. Springer.
Bonchi, F., Giannotti, F., Mainetto, G., and Pedreschi, D.
(1999). Using data mining techniques in fiscal fraud
detection. In Proc. 1st DaWaK, volume 1676 of LNCS,
pages 369–376. Springer.
Bowen, K. and Kowalski, R. A. (1982). Amalgamating lan-
guage and metalanguage. In Clark, K. and T¨arnlund,
S.-A., editors, Logic Programming, pages 153–172.
Academic Press.
Calvanese, D., De Giacomo, G., Lembo, D., Lenzerini, M.,
and Rosati, R. (2008). Inconsistency tolerance in P2P
data integration: an epistemic logic approach. Infor-
mation Systems, 33(4-5):360–384.
Calvanese, D., Kharlamov, E., Montali, M., and
Zheleznyakov, D. (2012). Inconsistency tolerance
in OWL 2 QL knowledge and action bases. In
Klinov, P. and Horridge, M., editors, Proc. OWL
Experiences and Directions Workshop, volume 849.
CEUR Electronic Workshop Proceedings. Available
Caruana, L. (2004). Wittgenstein and the status of contra-
dictions. In A. Coliva, E. P., editor, Wittgenstein To-
day, pages 223–232. Il Poligrafo, Padova.
Chihara, C. (1977). Wittgenstein’s analysis of the para-
doxes in his lectures on the foundations of mathemat-
ics. Philosophical Review, 86(3):365–381.
Chopra, S. and Parikh, R. (1999). An inconsistency tolerant
model for belief representation and belief revision. In
Proc. IJCAI, pages 192–197. Morgan Kaufmann.
Cuzzocrea, A., de Juan-Mar´ın, R., Decker, H., and Mu˜noz-
Esco´ı, F. D. (2012). Managing uncertainty in data-
bases and scaling it up to concurrent transactions. To
appear in Springer LNCS.
Decker, H. (2010). How to contain inconsistency or, why
Wittgenstein only scratched the surface. In Proc. 7th
European Conf. on Computing and Philosophy, pages
70–75. Dr. Hut.
Decker, H. (2011a). Answers that have integrity. In Schewe,
K.-D. and Thalheim, B., editors, Semantics in Data
and Knowledge Bases - 4th International Workshop
SDKB, volume 6834 of LNCS, pages 54–72. Springer.
Decker, H. (2011b). Causes for inconsistency-tolerant
schema update management. In Proc. 27th IDCE
Workshops, pages 157–161. IEEE CSP.
Decker, H. (2012). Measure-based inconsistency-tolerant
maintenance of database integrity. To appear in
Springer LNCS.
Decker, H. and Martinenghi, D. (2011). Inconsistency-
tolerant integrity checking. Transactions on Knowl-
edge and Data Engineering, 23(2):218–234.
Decker, H. and Mu˜noz-Esco´ı, F. D. (2010). Revisiting and
improving a result on integrity preservation by concur-
rent transactions. In Proc. OTM Workshops, volume
6428 of LNCS, pages 297–306. Springer.
Diamond, C. (1976). Wittgenstein’s Lectures on the Foun-
dations of Mathematics, Cambridge, 1939. Harvester,
Dung, P. M. (1995). An argumentation-theoretic founda-
tions for logic programming. J. Logic Programming,
Dunnei, P., Hunter, A., McBurney, P., Parsons, S., and
Wooldridge, M. (2009). Inconsistency tolerance in
weighted argument systems. In Proc. 8th Int. Conf.
on Autonomous Agents and Multiagent Systems (AA-
MAS 2009), volume 2, pages 851–858. International
Foundation for Autonomous Agents and Multiagent
Gupta, A., Sagiv, Y., Ullman, J. D., and Widom, J. (1994).
Constraint checking with partial information. In Pro-
ceedings of PODS 1994, pages 45–55. ACM Press.
Hewitt, C. (2012). Formalizing common sense for scal-
able inconsistency-robust information integration us-
ing direct logic reasoning and the actor model.
Hinrichs, T., Kao, J., and Genesereth, M. (2009).
Inconsistency-tolerant reasoning with classical logic
and large databases. In Proc. 8th SARA, pages 105–
112. AAAI Press.
Imam, F. and MacCaull, W. (2009). Integrating healthcare
ontologies: Inconsistency tolerance and case study.
In D. Ardagna, M. M. and Yang, J., editors, Busi-
ness Process Management Workshops (BPM 2008),
volume 17 of Lecture Notes in Business Information
Processing, pages 373–384. Springer.
Koogan Breitman, K., Felicssimo, C. H., and Cysneiros,
L. M. (2003). Semantic interoperability by align-
ing ontologies. In Galvo Martins, L. E. and Franch,
X., editors, Workshop em Engenharia de Requisitos
(WER03), pages 213–222.
Kowalski, R. (1988). Logic-based open systems. In Hoepel-
man, J., editor, Representation and Reasoning, pages
125–134, T¨ubingen. Max Niemeyer Verlag.
Kowalski, R. A. (1979). Logic for Problem Solving. Else-
M¨uller, H., Leser, U., and Freytag, J.-C. (200). Mining for
patterns in contradictory data. In Proc. 1st IQIS, pages
51–58. ACM SIGMOD.
Nuseibeh, B., Easterbrook, S., and Russo, A. (2000). Lever-
age inconsistency in software development. Com-
puter, 33(4):24–29.
Padmanabhan, B. and Tuzhilin, A. (2002). Knowledge re-
finement based on the discovery of unexpected pat-
terns in data mining. Decision Support Systems,
Qi, G., Haase, P., Schenk, S., Stadtm¨uller, S., and Hitzler,
P. (2009). Inconsistency-tolerant reasoning with net-
worked ontologies. Technical report, NeOn Deliver-
able D1.2.4.
Wright, C. (1980). Wittgenstein on the Foundations of
Mathematics. Duckworth, London.
Wrigley, M. (1980). Wittgenstein on inconsistency. Philos-
ophy, 55(214):471–484.
Yu, F., Qin, Z., and Jia, X.-L. (2003). Data mining applica-
tion issues in fraudulent tax declaration detection. In
Proc. 2nd Conf. Machine Learning and Cybernetics,
pages 2202–2206. IEEE.