Readings about the question:  It is said that resoning on the semantic web must be monotonic.  Why is this so, when human reasoning, which seems to have served us well, is nonmonotonic?

Compiled by Seth Russell
Last updated 2003.01.31

1973 Pat Hayes  The Frame Problem and Related Problems in AI. Artificial and Human Thinking [Summary]
... "The only way to ensure that all inferences remain consistent with the state of the world would be that the conditions for the applicability of all actions are stated exhaustively, leading to intractable descriptions. Alternatively, keeping preconditions reasonable requires being able to retract previously drawn conclusions upon receiving new information through observations."
1980 McDermott and J. Doyle Nonmonotonic logic 1, Artificial Intelligence [Summary]
... "The authors extend standard logic into a non-monotonic logic which allows new axioms to be added (through observations for example) that invalidate previously made inferences. Such non-monotonic reasoning capabilities seem crucial for systems with incomplete information about their environment."
... presents:  "The Case for Monotonic Logical Inference, with Nonmonotonic Procedures"
1986 John McCarthy Circumspection a Form of Nonmonotonic Reasoning
to do excerpts ...
1991 Guha Contexts: A Formalization and Some Application
..." Ideally, the issues related to nonmonotonic logic and hence the nonmonotonic mechanism used should be orthogonal to contexts."
..."If a consider statement cannot be proved, it is assumed to be false, i.e. we minimize the extent of consider. The mechanism of consider statements performs the job of contexts to a limited extent. The relationship between contexts and these consider statements is exactly like the relationship between contexts and nonmonotonic logic."
1993 Michael J. O'Donnell Against Nonmonotonic Logic (Draft)
Definition:   A logical inference relation $\vdash $ is monotonic if and only if, for all sets of propositions ${\bf S}$ and ${\bf T}$, and for all propositions $A$, if ${\bf S}\vdash A$ and ${\bf S}\subseteq{\bf T}$, then ${\bf T}\vdash A$.
... "That is, there are certainly cases in practice where the discovery of new knowledge requires a retraction of old assertions."
... "classical and intuitionistic logic seem to be too sensitive to errors, due to their conceptual foundation on the assertion of absolute truth rather than of rational belief based on fallible information, although I can find no rigorous discussion of this sensitivity in the literature. "
... "The right paraconsistent formal system can reduce the harm done by errors and contradictions in the hypotheses used by a reasoning agent, allowing useful reasoning to go on during the typically long time between the introduction of an error or contradiction and its detection. But, they still do not address the hard problem of retracting information when an error is finally detected."

Peter Suber Definition Nonmonotonic  Logic:  Logics in which the set of implications determined by a given group of premises does not necessarily grow, and can shrink, when new wffs are added to the set of premises.

Closed World Assumption CWA:   
When the semantic mapping between the language and the domain is incomplete or even missing, it may not be possible to determine whether a sentence is true or not. The closed world assumption is used provide a default solution in the absence of a better solution.

if you cannot prove P or ~P from a knowledge base KB, add ~P to the knowledge base KB.

The closed-world assumption simply declares that all relevant facts are stored in the database, so that any statement that is true about the actual world can be deduced from facts in the system.

1996 John Sowa  Nonmonotonic logic
..."I believe that nonmonotonic reasoning is very important, but I also believe that belief revision is a more natural and more general way to do it. With belief revision, you don't change the logic. Instead, you add a metalevel for extending or revising the axioms, followed by a level of conventional logic. The effect is exactly the same, but you get a clearer separation of the two stages."
.." The effect of considering nonmonotonic reasoning to be metalevel belief revision is that you keep the standard model theory for FOL. Even more importantly, you open up new techniques at the metalevel such as methods for evaluating relevance of possible axioms, measures of probability, etc. But all those additions are treated as heuristics for developing or discovering new theories, not as changes to the base logic."
... seeAlso follow up post to Delugach ... "The rules at the metalevel are rules about how to do reasoning. All standard rules of inference are metalevel rules. Negation as failure is an example of a belief revision kind of rule, which has the effect of adding a new axiom. If you like, you can implement completely new rules of inference, such as fuzzy logic for example, by changing the metalevel."
1997  Brewka Nonmonotonic Reasoning and Overview
..."Nonmonotonic reasoning, in its broadest sensel, is reasoning to conclusions on the basis of incomplete information.  Given more information, we are prepared to retract previously drawn inferences."
..."Nonmonotonic reasoning as a form of reasoining contrasts with standard deductive reasoning in the following way.  Suppose we know that all men are mortal and that Socretes is a man.  Then it follows, by simple syllogistic reasoning, that Socrates is mortal.  Further, nothing we can add to our knowledge will change that conclusion, given that we still hold the prenmises.  Deductive reasoning within a theory is 'local' in the sense that, having derived a conslusion from premises, we need not worry about any other sentences in the theory."
...
Obviously, nonmonotonic reasoning does not share this property. Why would we ever want a logic that is nonmonotonic?  Historically, the need for nonmonotonic reasoning was identified in the course of trying to slove knowledge representation problems in several application areas of Artificial Intelligence (AI)."
1997 Patrick Doyle QUAIL Qusetion of the day
Closed-World Assumption(CWM):  "Assumes all relevant information has been  specified. Any positive fact not specified is false."
1998 Pat Hayes Discussion about Ontologies for actions and change
... "The world is changing because that is what worlds do, by and large; certainly my own 'common-sense' world keeps changing all the time. If there were no change, our ordinary notions of time and tense would be incoherent."
... "when an observation contradicts a belief, we would seem to have contradictory beliefs, unless our logic is nonmonotonic."
... "...If we know ahead of time that actions always come out the way we want them to, then we could write change axioms which were guaranteed to be correct..."
1998 Frame Problem
..."A common response is to handle nonchanges implicitly by allowing the system to assume by default that a state persists, unless there is an axiom specifying that it is changed by an occurrence, given surrounding conditions.  Since such assumptions are not deducible from the axioms of change (even given surrounding conditions), and since the licensed conclusions are not cumulative as evidence is added, the frame problem helps motivate the development of special "NONMONOTONIC LOGICS" intended to minimize the assumptions that must be retracted given further evidence."

..."
Another approach to the frame problem seeks to remain within the strictures of classical (monotonic) logic (Reiter, 1991).  In most circumstances, it avoids the use of huge numbers of axioms about nonchanges, but at the cost of using hugely and implausibly bold axioms about nonchanges.  For example, it is assumed that all the possible causes of a certain kind of effect are known, or that all the actual events or actions operating on a given situation are known."
1998 Tim Berners-Lee Semantic Web - Inconsistent data
..."What, many people ask, will happen when this huge mass of classical logic meets its first inconsistncy? Surely, once you have one staement that A and another somewhere on the web that not A, then doesn't the whole system fall apart? Surely, then you can deduce anything you want?"
..."On the web, a fact may be asserted in an expression. That expression may be part fo a formula. The formula may ivolve negation, and may invove quotation. The whole formula is found by parsing some document . There is no a priori reason to believe any document on the web. The reason to believe a document will be found in some information (metadata) about the document. That metadata may be an endosement of the document - another RDF statement, which in turn was found another document, and so on."
..."Once any pair of conflicting statements can be deduced from information signed with a given key, then anything can be deduced from information signed with the key: the key is completely broken."
1999-2001 Tim Berners-Lee The Semantic Web as a language of logic
..." I notice that most of the semantic web applications I have been playing with involve rules and axioms which are not at all complete. Each application has its own sets of axioms and can be individually proved consistent (or not). So one way forward for standards would be to instantiate that, allowing each document and message on the semantic web to have a pointer to the vocabulary its uses, including its varieties of logical connectives and their assocaited axioms."
..."If there is a semantic web machine, then it is a proof validator, not a theorem prover. It can't find answers, it can't even check that an answer is right, but it can follow a simple explanation that an answer is right. The Semantic Web as a source of data should be fodder for automated reasoning systems of many kinds, but it as such not a reasoning system."
..."... Access Limited Logic is restricted (as I understand it) to relations r(a,b) available when r is accessed, and uses inference rules which only chain forward along such links. There is also a 'partitioning' of the Web by making partitioning the rules in order to limit complexity.  For the semantic web as a whole, then, we do require tractable ..."Electronic commerce needs a solid foundation in this way, and the development of the semantic web is (in 1999) essential to provide a rigid framework in which to define electronic commerce terms, before electronic commerce expands as a mass of vaguely defined semantics and ad hoc syntax which leaves no room for automatic treatment, and in which the court of law rather than a logical derivation settles arguments."
..."Keeping a language less powerful than first order predicate calculus is quite reasonable within an application, but not for the Web."
..."Decidability: A dream of logicians in the last century to find languages in which all sentences were either true or false, and provably so. This involved trying to restrict the language so as to avoid the possibility of (for example) self-contradictory statements which can not be categorized as a true or not true. On the Semantic Web, this looks like a very academic problem, when in fact one anyway operates with a mass of untrustworthy data at any point, and restricts what one uses to a limited subset of the web. Clearly one must not be able to derive a self-contradictory statement, but there is no harm in the language being powerful enough to express it. Indeed, endorsement systems must give us the power to say "that statement is false" and so loops which if believed prove self-contradictory will arise by accident or design. A typical response of a system which finds a self-contradictory statement might be similar to the response to finding a contradiction, for example, to cease to trust information from the same source (or public key)."
..."Pat Hayes mentioned a logic in which the law of the excluded middle does not exist - which si important as you can always considera paradox which is neither true nor false. .... 'is like full negation but rules out proofs by contradiction: basically, it insists on a higher standard of provability than classical negation' ... ."
1999 Alice M. AgoginoKnowledge Representation Using Nonmonotonic or Nonbinary Logics
...Nonmonotonic logic is concerned with reasoning that allows making assumptions that are not guaranteed to be correct (and thus involves a "nonsound" logical procedure), but are at least consistent with the statements in the knowledge base."
..."In order to solve everyday problems, we often resort to common-sense or default reasoning when the information required for a decision is incomplete or too complex to easily process. When new or more complete information is added, we may change our assumptions and make different decisions. Much of human learning is of this nonmonotonic nature. We create certain models of the world as children. As we mature and learn from experience, we change some parts of those models."
The Closed World Assumption (CWA): ..."when using the CWA, we assume that a logical sentence is not true if it is not implied. The CWA is nonmonotonic because this set of augmented beliefs will shrink as we add new logical sentences"...
Objections to Nonmonotonic Inference: ..."nonmonotonic systems suffer from the following two major problems. First, nonmonotonic systems can be excruciating slow because of the repeated need for consistency checking. Secondly, the non-decidability of nonmonotonic inference is such that any system using this logic will necessarily make mistakes from time to time (or loop indefinitely)"
Three-Valued Logic: ..."Three-valued logic provides the mechanism to draw conclusions when the information implies a conclusion and propagates lack of information, otherwise."
2001 Harry S. Delugach Nonmonotonic Logic Or How Many Logics Are There?
..."In a nonmonotonic system the addition of new facts can reduce the set of logical conclusions."
..."Humans use nonmonotonic reasoning constantly!"
..."First order logic although descriptively universal, is not effective at handling large classes of problems. If computers are going to handle common sense we need to be able to have some form of default reasoning. Nonmonotonic logic can be used in many domains where classical logic falls short, such as in the areas of default diagnosis, diagnosis, action, and temporal logic."
..."If we are told nothing about Tweety, other than Tweety is a bird, we assume that Tweety's feet are not in concrete, or Tweety's wings are not broken, this is the closed world assumption."
..."Humans regularly make assumptions and when new evidence appears those assumptions can be changed, causing a different answer, thus behaving nonmonotonicaly." .."Nonmonotonic logic does not have many essential properties of classical first order logic, specifically semidecidability. In classical logic, it is possible for a system to halt (be stuck in an infinite loop) trying to prove the negation of something for which there is insufficient information.
..."In nonmonotonic default logic rather than return with no answer the process returns with a wrong (default answer). ..."Nonmonotonic logic systems may miss the importance of probability. Probabilistic reasoners can also represent uncertainty, and in a different (probabilistic) way. These systems exhibit a different set of properties, with which nonmonotonic logic can not effectively deal with."
2000 Pat Hayes random thoughts on web logic
...."For the security  in B2B transactions which W3C wants, you need a nonmonotonic LOGIC whose proofs, once checked, stay checked.But Drew was also right in that there is no way that you or anyone else can manage without  making nonmonotonic INFERENCES, and that indeed people will make those inferences from your monotonic conclusions whether you like it or not (and if tested in court they will probably stand up under case law.) Fortunately there is a way around this apparent impasse, since one can represent the nonmonotonicity as what might be called fungible assumptions. " ...All the actual *reasoning* involved is monotonic (until the fuse burns.)"

then from www-rdf-logic/2000Sep/0009 in response to Peter F. Patel-Schneider:

..."Being a good oldfashioned logical type, I'm all for  good oldfashioned logics (as 'full' as we can manage), but I think  that we will need to modify our old ideas about semantics to accomodate to the web's messiness."
..."  think that people with a logical training have some new fields to conquer here, and I'd like to encourage us to get on with it rather than complain about a 'lack of logic'. Its our job to try to help get a suitable logic invented, right? Now, got any good ideas?"\
*** 2001 Stanford Encyclopedia of Philosophy  Non-Monotonic Logic

The import of monotony is that one cannot pre-empt conclusions by adding new premises. However, there are many inferences typical of everyday (as opposed to mathematical or formal) reasoning, that do not satisfy monotony. These are cases in which we reach our conclusions defeasibly (i.e., tentatively), reserving the right to retract them in the light of further information. Perhaps the clearest examples are derived from legal reasoning, in which defeasible assumptions abound. In the judicial system, the principle of presumption of innocence leads us to infer (defeasibly) from the fact that x is to stand trial, the conclusion that x is innocent; but clearly the conclusion can be retracted in the light of further information.

2001 Dialogue on rdf-logic: Why must the web be monotonic ?
Ian Horrocks:  www-rdf-logic/2001Jul/0064
This is certainly the case. If you say:
(sameClassAs WhiteWine (intersectionOf Wine (Restriction (onProperty
colour) (hasValue White))))
then being a wine with color White is both a necessary and sufficient
condition for being a WhiteWine. Moreover, adding further axioms (or
even RDF triples) to the ontology can never change this (otherwise we
would be non-monotonic).
Seth Russell:  www-rdf-logic/2001Jul/0065
I think I've heard it said that the web must be monotonic.  Have I misheard?
If not, then why must the web be monotonic?
Pat Hayes:  www-rdf-logic/2001Jul/0067
Good question. The answer is controversial, but seems to me to be 
clear. First, its not the Web that is monotonic (whatever that would
mean) but the reasoning from Web resources that must be monotonic.
And the reason is that it - the reasoning - needs to always take
place in a potentially open-ended situation: there is always the
possibility that new information might arise from some other source,
so one is never justified in assuming that one has 'all' the facts
about some topic (unless you have been explicitly told that you
have.) Nonmonotonic reasoning is therefore inherently unsafe on the
Web. In fact, nonmonotonic reasoning is inherently unsafe anywhere,
which is why all classical reasoning is monotonic; this isn't
anything particuarly new. But the open-ended assumption that seems to
underlie much thinking about reasoning on the semantic web makes the
issue a little more acute than it often is in many of the situations
where logic has been put to use in computer science.

For example, if you are reasoning with a particular database of
information, it is often assumed that the dbase is complete, in the
sense that if some item is missing, then it is assumed to be false:
if a hospital's databanks do not contain any record of a certain
patient, you can conclude that they weren't a patient at the hospital
(because if they had been, their record would be there.) Nonmonotonic
inference modes such as this (often called negation-as-failure, ie if
you fail to find P in the database, assume P is false) are widely
used because they are efficient and because closed worlds are so
common. It is used in Prolog, where it is often exactly what one
wants because the domains being described are recursively enumerable
and failure to prove amounts to knowing that no proof can exist. But
open-ended domains are not like this, and it is very dangerous to
rely on this kind of reasoning when one has no licence to assume that
the world is closed in the appropriate way. If there were ever an
open-ended domain it is surely the semantic web.

There is a way to combine the global security of monotonic reasoning
with the local advantages of nonmonotonic reasoning (eg when working
with hospital records on a website, say), which is to provide a way
to state the closed world assumptions explicitly. Take the
hospital-records example again, where you fail to find any record of
a patient and conclude that the person never was a patient. That is
a non-monotonic inference from just the patient records, but it is
monotonic from the patient records PLUS an explict statement of the
closed-world assumption, ie the statement that the set of records is
exhaustive. So if we have a way to refer to a set of assertions -
say, all those at a certain URL, or all those which use a certain
namespace, or by some other means - and a way to state the
closed-world assumptions that are being made about this set of
assertions - say, they they are exhaustive with respect to all
queries of a certain form - then the overall reasoning can be
considered monotonic, even though it proceeds locally by using
efficient nonmonotonic methods.

Right now, DAML+OIL and RDF have not entered into this area, but
'rules' languages need to consider it seriously, in my view. The
global advantages of monotonicity should not be casually tossed
aside, but at the same time the computational advantages of
nonmonotonic reasoning modes is hard to deny, and they are widely
used in the current state of the art. We need ways for them to
co-exist smoothly.
Seth Russell:  www-rdf-logic/2001Jul/0068
How about just stating that a given context is monotonic like the following?
{TheGroup containsMembers Pat, Tim, exp(membersOf RDFLogic). }
instanceOf MonotonicContext.
Pat Hayes:   www-rdf-logic/2001Jul/0069
The issue is how do we do reasoning in the global context, however. 
There really isnt anything useful to be gained from being told that a
small subworld is monotonic, since the need for monotonicity arises
when one wants to make inferences which will stay correct even in
some larger context. (That is the definition of monotonic inference:
if you can validly infer something from Q, then you can validly infer
it from Q plus anything else; the extra information can't invalidate
your earlier conclusion.) If we were restricted to some TheGroup
which is listed explicitly, or whose boundaries we can compute at a
reasonable cost, then we can use nonmonotonic inferences freely,
knowing that we are talking about this limited domain. But if we want
to publish our conclusions to a broader audience, it would be
irresponsible (IMHO*) to fail to indicate that they were derived from
this narrowly defined context, and rely on the nonmonotonic powers of
our listeners to rescue themselves from any errors into which they
might be led by relying on our conclusions in areas where they are no
longer valid. If, however, we publish them with an explicit
indication that they are derived from, and claimed to be valid only
with respect to, this narrower context, then our published claims
(including this rider) can be used monotonically by other reasoners
with perfect safety. If they are willing to accept the limits of the
context, they can continue to reason within the closed world. If they
do not, they can insert this rider as an explicit condition on any
further conclusions they might wish to draw from what we tell them.
Either way, this enables them to have the same confidence in what we
publish as we have ourselves, which is the best that one can do.

* PS. I became convinced of the importance of monotonicity for the
semantic web after trying, and failing, to persuade Tim B-L that
nonmonotonic reasoning was the proper way to go. He insisted that web
logic had to be monotonic, and he was right.


David Martin www-rdf-logic/2001Jul/0070
This idea of refering to a set of assertions ("say, all those at a
certain URL, or all those which use a certain namespace, or by some
other means") ties in with a comment that I've made on 2 different
threads, in connection with DAML-S expressiveness issues. I won't go
into detail here, but in considering some things we've wanted to
express, it has come up that it would be very useful to be able to state
a restriction with respect to "a set of assertions". For instance, it
would be useful to be able to say that, within a given namespace, a
property has cardinality 1 (without saying anything about property
instances outside of the namespace). (I suppose there's an issue about
what namespace a property instance belongs to, but for present purposes,
I don't think that needs to be dealt with here.)

Andrei S. Lopatenko www-rdf-logic/2001Jul/0073

Just a comment
I think that solutions when publish information with an explicit indication
that they are derived from becomes a real thing.
As example technology of application profiles
(http://archive.dstc.edu.au/RDU/staff/jane-hunter/www10/paper.html)
suggested by C. Lagoze and Hunter (project Harmony
http://metadata.net/harmony/) allows to redefine meaning of metadata
elements for application domain.
So each application must publish explicit information about its data and it
is very close to an explicit indication mentioned early
The communication between aplications can be established in only case when
each application accept the limits of the context of the other application

Enrico Franconi: www-rdf-logic/2001Jul/0071
... So, your observation should be read as follows: a monotonic logic can
be used to reason with a fixed KB which is (contextually)
nonmonotonic; at each update of the nonmonotonic KB, we should
nonmonotonically change the encoding in the monotonic logic. The
overall reasoning remains nonmonotonic.

Pat Hayes: www-rdf-logic/2001Jul/0081 [needs to be broken out with Enricos's dialogue above]

That depends on what you count the inference as being from. If the
closed-world assumption is made explicit as an antecedent of the
conclusion, then when a new patient is inserted, the original
*conclusion* may be incorrect, but the *reasoning* is still valid.
The incorrectness of the conclusion is due to the fact that you now
have a false premis in the (monotonic, and hence still valid)
argument. An emerging idea in the semantic web, as I understand it,
is that proofs will be published and be publicly checkable. The key
point is that these proofs should be monotonic, not that any
particular reasoner's behaviour should be.

Of course, one can take either route. However, I would point out that
when real databases rely on such closed world assumptions in a
context where the world is changeable, they usually take pains to
provide a mechanism for recording exactly this kind of dependence.
The use of transaction time in temporal databases, for example, is
exactly the kind of reference to the particular state of the database
at the time the query was made that is necessary to keep the
reasoning globally monotonic, even though nonmonotonic techniques are
in constant use. In general, any reasoning process which needs to be
'accountable' to future queries about its validity needs to keep an
audit trail of the assumptions it was using at the time the inference
was made; and that is another way of saying that it needs to be
globally monotonic, if necessary by recording the local assumptions
that warranted the use of nonmonotonic techniques. Tort logic cannot
afford to be globally nonmonotonic.

2002  rdfig# chat  between Guha and Seth
<Guha_> if A => C but A+B does not imply C, for any statements A, B and C, then the system is not monotonic.
16:44:04 <Seth> and that's all there is to it ??
16:45:49 * Seth feels a mentograph comming on
and here it is:
http://robustai.net/mentography/monotonic.gif
2002    Monotony
Seth Russell:  www-rdf-comments/2002OctDec/0005

Re: http://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2002Oct/0067.html

Where Patrick Stickler says:

"This either seems non-monotonic" ...

This unwritten law that states that logic on the semantic web must
always be non-monotonic is getting to sound more like a religious
incantation than a pragmatic rule of thumb. Personally I think it is
time to examine this liturgy in the light of day. Can anyone show me
how you expect to always preserve monotony in a world where new
knowledge can be discovered, facts can change, and anybody can say
anything about anything ?

Pertinent graphs:
http://robustai.net/mentography/monotonic.gif
Incidentally, Guha, did I get the graph right?

Frank Manola www-rdf-comments/2002OctDec/0008

Well, they certainly can be related unintentionally! For example, the
current discussion about RDF datatypes involves issues of monotonicity,
and also seems to strike some folks as being monotonous (since they
claim they keep hearing the same arguments over and over). More to your
original point, it seems to me what you want is the ability to control
or specify when you have monotonicity and when you don't. Kind of like
a database transaction mechanism

Seth Russell www-rdf-comments/2002OctDec/0009

Exactly! Whatever a *logical* RDF graph is, it is certainly is a
document or a cluster of documents - what it is *not* is the whole blody
semantic web. Yet we seem not to have any convention for RDF authors to
state that simple fact about their RDF documents and clusters of
documents. But there are many ways that this can be implemented without
breaking into any new specifications. Well we might need to define some
new properties, but nothing really major or tramatic.
Pat Hayes www-rdf-comments/2002OctDec/0016
If we have ways of stating the boundaries of 
documents/databases/whatever, and of referring to them (perhaps
implicitly) and saying explicitly that something follows from this
bounded thingie alone, then we could say a lot of things that we are
unable to say right now. And it wouldn't be rocket science to provide
for saying things like this. No argument there. But that is (or can
be) all monotonic. What is non-monotonic is using inference which
makes these 'closed-world' assumptions *without saying or recording
that it has made them*.
And the reason why that isn't a good idea on
the web (or indeed in any large-scale communicative context, such as
human society) is that unless your listener shares your unspoken
assumptions, they can draw incorrect and unintended conclusions from
what you tell them. That is why non-monotonicity is dangerous.

Of course people make mistakes, change their minds, correct their
mistakes, etc. BUt the fact that we use a vocabulary to talk about
this that talks of CHANGE is itself a tribute to the need to have a
monotonic underlying logic. If I tell you that Joe is a bird, you
conclude that Joe can fly, and I then tell you that Joe is a penguin,
who made the mistake? We might argue about this - I think you did, by
assuming more than I told you - but the nonmonotonic answer is, what
mistake? There never was a mistake. You concluded Joe could fly, now
you conclude he can't fly: so?. Nonmonotonic logic changes the
logical rules to accommodate to the current state of belief.

Monotonic logic says, you had to change your mind or you would now be
inconsistent.
I think that like the rest of us you probably do
actually think using a monotonic logic.
           [emphasis mine]
2002 Negation in CycL
..."Finally, each such inference implementation "decision" like
"Are we using the Unique Names Assumption?"
"Are we minimizing abnormalities?"
"Are we allowing assumption-based negation minimization?"
"Do we allow positive assumptions?"
is independently under the user's control via inference parameters
which the user can choose to turn on or off.
2002 RV Guha & Pat Hayes LBase: Semantics for Languages of the Semantic Web (not perminant url)
... " In this document, we use a version of first order logic (with equality and a few other standard constructions) as Lbase. This imposes a fairly strict monotonic discipline on the language, so that it cannot express local default preferences and several other commonly-used non-monotonic constructs. We expect that as the Semantic Web grows to encompass more and our understanding of the Semantic Web improves, we will need to replace this Lbase with more expressive logics. However, we expect that first order logic will be a proper subset of such systems and hence we will be able to smoothly transition to more expressive Lbase languages in the future."
..."Lbase does not provide a mechanism for expressing propositional attitudes or true second order constructs. "
..." At the moment, Lbase does not provide any facilities related to the representation of time and change."
2002 Jeremy Carroll Typed literals: current status
Example of the WG trying to keep rules monotonic
.."However, it was noted that these datatyped interpretations are monotonic with
respect to the set of datatypes conformed with. i.e.
if a entails b with respect to a set D of datatypes
and D' is a superset of D
then
a entails b with respect to D'"
2002 Pat Hayes (in private dialogue)
..."... I don't know of any way to build agents which can perform useful inferences without their being based on logic. There is a good, basic, simple reason for this: if agent1 concludes B from A, and then agent2 uses B to infer C, there may be no way for agent2 to find out how agent1 came to its conclusion. So the reliability of agent2's conclusions must be hostage to the validity of agent1's reasoning. If the inference chains are always rather short then this might not matter, but in general it does matter. Having monotonic entailment as a basic convention for SW transactions provides at least the possibility of having reliable entailments which can be passed between agents without their needing to go back to first principles at every stage. "

Nov 2002 Pat hayes  Re: MT for imports (was: Re: Imports Proposal)
But in any case, so far NONE of the semantics refers to ANY kind of 
change. We are breaking new ground here. This is not a lightweight
area to get involved in. It gets us into issues of temporal
reference, common time ontologies, distinctions between transaction
time and reference time, truth maintenance strategies, nonmonotonic
justification reasoning (nobody told me it had changed, so I assumed
that it hadn't) indexicality, all kinds of stuff. Im not saying that
imports necessarily involves all this, but the whole area of handling
changing data does. Just blandly assuming that we have been in this
world all the time isn't being realistic, seems to me. We really
havn't.
Jan 2003 Pat Hayes RDF Semantics

Exerpt from the Glossary for Nonmonotonic
"The relationship between monotonic and nonmonotonic inferences is often subtle. For example, if a closed-world assumption is made explicit, e.g. by asserting explicitly that the corpus is complete and providing explicit provenance information in the conclusion, then closed-world reasoning is monotonic; it is the implicitness that makes the reasoning nonmonotonic. Nonmonotonic conclusions can be said to be valid only in some kind of 'context', and are liable to be incorrect or misleading when used outside that context. Making the context explicit in the reasoning and visible in the conclusion is a way to map them into a monotonic framework."