# Non-monotonic Logic

*First published Tue Dec 11, 2001; substantive revision Wed Apr 14, 2010*

The term “non-monotonic logic” covers a family of formal frameworks
devised to capture and represent *defeasible inference*, i.e.,
that kind of inference of everyday life in which reasoners draw
conclusions tentatively, reserving the right to retract them in the
light of further information. Such inferences are called
“non-monotonic” because the set of conclusions warranted on the basis
of a given knowledge base, given as a set of premises, does not
increase (in fact, it can shrink) with the size of the knowledge base
itself. This is in contrast to standard logical frameworks (e.g.,
classical first-order) logic, whose inferences, being deductively
valid, can never be “undone” by new information.

- 1. Abstract consequence relations
- 2. Skeptical or credulous?
- 3. Non-monotonic formalisms
- 4. Conclusion
- Bibliography
- Academic Tools
- Other Internet Resources
- Related Entries

## 1. Abstract Consequence relations

Classical first-order logic (henceforth: FOL) is monotonic: if a
sentence φ can be inferred in FOL from a set Γ of premises,
then it can also be inferred from any set Δ of premises
extending Γ. In other words, FOL provides a
relation
⊨ of *logical consequence* between
sets of premises and single sentences with the property that if
Γ
⊨ φ and Γ ⊆ Δ
then Δ
⊨
φ. This follows
immediately from the nature of the relation
⊨,
for Γ
⊨
φ
holds precisely when φ is true on every interpretation on
which all sentences in Γ are true (see the entry on
classical logic
for details on the
relation
⊨).

Consider the formal properties of a consequence relation from an abstract point of view. Let be any relation between sets of premises and single sentences. The following properties are all satisfied by the consequence relation ⊨ of FOL:

**Supraclassicality**: if Γ ⊨ φ then Γ φ.**Reflexivity**: if φ ∈ Γ then Γ φ;**Cut**: If Γ φ and Γ, φ ψ then Γ ψ;**Monotony**: If Γ φ and Γ ⊆ Δ then Δ φ.

Supraclassicality just requires that be an extension of ⊨, i.e., that if φ follows from Γ in FOL, then it must also follow according to . (The relation ⊨ is trivially supraclassical).

The most straightforward of the remaining conditions is reflexivity,
which requires that all sentences in Γ be inferable from
Γ. This translates to the requirement that if φ belongs to
the set Γ, then φ is a consequence of Γ. Reflexivity
is a rather minimal requirement on a relation of logical consequence:
It is hard to imagine in what sense a relation that fails to satisfy
reflexivity, can still be considered a *consequence*
relation.

Cut, a form of transitivity, is another crucial feature of
consequence relations. Cut is a conservativity principle: if φ is a
consequence of Γ, then ψ is a consequence of Γ together
with φ only if it is already a consequence of Γ alone. In
other words, adjoining to Γ something which is already a
consequence of Γ does not lead to any *increase* in
inferential power.

Cut is best regarded as a condition on the “length” of a proof (where “length” is to be understood in some intuitive sense related to the complexity of the argument supporting a given conclusion). When viewed in these terms, Cut amounts to the requirement that the length of the proof does not affect the degree to which the assumptions support the conclusion. Where φ is already a consequence of Γ, if ψ can be inferred from Γ together with φ, then ψ can also be obtained via a longer “proof” that proceeds indirectly by first inferring φ directly from Γ.

In contrast, in many forms of probabilistic reasoning the degree to which the
premises support the conclusion is inversely correlated to the length
of the proof. For this reason, many forms of probabilistic reasoning
fail to satisfy Cut. To see this, we adapt a well-known
counter-example: let *Ax* abbreviate “*x* is a
Pennsylvania Dutch”, *Bx* abbreviate “*x* is a native
speaker of German”, and *Cx* abbreviate “*x* was born in
Germany”. Further, let Γ comprise the statements “Most
*A*s are *B*s,” “Most *B*s are *C*s,” and
*Ax*. Then Γ supports *Bx*, and Γ together
with *Bx* supports *Cx*, but Γ by itself does not
support *Cx*. (Here statements of the form “Most *A*s are
*B*s” can be interpreted probabilistically, as saying that the
conditional probability of *B* given *A* is, say, greater
that 50%.) To the extent that Cut is a necessary feature of a
well-behaved consequence relation, examples of inductive reasoning such
as the one just given cast some doubt on the possibility of coming up
with a well-behaved relation of probabilistic consequence.

For our purposes, Monotony is the central property. Monotony states
that if φ is a consequence of Γ then it is also a consequence
of any set containing Γ (as a subset). 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 new evidence.

Other examples are driven by typicality considerations. If being
a *B* is a typical trait of *A*'s, then from the fact
that *x* is an *A* we infer the conclusion
that *x* is a *B*. But the conclusion is defeasible, in
that *B*-ness is not a necessary trait of *A*'s, but
only (perhaps) of 90% of them. For example, mammals, by and large,
don't fly, so that lack of flight can be considered a typical trait of
mammals. Thus, when supplied with information that *x* is a
mammal, we naturally infer that *x* does not fly. But this
conclusion is defeasible, and can be undermined by the acquisition of
new information, for example by the information that *x* is a
bat. This inferential process can be further iterated. We can learn,
for instance, that *x* is a baby bat that (as such) does not
yet know how to fly .

Defeasible reasoning, just like FOL, can follow complex patterns.
However, such patterns are beyond reach for FOL, which is, by its very
nature, monotonic. The challenge then is to provide for defeasible
reasoning what FOL provides for formal or mathematical reasoning, i.e.,
an account that is both *materially adequate* and *formally precise*.

It follows that monotony has to be abandoned, if we want to give a formal account of these patterns of defeasible reasoning. But then the question naturally arises of what formal properties of the consequence relation are to replace monotony. Two such properties, given below, have been considered in the literature, where is an arbitrary consequence relation:

**Cautious Monotony**: If Γ φ and Γ ψ, then Γ, φ ψ.**Rational Monotony**: If it's not the case that Γ¬ φ, and moreover Γ ψ, then Γ, φ ψ.

Both Cautious Monotony and the stronger principle of Rational Monotony are special cases of Monotony, and are therefore not in the foreground as long as we restrict ourselves to the classical consequence relation ⊨ of FOL.

Although superficially similar, these principles are in reality
quite different. Cautious Monotony is the converse of Cut: it states
that adding a consequence φ back into the premise-set Γ does
not lead to any *decrease* in inferential power. Cautious
Monotony tells us that inference is a cumulative enterprise: we can
keep drawing consequences that can in turn be used as additional
premises, without affecting the set of conclusions. Together with Cut,
Cautious Monotony says that if φ is a consequence of Γ then
for any proposition ψ, ψ is a consequence of Γ if and
only if it is a consequence of Γ together with φ. It has been
often pointed out, most notably by
Gabbay (1985),
that Reflexivity, Cut and Cautious Monotony are critical properties for
any well-behaved non-monotonic consequence relation.

The status of Rational Monotony is much more problematic. As we observed, Rational Monotony can be regarded as a strengthening of Cautious Monotony, and like the latter it is a special case of Monotony. However, there are reasons to think that Rational Monotony might not be a correct feature of a non-monotonic consequence relation, and in fact there is a counter-example to Rational Monotony, which is due to Robert Stalnaker (see Stalnaker (1994)). Consider the three composers: Verdi, Bizet, and Satie, and suppose that we initially accept (correctly but defeasibly) that Verdi is Italian, while Bizet and Satie are French. We assume that these defeasible conclusions are built into whatever inferential mechanism implements the non-monotonic relation .

Suppose now that we are told by a reliable (but not infallible!)
source of information that that Verdi and Bizet are compatriots. This
leads us no longer to endorse either the proposition that Verdi is Italian
(because he could be French), or that Bizet is French (because he
could be Italian); but we would still draw the defeasible consequence
that Satie is French, since nothing that we have learned conflicts with
it. By letting I(*v*), F(*b*), and F(*s*)
represent our initial beliefs about the composers' nationalities, and
C(*v*,*b*) represent the proposition that Verdi and Bizet
are compatriots, the situation could be represented as follows:

C(v,b) F(s),

Now consider the proposition C(*v*,*s*) that Verdi and
Satie are compatriots. Before learning that C(*v*,*b*) we
would be inclined to reject the proposition C(*v*,*s*)
because we endorse and I(*v*) and F(*s*), but after
learning that Verdi and Bizet are compatriots, we can no longer endorse
I(*v*), and therefore no longer reject C(*v*,*s*).
Then the following *fails*:

C(v,b) ¬ C(v,s).

However, if we added C(*v*,*s*) to our stock of beliefs,
we would lose the inference to F(*s*): in the context of
C(*v*,*b*), the proposition C(*v*,*s*) is
equivalent to the statement that all three composers have the same
nationality. This leads us to suspend our assent to the proposition
F(*s*). In other words, and contrary to Rational Monotony, the
following also *fails*:

C(v,b), C(v,s) F(s).

The previous discussion gives a rather clear picture of the desirable features of a non-monotonic consequence relation. Such a relation should satisfy Supraclassicality, Reflexivity, Cut, and Cautious Monotony.

## 2. Skeptical or credulous?

A separate issue from the formal properties of a non-monotonic
consequence relation, although one that is strictly intertwined with
it, is the issue of how *conflicts* between potential defeasible
conclusions are to be handled.

There are two different kinds of conflicts that can arise within a given non-monotonic framework: (i) conflicts between defeasible conclusions and “hard facts,” some of which possibly newly learned; and (ii) conflicts between one potential defeasible conclusion and another (many formalisms, for instance, provide some form of defeasible inference rules, and such rules might have conflicting conclusions). When a conflict (of either kind) arises, steps have to be taken to preserve or restore consistency.

All non-monotonic logics handle conflicts of the first kind in the same way: indeed, it is the very essence of defeasible reasoning that conclusions can be retracted when new facts are learned. But conflicts of the second kind can be handled in two different ways: one can draw inferences either in a “cautious” or “bold” fashion (also known as “skeptical” or, respectively, “credulous”). These two options correspond to significantly different ways to construe a given body of defeasible knowledge, and yield different results as to what defeasible conclusions are warranted on the basis of such a knowledge base.

The difference between these basic attitudes comes to this. In the presence of potentially conflicting defeasible inferences (and in the absence of further considerations such as specificity — see below), the credulous reasoner always commits to as many defeasible conclusions as possible, subject to a consistency requirement, whereas the skeptical reasoner withholds assent from potentially conflicted defeasible conclusions.

A famous example from the literature, the so-called “Nixon diamond,” will help make the distinction clear. Suppose our knowledge base contains (defeasible) information to the effect that a given individual, Nixon, is both a Quaker and a Republican. Quakers, by and large, are pacifists, whereas Republicans, by and large, are not. The question is what defeasible conclusions are warranted on the basis of this body of knowledge, and in particular whether we should infer that Nixon is a pacifist or that he is not pacifist. The following figure provides a schematic representation of this state of affairs in the form a (defeasible) network:

The credulous reasoner has no reason to prefer either conclusion (“Nixon is a pacifist;” “Nixon is not a pacifist”) to the other one, but will definitely commit to one or the other. The skeptical reasoner recognizes that this is a conflict not between hard facts and defeasible inferences, but between two different defeasible inferences. Since the two possible inferences in some sense “cancel out,” the skeptical reasoner will refrain from drawing either one.

Defeasible inheritance networks can also be interpreted as providing
a representation of that particular kind of inference referred to as
“stereotypical.” *Stereotypes* can be viewed as generalizations
concerning particular populations; while they might be sometimes useful
as such, their applicability to individual cases is problematic. In
fact, the particular point of view according to which stereotypes apply
to groups but not to individuals meshes particularly well with the
skeptical approach to defeasible inheritance.

Consider again the Nixon Diamond above. Part of what it says is that there are stereotypes to the effect that Quakers are Pacifists and Republicans are not. But of course the whole point of the skeptical interpretation of the diagram is that one cannot reliably use such information to reach conclusions about a particular individual.

The point can be presented quantitatively as well. Consider the following Venn diagram, where the numbers represent the population inhabiting a particular region:

The stereotypes are supported, from a probabilistic point of view.
In fact, the probability of *P* given *Q* is 95/120, or
about 79%. Similarly, the probability of *not-P* given
*R* is about 79%. But of course the probability of *P*
given *Q* *and* *R* is only 50%. It seems like no
particular conclusions can be drawn from the fact that a particular
individual is both a Quaker and a Republican.

Indeed, whereas many of the early formulations of defeasible reasoning have been credulous, skepticism has gradually emerged as a viable alternative, which can, at times, be better behaved. Arguments have been given in favor of both skeptical and credulous inference. Some have argued that credulity seems better to capture a certain class of intuitions, while others have objected that although a certain degree of “jumping to conclusion” is by definition built into any non-monotonic formalism, such jumping to conclusions needs to be regimented, and that skepticism provides precisely the required regimentation. (A further issue in the skeptical/credulous debate is the question of whether so-called “floating conclusions” should be allowed; see Horty (2002) for a review of the literature and a substantial argument that they should not.)

## 3. Non-monotonic formalisms

One of the most significant developments both in logic and artificial intelligence is the emergence of a number of non-monotonic formalisms, which were devised expressly for the purpose of capturing defeasible reasoning in a mathematically precise manner. The fact that patterns of defeasible reasoning have been accounted for in such a rigorous fashion has wide-ranging consequences for our conceptual understanding of argumentation and inference.

Pioneering work in the field of non-monotonic logics began with the
realization that FOL is inadequate for the representation of
defeasible reasoning. Such a realization was accompanied by the effort
to reproduce the success of FOL in the representation of mathematical,
or formal, reasoning. Among the pioneers of the field in the late
1970's were (among others) J. McCarthy, D. McDermott & J. Doyle,
and R. Reiter (see
Ginsberg (1987)
for a collection of
early papers in the field and
Gabbay et al (1994)
for a more recent collection of excellent survey papers). In 1980, the
*Artificial Intelligence Journal* published an issue (vol. 13,
1980) dedicated to these new formalisms, an event that has come to be
regarded as the “coming of age” of non-monotonic logic.

If one of the goals of non-monotonic logic is to provide a
materially adequate account of defeasible reasoning, it is important to
rely on a rich supply of examples to guide and hone intuitions.
Database theory was one of the earliest sources of such examples,
especially as regards the *closed world assumption*. Suppose a
travel agent has access to a flight database, and needs to answer a
client's query about the best way to get from Oshkosh to Minsk. The
agents queries the database and, not surprisingly, responds that there
are no direct flights. How does the travel agent know?

It is quite clear that, in a strong sense of “know,” the travel
agent does not *know* that there are no such flights. What is at
work here is a tacit assumption that the database is *complete*,
and that since the database does not list any direct flights between
the two cities, then there are none. A useful way to look at this
process is as a kind of *minimization*, i.e., an attempt to
minimize the extension of a given predicate (“flight-between,” in this
case). Moreover, on pain of inconsistencies, such a minimization needs
to take place not with respect to what the database explicitly contains
but with respect to what it implies.

The idea of minimization is at the basis of one of the earliest
non-monotonic formalisms, McCarthy's *circumscription*.
Circumscription makes explicit the intuition that, all other things
being equal, extensions of certain predicates should be
*minimal*. Again, consider principles such as “all normal birds
fly”. Implicit in this principle is the idea that the extension of the
abnormality predicate should be minimal, and that specimens should not
be considered to be abnormal unless there is positive information to
that effect. McCarthy's idea was to represent this formally, using
second-order logic (SOL). In SOL, in contrast to FOL, one is allowed to
explicitly quantify over predicates, forming sentences such as
∃*P*∀*xPx* (“there is a universal
predicate”) or ∀*P*(*Pa ↔Pb*) (“*a*
and *b* are indiscernible”).

In circumscription, given predicates *P* and *Q*, we
abbreviate ∀*x*(*Px* → *Qx*) as
*P≤Q*; similarly, we abbreviate *P≤Q* & ¬
*Q≤P* as *P<Q*. If *A*(*P*) is a
formula containing occurrences of a predicate *P*, then the
circumscription of *P* in *A* is the second-order
sentence *A**(*P*):

A(P) & ¬∃Q[A(Q) &Q<P]

*A**(

*P*) says that

*P*satisfies

*A*, and that no smaller predicate does. Let

*Px*be the predicate “

*x*is abnormal,” and let

*A*(

*P*) be the sentence “All normal birds fly.” Then the sentence “Tweety is a bird,” together with

*A**(

*P*) implies “Tweety flies,” for the circumscription axiom forces the extension of

*P*to be empty, so that “Tweety is normal” is automatically true.

In terms of consequence relations, circumscription allows us to
define, for each predicate *P*, a non-monotonic relation
*A*(*P*) φ
that holds precisely when *A**(*P*)
⊨ φ.
(This basic form of circumscription has been
generalized, for, in practice, one needs to minimize the extension of a
predicate, while allowing the extension of certain other predicates to
vary.) From the point of view of applications, however, circumscription
has a major computational shortcoming, which is due to the nature of
the second-order language in which circumscription is formulated. The
problem is that SOL, contrary to FOL, lacks a complete inference
procedure: the price one pays for the greater expressive power of
second-order logic is that there are no complete axiomatizations, as we
have for FOL. It follows that there is no way to list, in an effective
manner, all SOL validities, and hence to determine whether
*A*(*P*)
φ.

Another non-monotonic formalism inspired by the intuition of
minimization of abnormalities is *non-monotonic inheritance*.
Whenever we have a taxonomically organized body of knowledge, we
presuppose that subclasses inherit properties from their superclasses:
dogs have lungs because they are mammals, and mammals have lungs.
However, there can be exceptions, which can interact in complex ways.
To use an example already introduced, mammals, by and large, don't fly;
since bats are mammals, in the absence of any information to the
contrary, we are justified in inferring that bats do not fly. But if then
we learn that bats are exceptional mammals, in that they do fly, the
conclusion that they don't fly is retracted, and the conclusion that
they fly is drawn instead. Things can be more complicated still, for in
turn, as we have seen, baby bats are exceptional bats, in that they do
not fly (does that make them unexceptional mammals?). Here we have
potentially conflicting inferences. When we infer that Stellaluna,
being a baby bat, does not fly, we are resolving all these potential
conflicts based on a *specificity* principle: more specific
information overrides more generic information.

Non-monotonic inheritance networks were developed for the purpose of
capturing taxonomic examples such as the one above. Such networks are
collections of nodes and directed (“IS-A”) links representing taxonomic
information. When exceptions are allowed, the network is interpreted
*defeasibly*. The following figure gives a network representing
this state of affair:

In such a network, links of the form *A* → *B*
represent the fact that, typically and for the most part, *A*'s
are *B*'s, and that therefore information about *A*'s is
more specific than information about *B*'s. More specific
information overrides more generic information. Research on
non-monotonic inheritance focuses on the different ways in which one
can make this idea precise.

The main issue in defeasible inheritance is to characterize the set
of assertions that are supported by a given network. It is of course
not enough to devise a representational formalism, one also needs to
specify how the formalism is to be interpreted. Such a characterization
is accomplished through the notion of an *extension* of a given
network. There are two competing characterizations of extension for
this kind of networks, one that follows the credulous strategy and one
that follows the skeptical one. Both proceed by first defining the
*degree* of a path through the network as the length of the
longest sequence of links connecting its endpoints; extensions are then
constructed by considering paths in ascending order of their degrees.
We are not going to review the details, since many of the same issues
arise in connection with default logic (which is treated to greater
length below), but
Horty (1994)
provides an
extensive survey. It is worth mentioning that since the notion of
degree makes sense only in the case of acyclic networks, special issues
arise when networks contain cycles (see
Antonelli (1997),
(2005)
for a treatment of inheritance
on cyclic networks).

Although the language of non-monotonic networks is expressively
limited by design (in that only links of the form “IS-A” — and
their negations — can be represented in a natural fashion), such
networks provide an extremely useful setting in which to test and hone
one's intuitions and methods for handling defeasible information. Such
intuitions and methods are then applied to more expressive
formalisms. Among the latter is Reiter's *Default Logic*,
perhaps the most flexible among non-monotonic frameworks.

In Default Logic, the main representational tool is that of a
*default rule*, or simply a *default*. A default is a
*defeasible inference rule* of the form

γ : θ τ

(where γ, θ, τ are sentences in a given language,
respectively called the *pre-requisite*,
the *justification* and the
*conclusion* of the default). The interpretation of the default
is that if γ is known, and there is no evidence that θ
might be false, then τ can be inferred.

As is clear, application of the rule requires that a consistency condition be satisfied. What makes meeting the condition complicated is the fact that rules can interact in complex ways. In particular, it is possible that application of some rule might cause the consistency condition to fail for some, not necessarily distinct, rule. For instance, if θ is ¬τ then application of the rule is in a sense self-defeating, in that application of the rule itself causes the consistency condition to fail.

Reiter's default logic uses its own notion of an *extension*
to make precise the idea that the consistency condition has to be met
both before and after the rule is applied. Given a set Γ of
defaults, an extension for Γ represents a set of inferences that
can be *reasonably* and *consistently* drawn using
defaults from Γ. Such inferences are those that are warranted on
the basis of a maximal set of defaults whose consistency condition is
met both before and after their being triggered.

More in particular (and in typical circular fashion), an extension
for Γ is a maximal subset Δ of Γ the conclusions of
whose defaults both imply all the pre-requisites of defaults in Γ
and are consistent with all the justifications of defaults in Γ.
This definition can be made both less mysterious and more precise as
follows. By a *default theory* we mean a pair
(*W*,Δ), where Δ is a (finite) set of defaults, and
*W* is a set of sentences (a world description). The idea is
that *W* represents the strict or background information,
whereas Δ specifies the defeasible information. Given a pair
(*T*_{1},*T*_{2}) of sets of sentences, a
default such as the
above
is *triggered*
by (*T*_{1},*T*_{2}) if and only if
*T*_{1} ⊨ γ and it's not the case
that *T*_{2}⊨ ¬θ (i.e., θ is
consistent with
*T*_{2}). (Notice how this definition is built “on top”
of ⊨: we could, conceivably, employ
a different consequence relation here, e.g., relevant, intuitionistic, etc..)

Finally we say that a set of sentences *E* is an
*extension* for a default theory (*W*,Δ) if and
only if

E=E_{0}∪E∪ ... ∪_{1}E∪... ,_{n}

where: *E*_{0} = *W*, and

E_{n+1}=E∪ {τ : the default (γ : θ) / τ ∈ Δ is triggered by (_{n}E,_{n}E) }.

It is important to notice the occurrence of the limit *E* in the
definition of *E*_{n+1}: the condition above is
not a garden-variety recursive defintion, but a truly circular
characterization of extensions.

This circularity can be made explicit by giving an equivalent
definition of extension as solution of fixpoint equations. Given a
default theory, let **S** be an operator defined on sets of
sentences such that for any such set *S*,
**S**(*S*) is the smallest set containing
*W*, deductively closed (i.e., such that if
**S**(*S*) ⊨ φ
then φ
∈ **S**(*S*)), and such that if a default with
consequent τ is triggered by (*S*,*S*) then
τ∈**S**(*S*). Then one can show that
*E* is an extension for (*W*,Δ) if and only if
*E* is a fixed point of **S**, i.e., if
**S**(*E*) = *E*.

For any given default theory, extensions need not exist, and even
when they exist, they need not be unique. Let us consider a couple of
examples. Our first example is a default theory that has no extension:
let *W* contain the sentence γ and let Δ comprise the
single default

γ : θ ¬θ

If *E* were an extension, then the default above would have to
be either triggered or not triggered by it, and either case is
impossible. For if the default were triggered, then the consistency
condition would fail, and if if were not triggered then the consistency
condition would hold, and hence the default would have to be triggered
by maximality of extensions.

Let us now consider an example of a default theory with multiple
extensions. As before, let *W* contain the sentence γ, and
suppose Δ comprises the following two defaults

γ : θ ¬τ

and

γ : τ ¬θ

This theory has exactly two extensions, one in which the first default is triggered and one in which the second one is. It is easy to see that at least one default has to be triggered in any extension, and that both defaults cannot be triggered by the same extension.

These examples are enough to bring out a number of features. First,
it should be noted that neither one of the two characterizations of
extension for default logic (i.e., the fixpoint definition and the
pseudo-iterative one) given above gives us a way to “construct”
extension by means of anything resembling an iterative process.
Essentially, one has to “guess” a set of sentences *E*, and then
verify that it satisfies the definition of an extension.

Further, the fact that default theories can have zero, one, or more
extensions raises the issues of what inferences one is warranted in
drawing from a given default theory. The problem can be presented as
follows: given a default theory (*W*,Δ), what sentences
can be regarded as *defeasible consequences* of the theory? At
first sight, there are several options available.

One option is to take the union of the extensions of the theory, and
consider a sentence φ a consequence of a given default theory if
and only if φ belongs to *any* extension of the theory. But
this option is immediately ruled out, in that it leads to endorsing
contradictory conclusion, as in the second example above. It is widely
believed that any viable notion
of defeasible consequence for default
logic must have the property that the set

{φ : (W,Δ) φ }

must be consistent whenever *W* is. Once this option is ruled
out, only two alternatives are left:

The first alternative, known as the “credulous” or “bold” strategy,
is to pick an extension *E* for the theory, and say that φ
is a defeasible consequence if and only if φ ∈ *E*. The
second alternative, known as the “skeptical” or “cautious” strategy, is
to endorse a conclusion φ if and only if φ is contained in
*every* extension of the theory.

Neither the credulous or the skeptical strategy is completely satisfactory. The
problem with the credulous strategy is that the choice of *E* is
arbitrary: with the notion of extension introduced by Reiter,
extensions are *orthogonal*: of any two distinct extensions,
neither one contains the other. Hence, there seems to be no principled
way to pick an extension over any other one. This has led a number of
researchers to endorse the skeptical strategy as a viable approach to
the problem of defeasible consequence. But as showed by Makinson,
skeptical consequence, as based on Reiter's notion of extension, fails
to be cautiously monotonic. To see this, consider the default theory
(*W*, Δ), where *W* is empty, and Δ comprises
the following two defaults:

: θ θ

and

θ ∨ γ : ¬θ ¬θ

This theory has only one extension, coinciding with the deductive
closure of {θ}. Hence, if we define defeasible consequence by
putting (*W*, Δ)
φ
if and only if φ belongs to every
extension of (*W*, Δ), we have (*W*, Δ)
θ, as well as
(*W*, Δ)
θ∨γ
(by the deductive closure of extensions).

Now consider the theory with Δ as before, but with *W*
containing the sentence
θ∨γ.
This theory has two extensions: one the same as before, but also
another one coinciding with the deductive closure of {¬θ},
and hence not containing θ. It follows that the intersection of
the extensions no longer contains θ, so that ({
θ∨γ}, Δ)
θ now *fails*, against cautious
monotony. (Notice that the same example establishes a counter-example
for Cut for the credulous strategy, when we pick the extension
of
({θ∨ γ}, Δ) that
contains ¬θ.)

(The skeptical approach is also, conceivably, subject to
considerations related to *floating conclusions*, as we already
mentioned. But missing these conclusion might indeed be a desirable
feature, as argued by
Horty (2002)).
It is clear that the issue of how to define a non-monotonic
consequence relation for default logic is intertwined with the way that
*conflicts* are handled. The problem of course is that in this
case neither the skeptical nor the credulous strategy yield an adequate
relation of defeasible consequence.

In Antonelli (1999)
a notion of *general extension* for default logic is
introduced, showing that this notion yields a well-behaved relation of
defeasible consequence that satisfies all four requirements of
Supraclassicality, Reflexivity, Cut, and Cautious Monotony. (The same
treatment can be found in
Antonelli (2005)). The idea can be explained in
a particularly simple way in the case of *pre-requisite free*
default theories (called “categorical” in
Antonelli (1999) and
Antonelli (2005)).
A general extension for such a default theory is a *pair*
(Γ^{+}, Γ^{-}) of sets of defaults (or
conclusions thereof) that simultaneously satisfies *two*
fixpoint equations. The set Γ^{+} comprises (conclusions
of) defaults that are explicitly triggered, and the set
Γ^{-} comprises (conclusions of) defaults that are
explicitly ruled out. The intuition is that defaults that are not
ruled out can still prevent other defaults from being triggered
(although they might not be triggered themselves). We thus obtain a
“3-valued” approach (not unlike that of
Kripke's (1975)
theory of truth), in virtue of which general extensions are
now endowed with a non-trivial (i.e., not “flat”) algebraic
structure. It is then possible to pick out, in a principled way, a
particular general extension (for instance, the unique least one,
which is always guaranteed to exist) on which to base a notion of
defeasible consequence. As mentioned, such a notion of consequence
satisfies all four *desiderata* of Supraclassicality,
Reflexivity, Cut, and Cautious Monotony.

A different set of issues arises in connection with the behavior of
default logic from the point of view of computation. For a given
semi-decidable set Γ of sentences, the set of all φ that are
a consequence of Γ in FOL is itself semi-decidable (see the entry
on
computability and complexity).
In
the case of default logic, to formulate the corresponding problem one
extends (in the obvious way) the notion of (semi-)decidability to sets
of defaults. The problem, then, is to decide, given a default theory
(*W*, Δ) and a sentence φ whether
(*W*, Δ) φ,
where
is
defined, say,
skeptically. Such a problem is not even semi-decidable, the essential
reason being that in general, in order to determine whether a default
is triggered by a pair of sets of sentences, one has to perform a
consistency check, and such checks are not computable.

But the consistency checks are not the only source of complexity in default logic. For instance, we could restrict our language to conjunctions of atomic sentences and their negations (making consistency checks feasible). Even so, the problem of determining whether a given default theory has an extension would still be highly intractable (NP-complete, to be precise, as shown by Kautz & Selman (1991)), seemingly because the problem requires checking all possible sequences of firings of defaults.

Default logic is intimately connected with certain *modal*
approaches to non-monotonic reasoning, which belong to the family of
*autoepistemic logics*. Modal logics in general have proved to
be one of the most flexible tools for modelling all sorts of dynamic
processes and their complex interactions (see the entry
“modal logic”,
this Encyclopedia). Beside the
applications in knowledge representation, which we are going to treat
below, there are modal frameworks, known as *dynamic logics*,
that play a crucial role, for instance, in the modelling of serial or
parallel computation. The basic idea of modal logic is that the
language is interpreted with respect to a given set of *states*,
and that sentences are evaluated relative to one of these states. What
these states are taken to represent depends on the particular
application under consideration (they could be epistemic states, or
states in the evolution of a dynamical system, etc.), but the important
thing is that there are *transitions* (of one or more different
kinds) between states.

In the case of one transition that is both *transitive*
(i.e., such that if *a*→*b* and
*b*→*c* then *a*→ *c*) and
*euclidean* (if *a* → *b* and *a*
→ *c* then *b* → *c*), the resulting
modal system is referred to as K45. Associated with each kind of state
transition there is a corresponding modality in the language, usually
represented as a box
□.
A sentence of the
form
□*A* is true at a state
*s* if and only if *A* is true at every state
*s*′ reachable from *s* by the kind of transition
associated with
□
(see
Chellas (1980)
or
Hughes and Cresswell (1996)
for comprehensive introductions to modal
logic).

In autoepistemic logic, the states involved are epistemic states of
the agent (or agents). The intuition underlying autoepistemic logic is
that we can sometimes draw inferences concerning the state of the world
using information concerning our own knowledge or ignorance. For
instance, I can conclude that I do not have a sister given that if I
did I would probably know about it, and nothing to that effect is
present in my “knowledge base”. But such a conclusion is defeasible,
since there is always the possibility of learning new facts. In order
to make these intuitions precise, consider a modal language in which
the necessity operator
□
is interpreted
as “it is known that”. As in default logic or defeasible inheritance,
the central notion in autoepistemic logic is that of an
*extension* of a theory *S*, i.e., a consistent and
self-supporting set of beliefs that can reasonably be entertained on
the basis of *S*. Given a set *S* of sentences, let
*S*_{0} be the subset of *S* composed of those
sentences containing no occurrences of
□;
further,
let the *positive introspective closure*
PIC(*S*_{0}) of *S*_{0} be the set

{□φ : φ ∈S_{0}},

and the *negative introspective closure*
NIC(*S*_{0}) of *S*_{0} the set

{¬□φ : φ ∉S_{0}}.

The set PIC(*S*_{0}) is called the introspective closure
because it explicitly contains positive information about the agent's
epistemic status: PIC(*S*_{0}) expresses what is known
(similarly, NIC(*S*_{0}) contains negative information
about the agent's epistemic status, stating explicitly what is
*not* known).

With these notions in place, we define an extension for *S*
to be a set *T* of sentences such that:

T= { φ : φ follows (in K45) fromS∪ PIC(T_{0}) ∪ NIC(T_{0}) }

Autoepistemic logic provides a rich language, with interesting mathematical properties and connections to other non-monotonic formalisms. It is faithfully intertranslatable with Reiter's version of default logic, and provides a defeasible framework with well-understood modal properties.

## 4. Conclusion

There are three major issues connected with the development of logical
frameworks that can adequately represent defeasible reasoning:
(*i*) material adequacy; (*ii*) formal properties; and
(*iii*) complexity. Material adequacy concerns the question of
how broad a range of examples is captured by the framework, and the
extent to which the framework can do justice to our intutions on the
subject (at least the most entrenched ones). The question of formal
properties has to do with the degree to which the framework allows for
a relation of logical consequence that satisfies the above mentioned
conditions of Supraclassicality, Reflexivity, Cut, and Cautious
Monotony. The third set of issues has to do with computational
complexity of the most basic questions concerning the framework.

There is a potential tension between (*i*) and (*ii*):
the desire to capture a broad range of intuitions can lead to *ad
hoc* solutions that can sometimes undermine the desirable formal
properties of the framework. In general, the development of
non-monotonic logics and related formalisms has been driven, since its
inception, by consideration (*i*) and has relied on a rich and
well-chosen array of examples. Of course, there is some question as to
whether any single framework can aspire to be universal in this
respect.

More recently, researchers have started paying attention to
consideration (*ii*), looking at the extent to which
non-monotonic logics have generated well-behaved relations of logical
consequence. As
Makinson (1994)
points out,
practitioners of the field have encountered mixed success. In
particular, one abstract property, Cautious Monotony, appears at the
same time to be crucial and elusive for many of the frameworks to be
found in the literature. This is a fact that is perhaps to be traced
back, at least in part, to the above-mentioned tension between the
requirement of material adequacy and the need to generate a
well-behaved consequence relation.

The complexity issue appears to be the most difficult among the ones that have been singled out. Non-monotonic logics appear to be stubbornly intractable with respect to the corresponding problem for classical logic. This is clear in the case of default logic, given the ubiquitous consistency checks. But besides consistency checks, there are other, often overlooked, sources of complexity that are purely combinatorial. Other forms of nonmonotonic reasoning, besides default logic, are far from immune from these combinatorial roots of intractability. Although some important work has been done trying to make various non-monotonic formalism more tractable, this is perhaps the problem on which progress has been slowest in coming.

## Bibliography

The following list of references is of course far from exhaustive, but it is only meant to provide entry points to the literature.

- Antonelli, G.A., 1997, “Defeasible inheritance
over cyclic networks”,
*Artificial Intelligence*, vol. 92 (1), pp. 1–23. - –––, 1999, “A directly
cautious theory of defeasible consequence for default logic via the
notion of general extension”,
*Artificial Intelligence*, 109 (1–2): 71–109. - –––, 2005,
*Grounded Consequence for Defeasible Logic*, Cambridge: Cambridge University Press. - Chellas, B., 1980,
*Modal Logic: an introduction*, Cambridge: Cambridge University Press. - Gabbay, D. M., 1985, “Theoretical foundations
for nonmonotonic reasoning in expert systems”, in K. Apt (ed.),
*Logics and Models of Concurrent Systems*, Berlin and New York: Springer Verlag, pp. 439–459. - Gabbay, D., Hogger, C., and Robinson, J.,
(eds.), 1994,
*Handbook of Logic in Artificial Intelligence and Logic Programming*, volume 3, Oxford and New York: Oxford University Press. - Ginsberg, M., (ed.), 1987,
*Readings in Nonmonotonic Reasoning*, Los Altos, CA: Morgan Kauffman. - Hughes, G. and Cresswell, M.,
1996,
*A New Introduction to Modal Logic*, London: Routledge. - Horty, J.F., 1994, “Some direct theories of
nonmonotonic inheritance”, in Gabbay
*et al*. (1994), pp. 111–187. - –––, 2002, “Skepticism and
floating conclusions”,
*Artificial Intelligence Journal*, 135: 55–72. - Kautz, H., and Selman, B., 1991, “Hard
problems for simple default logic”,
*Artificial Intelligence Journal*, 49: 243–279 - Kripke, S., 1975, “Outline of a theory of
truth”,
*The Journal of Philosophy*, 72: 690–716. - Lifschitz, V., 1993, “Circumscription”, in Gabbay, Hogger, and Robinson (eds.) 1994.
- Makinson, D., 1994, “General patterns in
nonmonotonic reasoning”, in Gabbay
*et al*. (1994), pp. 35–110. - McCarthy, J., 1980, “Circumscription — A
Form of Non-Monotonic Reasoning”,
*Artificial Inteligence*, 13: 27–39. - –––, 1986, “Applications of
Circumscription to Formalizing Common Sense Knowledge”,
*Artificial Inteligence*, 28: 89–116. - –––,
1990,
*Formalization of common sense. Papers by John McCarthy edited by V. Lifschitz*, Ablex. - Stalnaker, R., 1994, “Nonmonotonic consequence
relations”,
*Fundamenta Informaticae*, 21: 7–21.

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