Default Logic, Natural Language and 
Generalized Quantifiers 
Patrick Saint-Dizier 
1RISA-INRIA, CAMPUS de BEAULIEU, 35042 RENNES -Cedex FRANCE 
Abstract 
Tha use of default logic to represent various linguistic 
constru,'tions is explored in this paper. Default logic is 
then integrated into a theory of natural language seman- 
flee, namely Generalized Quantifiers. Finally, propertiee 
of haterest to the AI community such as the characteri- 
zation of troth peraietence and come inferential patterns 
are emphasized. 
1 Introduction 
Default logic \[Reiter 80\] is now commonly used in artificial in- 
telligence systems and turns out to be one of the major research 
fields in knowledge representation. One of the first motivations 
of default logic wan to give a more accurate semantic represem 
tation to natural language statements of the form: 
Birds fl~: 
Most birds fly. 
or~ arguably: 
Typicall2o; birds lty~ 
The idea was to define a representation that permits ex~ 
ceptions to be coherent togeLher with a general statement or 
law. -Very soon this linguistic motivation becanm somewhat ne- 
gh;cted, yielding the way to theoretical investigations in non~ 
monotonic reasoning and to the elaboration of automatic the~ 
orem prover,. For a good introduction see \[Besnard 89\]. The 
only works we are aware of in natural language processing are 
the use of default logic to solve anaphoras \[Dunin-Keplitz 84\], 
to model dialogs \[Joshi et al. 87\] and to derive presuppositions 
from sentences \[Mercer 88\]. 
In this contribution, we come back to the very first moti~ 
ration of default logic and explore its integration into natural 
language semantics, as introduced in \[Saint-Dizier 87\]. More 
precisely, we propose an integration of default logic into Gent. 
eralized Quontifiers framework \[Barwise and Cooper 81\]. This 
leads us to reformulate default logic within the framework of 
set theory defined on universes. We first propose several types 
of linguistic expressions for which a representation by a default 
rule turns out to be relevant. Next, we present some basic 
properties about truth persistence of statements represented 
by default logic and, finally, we propose several inferential pat~ 
terns which permit to derive new default rules on the one hand 
and new linguistic expressions on the other hand. 
2 Default logic for natural language 
semantlcs 
2.1 Introduction to default logic 
A default rule has the following general form: 
P : R 
C 
The meaning of this deduction rule is that given x such that. 
P is true, if there are no contradiction to R being true in the 
current world, then infer C. P is called the prerequisite and C 
is the conclusion. R is a condition which has to be coherent 
with the current world. 
Such a rule can be used to represent general laws or state- 
ments such as: 
Most ravens are black. 
or 
All ravens are black. 
as well as contingent facts such as: 
Most lights are on. 
The first sentence above is represented as: 
raven(X) : black(X) 
black(X) 
(variables are represented by capital letters) 
The difference between general lawn and contingent state-- 
ments is however fundanmntal but it cannot be captured 
straightforwardly within default logic. Indeed, in the first ex- 
ample, blackness is taken as a property inherent to ravens, not 
completely based on observation but also on induction. Default 
logic permits this property to remain inherent to ravens even 
if there are exceptions. The second sentence is strictly based 
on observation and is valid only on a certain time interval t 
at a given location 1. The distinction between contingent facts 
and general laws can however be context-dependent. Suppose 
the above sentence is stated in the context of a factory work~ 
ing 24 hours every day then the fact that lights are on can be 
interpreted as a law imposed, for instance, for security reasons, 
There is another difference: the first sentence may be true 
even if there are no ravens in the current world whereas the 
second sentence requires a non-empty denotation for the part 
of noun phrase following most, in other terms, the presence in 
our example of at least two lights. 
Default logic permits to assign a property (C) to a precise 
individual in a given world, modulo some coherence conditions 
(R). This can be contrasted with filzzy and probabilistic 10g.= 
ics where a set of individuals is characterized as a whole by a 
property. Thus, no property can be deduced for a given indi.- 
viduM in this set with a full degree of certainty. Our choice 
of default logic is then motivated by the need of being able to 
make inferences about a precise individual and to provide pre- 
cise responses. This does not exclude fuzzy and probabilistie 
logics, but it simply has different motivations and uses. 
555 
Using default logic also permits to skip over some recalci- 
trant representational problems of fuzzy logic, but, on the other 
hand, it cannot take into account slight differences between, for 
example: 
Most birds fly Versus Almost all birds tty. 
A majority of birds fly versus A high majority of birds Ay. 
The question however is to know what is the real nature of 
the difference between those statements and if it is more than 
a connotative meaning. 
At a formal level, it turns out that additional inferential 
patterns can be formulated when using default reasoning. Fur- 
thermore, due to a certain stability, a default rule used in a 
semantic representation of a sentence confers to this represen- 
tation useful properties~in a knowledge base context such as: 
conservativity, extensibility and some forms of monotonicity. 
Because of their generality and stability, default rules can also 
play a prominent role to restore consistency in knowledge bases. 
2.2 Classes of words and constructions rep- 
resented by default logic 
In a sentence, a wide diversity of types of words and lin- 
guistic constructions confer a certain degree of generality (ei- 
ther contingent or permanent) to the statement in which they 
are included. In this class fall words like context-dependent 
determiners, qnantiflcational adverbs, propositional adjectives 
and adverbs and some very specific adjectives (worse, perfect, 
ideal,...). Constructions like superlatives and some agentless 
passive constructions also belong to this class. It turns out that 
these words or constructions can be represented in a number 
of contexts with a greater precision and adequacy by default 
logic. We now briefly examine some simple examples. 
2.2.1 Context-dependent determiners 
Determiners like: most, many, several, few and probably de- 
terminers like: a majority when they have a universal meaning 
can be represented by default logic: 
Many workers have a car. 
John knows few bird names. 
The delegates of most workers have arrived. 
Now, a majority o£ students have a car. 
Notice the variety of syntactic positions in which those de- 
terminers can appear. Such determiners can also be included 
into propositional statements: 
John believes that few birds can fly. 
Determiners introducing some generic statements also fall 
in this category: 
A car has four wheels. 
which is equivalent to: 
Most cars have four wheels. 
However, a generic statement like: 
A car has at least one door. 
should rather be universally quantified. 
A context-dependent determiner in a situation where it has 
a relative meaning cannot in general be represented by default 
logic. By relative meaning, we mean, for example, sentences 
where the context-dependent determiner is in the scope of a 
quantification introducing distributivity: 
The owners of several cars pay an additional tax. 
Relative meaning can also be introduced by an implicit re- 
striction on the set of instances refered to, as in: 
John met many people to-day. 
~her~ the set of people that John can meet in a day is 
imp~'ieit~y r~stricted to a (small) subset of all the people in the 
curxent world. 
The latter example shows that default logic cannot be used 
to represent determiners where the set of elements refered to 
by the noun phrase following that determiner is restricted by 
a constituent outside that noun phrase. Futhermore, th~tele- 
ment introducing the restriction can be explicit (e.g. t~lay) 
or implicit as in: 
John ate many apples. 
where the set of apples in which John ate many apples can 
be implicitlyrestricted by the context (e.g. in the basked, in the 
frig...) or by the semantics of the verb to eat (a human cannot 
eat more than a certain quantity of food per day). An informal 
criterion to determine whether a determiner can be represented 
by default logic is to substitute all or no respectively for most, 
many.., and few and to check if it is possible to build a coherent 
world from the cnrent world with less exceptions (i.e. a more 
uniform world as in \[Delgrande 87\]) so that the universally 
quantified statement is true. If there is such a world then the 
original determiner can be represented by a default rule. 
For example, consider the sentence: 
John met many people to-day. 
uttered in a world where John is the manager of a company, 
people refers to the staff of the company and John met every- 
body in the company except n people (n being small). If we 
substitute all for many, we obtain the sentence: 
John met all people to-day. 
and if we can build a consistent world W' from W where 
John met those n people in addition to the others then many 
can be represented by a default rule. 
If John is a politician and people refers to the people who 
want to vote for him (suppose this number is large), then it is 
not possible to consistently build a world in which John met 
every people in a single day, in particular if those people are 
scattered throughout the whole country. In this latter case, 
many cannot be represented by a default rule. 
2.2.2 Quantiflcationa\] adverbs 
Adverbs introducing a form of universality or contributing to 
it can originate a default logic based representation: 
John sings rarely. 
Mary is incidentally (very rarely) on time. 
John meets Sue almost every day. 
John often travels by bus rather than by subway in Paris. 
For default logic to be used, the concept or the quality mod- 
ified by the adverb has to be countable and quantifiable. The 
first example can be paraphrased as: 
At most times t (or preferably, on most time intervals t), 
by default John does not sing. 
The representation of sentence 2 requires the use of events 
and is far more complex. Roughly speaking, we can paraphrase 
this sentence by: 
In most eases when Mary has something to do, she does it 
later than the scheduled time for this work. 
This can be formulated in an equiwlent way in terms of de- 
fault rule. If el is the event associated with the scheduled time 
for the beginning of the activity and if e2 is the event of Mary 
beginning to do this activity, then if there is no contradiction 
with e2 > el then infer that e2 > el. 
Sentence 3 requires events to be associated with, for exam- 
ple, a date and a duration. It can be paraphrased by: 
If it is not inconsistent that John meets Sue to-day (e.g. 
John is in the same location as Sue, etc...), then infer that 
there is a time interwl t during which John meets Sue. 
Various factors llke temporal factors can restrict the scope 
of a default: 
From 8 to 10 am, Mary is very rarely in her omce. 
556 
but they still have a universal meaning over that restricted 
domain. 
2.2.3 Adjectives 
Adjectives in the superlative, in particular when the statement 
is not absolutely incompatible with a small number of excep- 
tions can be represented by a default rule. This permits to 
avoid a too strong and definitive formulation. Consider: 
The Mon~-Blanc is (one of) the highest mountains. 
Represe:fl;ing this sentence by default logic permits: 
s to assume that a mountain whose elevation is unknown 
is lower than the Moat-Blanc, 
s to accept without introducing inconsistencies that there 
are mountains with an elevation explicitly known and 
greater than the Mont-Blanc elevation. 
This approach is particularly relevant for superlatives: it turns 
out indeed that most superlatives are not completely universal. 
They are often true in a coherent subset of the current world. 
In the same range of ideas, adjectives such as ideal, worse of 
perfect turn out to be, in most contexts, implicit superlatives 
ranging over several properties. In: 
Jim is the ideal hiker. 
idea\] couht mean that Jim has most of the qualities of a hiker 
in something close to the superlative. Thus, such a sentence can 
be represented by a conjunction of properties in the superlative, 
each of them being separately represented by a default rule. 
The range of properties refered to by adjectives llke ideal or 
worse can be restricted (or a specific property can be empha- 
sized) as in: 
Mary is the worse person to work with: she is (almost) never 
on time. 
where the quality refered to is to be never on time. In this 
case, the representation becomes much simpler. 
2.2.4 Agentless passive constructions 
Agentless passive constructions of verbs like to admire, to hate, 
to neglect or ~o laugth at, in some contexts lend very well them- 
selves to a representation with defaults. For example, a sen- 
tence like: 
Mary is admired. 
can be paraphrased by: 
Most persons who know Mary admire her. 
2.2.5 Propositional adverbs and adjectives 
Adverbs and adjectives such as probable, likely, unlikely and 
certainly in some contexts modify the certainty of a statement, 
as in the sentence: 
It is unlikely that computer science students know baroque 
music composers. 
These adverbs and adjectives are basically interpreted with 
an intensional meaning. However, representing them by default 
logic is also very relevant if we do not want to stress only on 
the likelihood of the statement but if we also want to have a 
more precise reading, for example if we want to state that: 
If X is a computer science student and if it is not consistent 
to say that X does not know any baroque music composer then 
infer that X does not know any baroque music composer. 
An interesting problem at this level would be to investigate 
the possibility of using default reasoning paired with intension- 
ality. Using possible world semantics could be a mean to solve 
this problem, but this is still unclear to us. 
3 Integrating default logic into Gen- 
eralized Quantiflers 
In the examples we have given above, we have restricted our 
attention to the most straightforward uses of default logic. In 
particular, the expressions or concepts used are: 
• completely defined (no pronominal references, ...), 
• extensional, 
• discrete (no continuous uses as in "a lot of snow"). 
None of these restrictions are, however, essential. Another more 
important restriction is that universes are supposed to be finite 
at each time t, in order to make our notations below computa- 
tionally tractable. 
3.1 Generalized quantifiers 
To deal with semantic representations, we adopt the spirit of 
the Generalized Quantifiers framework \[Barwise and Cooper 
81\]. This approach is, in fact, of much interest because it is 
quite close to current research in knowledge bases. A general- 
ized quantifier Q denotes a relation among sets of entities in a 
world W. It is noted as: 
QAB 
where A and B are linguistic expressions or their set- 
theoretic equivalents. For example: 
All ravens are black 
is noted as: 
AU (ravens) (are black). 
All establishes a relation between the set of ravens and the 
set of entities which are black. Let us note the denotation 
of A in world W (H A \]lw ) A' and that of B ( I\[ 13 \]\[w), 
B'. Then, every generalized quantifier Q can be represented 
by a corresponding numerical relation RQ \[Van Benthem 86\], 
\[Vesterstahl 84,85\] with the following definition: 
Q A B ~ R~(I A' - B' I, IA' rl B' I). 
In Generalized Quantifiers, determiners are studied in a 
principled way by looking at their semantic properties. This 
study appears to have enough logical foundations to motiw~.tc 
theoretical investigations. Generalized Quantifiers also turn 
out not to be limited to representing determiners but extends 
to the semantics of other structures such as conditionals \[Van 
Benthem 86\]. 
We now turn to informally introduce default logic into the 
Generalized Quantifiers framework. A default rule is basically 
used to conclude a formula C for a given entity e, satisfying a 
prerequisite P, modulo R, However, it is also possible to charac- 
terize the set E of elements e satisfying P and for which C can 
be concluded modulo the coherence control on 1L If we view a 
default rule as a ternary relation: 
by-def P R (7. 
or, simply as a binary relation since, in our context, R and 
C are identical: 
by-def P C. 
then \[\[ P \]lw and I\[ C \]\[w can be defined and the above 
relation is a relation among sets of individuals over world W 
in a way similar Vo the determiners accounted for within the 
Generalized Quantiflers framework. In addition, a third set 
should be mentioned, which is the set of exceptions: 
, = lIP ^ -,C\]lw. 
which introduces another interesting type of relation, out of 
the scope of the present contribution. 
557 
3.2 Some examples 
As an illustration, we now present possible representations 
for some of the examples given above in the previous section. 
Those representations have a knowhdge-base orientation rather 
than a pure formal semantics one. 
(a) Many workers have a car. 
is represented by: 
by-def(worker(X),car(Y) A to-have(X,Y)). 
(b) John knows few bird names. 
is represented by: 
by-def(bird(Y) A name-of(X,Y), 
-1 to-know(john, name-of(X,Y)). 
(c) John sings rarely. 
is represented by: 
by-clef(time(T), ~ to-sing(john,T)). 
T is a precise time or, preferably, a time interval. 
(d) John meets Sue almost every day. 
is represented as follows: 
by-def(day(D),(3 H1,H2, H2 > H1, 
to-meet (john,sue,time(D,H1,H2)))). 
(e) John often travels by bus rather than by subway. 
is represented by: 
by-def((bus(X) A subway(Y) A to-travel(john,Z) ^ ((Z=X) 
v (z=Y))), z=x ). 
This logical representation means that if John travels by 
Z which can be either a bus or a subway (X or Y), then, by 
default, John travels by bus (i.e. by X). 
(f) The Mont-Blane is one Of the highest mountains. 
is represented as follows: 
by-def((mountain(X) A -,(X = mont-blanc) A eleva- 
tion(X,Y) A elevation(mont-blanc,M)), Y < M ). 
(g) Mary is admired. 
is represented by: 
by-def(to-know (X,mary),to-admire(X,mary)). 
We could also add that X has the ability to admire. 
4 Stability of statements represented 
by default rules 
By stability of a statement, we mean the characterization of 
conditions under which a statement remains true when the cur- 
rent world is updated. In our framework, stability means the 
characterization of the conditions under which the set of ele- 
ments x that satisfy by-def A B remains unchanged, i.e. any 
deduction made from that default rule for any individual x be- 
fore the updating remains true after the updating. 
Representations with defaults appear to have slightly differ- 
ent properties than their more classical counterparts. Among 
those properties, we now present some of those which are of 
much interest to knowledge representation systems. The prop- 
erties listed below are central to the field of Generalized Quan- 
tifiers. 
4.1 Conservativity 
For all A, B being linguistic expressions (or their set-theoretlc 
equivalents): 
by-def A B ~ by-def A (.4 A B) (noted CONS) 
The equivalence is straightforward in virtue of the very na- 
ture of the prerequisite A. 
4.2 Extension 
Let us first introduce the notion of irrelevance. The idea is 
that propositions irrelevant to a default statement should have 
55B 
no consequence on any inference involving that statement. This 
idea was developped by the philosophical community and, more 
recently by \[Delgrande 87\]. In our framework, if we consider 
the statement: by-defA B, then, intuitively, C is irrelevant for 
this statement if knowing C does not alter the set of individuals 
for Which that statement holds. We differenciate right and left 
irrelevance: 
Let W, W' and W" be worlds defined on universe U. 
Let A, B and C be linguistic expressions or their set- 
theoretic equivalents. We consider the statement by-def A B. 
Then we have the following properties: 
- left-irrelevance: 
C is left-irrelevant iff: W' = W U {C} then 
I\[ A \]lw = I\[ A \]lw'. 
- right-irrelevance: 
Let W" = W U {C'}, then C' is right-irrelevant iff: 
{ • ~ w I co~si~t~,~tw(B(..~..))) = 
( y ~ w" I co~te,,tw.(B(..y..)) }. 
consistentw(S) is a predicate which is true if in world W 
the statement S is consistent. 
If W contains disjunctions of formulae, then it is neces- 
sary to consider all maximal extensions E of W to define right- 
irrelevance. The following condition has to be true: 
For all E, maximal extension of W such that: 
(1) E'=EU {C }. 
(2) E is consistent in W. 
(3) { x c E lco~siae~t~(n(..~..)) } = 
{ y ~ E'I consistentw,BC..tl..)) }. 
Then, we say that C is irrelevant to by-def A B if it is 
both left and right irrelevant. Then the property of extension 
follows: 
For all A, B, for all W, W' such that W' is an extension of 
W where only irrelevant statements to by-def A B have been 
added, then: 
by - defw A B ~ by - defw, A B. (noted EXT). 
4.3 Monotonicity 
For all A, B, W, W' : 
(a) Let W' be an extension of W such that W N W' is a 
set of left-irrelevant statements then: 
by- defw, A B :=¢. by- defw A B (downward left- 
monotonicity, noted ~MON ). 
(b) Let W" be an extension of W such that W N W n is a 
set of right irrelevant statements then: 
by - defw, A B ==~ by - defw A B (downward right- 
monotonicity, noted MON.L ). 
Not surprisingly (default logic is a non-monotonic logic), 
upward monotonicity does not hold in general. 
5 Inferential patterns 
Some inferential patterns within Generalized Quantifiers 
\[Vesterstahl 84,85\], \[Van Benthem 86\] also hold, with some 
restrictions, for default logic. Some additional patterns can 
be formulated, given the specificities of default logic. These 
patterns permit to derive new rules from previous ones and 
to generate new linguistic expressions. Here are some simple, 
basic inferential patterns: 
5.1 Restricted transitivity 
For all A, B, C linguistic expressions: 
• (by-def A (is B)) A ((all B) C) :=~ (hy-def A C). 
(is B), (all B), ... are recta-linguistic expressions corre- 
sponding to well-formed linguistic expressions explicitly con- 
taining the verb to be or the determiner all. This inferential 
pattern can be used, for instance, to deduce (2) from (1): 
(1) Most animals are mammals and All mammals feed their 
babies. 
(2) Most animals feed their babies. 
via the following instanciation of the pattern: 
by-def(an~mal(A),mammal(A)) A (ali(A'),nmmmal(A') A 
(baby-of(B,A') A to-fecd(A',B))) ==~ 
by-def(an2mai(A), baby-of(B,A) A to-feed(A,B)). 
Notice that A' is bound to A in the consequent, because the 
two formulae are merged. 
The denotation of B need not be included in the denotation 
of A. For example: 
Most workers are union members and All union members 
are on strike entail: Mast workers are on strike. 
Here is another restricted transitivity pattern: 
• (by-def X (is B)) A ((no B) C) ~ (by-clef A -I C). 
or, equivalently: 
by-def(A (is B)) A ((all B), - C) ~ by-def A -i C. 
Thus, (4) can de deduced from (3): 
(3) Most animals are mammals and No mammal can t/y. 
(4) Most animals cannot fly. 
Notice that in the patterns already stated, the determiner at 
the origin of the default remains unchanged in the conclusion. 
Finally, here is the last restricted transitivity pattern: 
• ((all A) (are B)) A (by-def B C) ==~ (by-def A C). 
Then, for example: 
(5) All raammals are animals and Most animals are vege- 
tarians entails (6) Most mammals are vegetarians. 
This latter pattern is however weaker than the previous 
ones. Nothing, indeed, excludes that there exist models M~ in 
which no n~mnnals are vegetarians since in the premises noth- 
ing is said about the intersection of the set of mammals and 
the set of vegetarians. If the intersection is empty, then the 
default rule will simply be never applied. In the premises of 
the three tlrst inferential patterns, there is a guarantee that 
the intersection of \[\[ A \]lle and I\[ C \]lw is not empty, provided 
that I\[ A \]Iv/, \[\[ B \]lw and \[\[ C \]lie are non-empty sets. As a 
consequence, this latter inferential pattern is valid but its corre- 
sponding paraphrase cannot be directly derived. For example, 
for (6} we could have: 
(6a) Most mammals are vegetarians. 
(6b) Several mammals are vegetarians. 
(6c) Some mammals are vegetarians. 
The determiner some is more neutral and will be prefered 
in this type of situation. 
5.2 Distrlbutivity 
If B and C are independent properties then: 
(hy-def A (B A C) ==~ (by-def A B) A (by-def A C) 
(by.-def A B) 
==~ (by-clef A C). 
Thus, for example: 
Most workers own a car and are married entails: 
Most workers own a car and Most workers are married. 
The reverse pattern: 
(by-def A B) A (by-defX' C) ~ by-def A (B A C) 
where A' is a copy of A with different variables, also holds 
but it is somewhat weaker in the sense that the denotation of 
B A C ill W is included in the denotation of B in W and 
in that of C in W. Thus, the same remark as for the previous 
pattern holds: the determiner at the origin of the default rule 
in the pre~rdses is not preserved and another context-dependent 
determiner can be more appropriate, depending on how much 
the default has been weaken, i.e. on how much the number of 
exceptions to the default rule has increased. This number is 
characterized by the cardinal of the following set: 
(l\[ A ^ B\]Iw n I\[B\]lw) u (1\[ A A C \]lw n lie\]lie). 
In this case, we also adopt the determiner some as a neutral 
representation. 
5.3 Contraposition 
For all linguistic expressions A, B and C such that 
llAllw c Iic\]l*, llBllie c licllw, then: 
by-def A B ==Y by-clef (C A ~ B) (C A -- A). 
For example: 
(7) Most workers are union members entails: 
(8) Most people who are not union members are people who 
are not workers. 
If (3 is the set of all entities of the current world W then 
contraposition permits to express that: 
by-def A B ~ by-def ~B -~A . (Extended contraposition). 
This property goes beyond a theorem given in \[van Eijck 
84\] which states that: 
A quantifier Q observes contrapasition iff Q is of the form: 
'at most k A are not B'. 
in the sense that (1) k is not implicitly intended to be quite 
small (with respect to the size of the world) and (2) k can be 
null. In fact, the value of k turns out to be irrelevant since 
each time a default rule is used a test of coherence is made on 
a formula. 
5.4 Cosymmetry 
As shown in \[van Eijck 84\], if C = B, then we have the property 
of cosymmetry: 
by-def A (A A -~ B) ==~ by-def B (B A -~ A). 
Furthermore, since reflexivity holds for all A: 
by-def A A 
we have: 
by-def A B ¢=e~ by-def A ( A A B ). 
Suppose that: B = -1 B', then: 
by-def A B ~ by-def A ( A A -~ B' ) 
and, finally, from this result and the definition of cosymme- 
try, we have: 
by-def A -~ B ~ by-def B -~ A. 
Notice that, due to symmetry of formulae, there is an equiv- 
alence instead of an implication. 
For example: 
(8) Most teenagers are not married is logically equivalent to 
(9) Most married people are not teenagers.. 
This latter result can also be used to build passive forms 
from their affirmative counterparts. 
5.5 Subalternacy 
From complex relations holding between different classes of de- 
terminers, the property of subalternaey has emerged and turns 
out to be relevant for statements represented by default logic. 
This property states that: 
For all A, B, by-def A B ==~ ~ by-def A (A A -1 B). 
For the same reasons as above, this expression can be sim- 
plified and becomes: 
by-def A B ~ -~ by-def A -~ B. 
(10) Most birds fly entails 
(11) It is false that most birds cannot fly. 
or, using eontrapasition and if few is the opposite of most: 
(12) Few birds does not fly. 
559 
6 Epilogue 
Default logic permits to represent with a greater accuracy and 
relevance several types of words and constructions. We have 
proposed here to integrate it into the formal framework of Gen- 
eralized Quantifiers. Default logic also exhibits a number of 
particular properties at the level of the characterization of the 
truth persistence of a statement represented by default logic 
in a knowledge base being updated. Next, new and revised 
inferential patterns are introduced and illustrated. These pat- 
terns permit to derive new default rules and to construct new 
linguistic expressions from previous ones. 
The forthcoming works include: 
(1) The extension and investigation of those linguistic ex- 
pressions that can be represented by default logic. 
(2) A formal characterization of contextual situations where 
default logic can (or must) be used, permitting thus the spec- 
ification of contextual semantic compositional rules, coherent 
with the representation defined in \[saint-Dizier 86\]. 
(3) The establishment of a link with the logic of presuppo- 
sitions. 
(4) The integration of default logic into other formal theo- 
ries of natural language semantics, in particular into the frame- 
work of Situations Semantics \[Barwise and Perry 83\]. 
Acknowledgements I am very grateful to Philippe 
Besnard, Mario Borillo and Jim Delgrande for their useful com- 
ments on this work. I thank also Dag Vesterstahl for providing 
me several of his publications mentioned below. This work was 
supported by the French INRIA and the PRC CNRS Commu- 
nication Homme-machine, both civilian public institutions. 

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