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<Paper uid="C88-2117">
  <Title>Default Logic, Natural Language and Generalized Quantifiers</Title>
  <Section position="2" start_page="0" end_page="556" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Default logic \[Reiter 80\] is now commonly used in artificial intelligence 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.</Paragraph>
    <Paragraph position="1"> 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 negh;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 \[Mercer 88\].</Paragraph>
    <Paragraph position="2"> 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.</Paragraph>
    <Paragraph position="3"> 2 Default logic for natural language semantlcs</Paragraph>
    <Section position="1" start_page="0" end_page="555" type="sub_section">
      <SectionTitle>
2.1 Introduction to default logic
</SectionTitle>
      <Paragraph position="0"> A default rule has the following general form:</Paragraph>
      <Paragraph position="2"> The meaning of this deduction rule is that given x such that.</Paragraph>
      <Paragraph position="3"> 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.</Paragraph>
      <Paragraph position="4"> Such a rule can be used to represent general laws or statements such as: Most ravens are black.</Paragraph>
      <Paragraph position="5"> or All ravens are black.</Paragraph>
      <Paragraph position="6"> as well as contingent facts such as: Most lights are on.</Paragraph>
      <Paragraph position="7"> The first sentence above is represented as:</Paragraph>
      <Paragraph position="9"> (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 example, 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.</Paragraph>
      <Paragraph position="10"> 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 precise responses. This does not exclude fuzzy and probabilistie logics, but it simply has different motivations and uses.</Paragraph>
      <Paragraph position="11">  Using default logic also permits to skip over some recalcitrant 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.</Paragraph>
      <Paragraph position="12"> A majority of birds fly versus A high majority of birds Ay.</Paragraph>
      <Paragraph position="13"> 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.</Paragraph>
      <Paragraph position="14"> At a formal level, it turns out that additional inferential patterns can be formulated when using default reasoning. Furthermore, due to a certain stability, a default rule used in a semantic representation of a sentence confers to this representation useful properties~in a knowledge base context such as: conservativity, extensibility and some forms of monotonicity.</Paragraph>
      <Paragraph position="15"> Because of their generality and stability, default rules can also play a prominent role to restore consistency in knowledge bases.</Paragraph>
    </Section>
    <Section position="2" start_page="555" end_page="556" type="sub_section">
      <SectionTitle>
2.2 Classes of words and constructions rep-
</SectionTitle>
      <Paragraph position="0"> resented by default logic In a sentence, a wide diversity of types of words and linguistic constructions confer a certain degree of generality (either 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.</Paragraph>
      <Paragraph position="1">  Determiners like: most, many, several, few and probably determiners like: a majority when they have a universal meaning can be represented by default logic: Many workers have a car.</Paragraph>
      <Paragraph position="2"> John knows few bird names.</Paragraph>
      <Paragraph position="3"> The delegates of most workers have arrived.</Paragraph>
      <Paragraph position="4"> Now, a majority oPS students have a car.</Paragraph>
      <Paragraph position="5"> Notice the variety of syntactic positions in which those determiners can appear. Such determiners can also be included into propositional statements: John believes that few birds can fly.</Paragraph>
      <Paragraph position="6"> Determiners introducing some generic statements also fall in this category: A car has four wheels.</Paragraph>
      <Paragraph position="7"> which is equivalent to: Most cars have four wheels.</Paragraph>
      <Paragraph position="8"> However, a generic statement like: A car has at least one door.</Paragraph>
      <Paragraph position="9"> should rather be universally quantified.</Paragraph>
      <Paragraph position="10"> 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.</Paragraph>
      <Paragraph position="11"> Relative meaning can also be introduced by an implicit restriction on the set of instances refered to, as in: John met many people to-day.</Paragraph>
      <Paragraph position="12"> ~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.</Paragraph>
      <Paragraph position="13"> 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~telement introducing the restriction can be explicit (e.g. t~lay) or implicit as in: John ate many apples.</Paragraph>
      <Paragraph position="14"> 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.</Paragraph>
      <Paragraph position="15"> uttered in a world where John is the manager of a company, people refers to the staff of the company and John met everybody 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.</Paragraph>
      <Paragraph position="16"> 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.</Paragraph>
      <Paragraph position="17"> 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.</Paragraph>
      <Paragraph position="18">  Adverbs introducing a form of universality or contributing to it can originate a default logic based representation: John sings rarely.</Paragraph>
      <Paragraph position="19"> Mary is incidentally (very rarely) on time.</Paragraph>
      <Paragraph position="20"> John meets Sue almost every day.</Paragraph>
      <Paragraph position="21"> John often travels by bus rather than by subway in Paris. For default logic to be used, the concept or the quality modified 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.</Paragraph>
      <Paragraph position="22"> 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.</Paragraph>
      <Paragraph position="23"> This can be formulated in an equiwlent way in terms of default 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 &gt; el then infer that e2 &gt; el.</Paragraph>
      <Paragraph position="24"> Sentence 3 requires events to be associated with, for example, 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.</Paragraph>
      <Paragraph position="25"> Various factors llke temporal factors can restrict the scope of a default: From 8 to 10 am, Mary is very rarely in her omce.</Paragraph>
      <Paragraph position="26">  but they still have a universal meaning over that restricted domain.</Paragraph>
      <Paragraph position="27">  Adjectives in the superlative, in particular when the statement is not absolutely incompatible with a small number of exceptions 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.</Paragraph>
      <Paragraph position="28"> 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.</Paragraph>
      <Paragraph position="29"> 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.</Paragraph>
      <Paragraph position="30"> 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 emphasized) as in: Mary is the worse person to work with: she is (almost) never on time.</Paragraph>
      <Paragraph position="31"> where the quality refered to is to be never on time. In this case, the representation becomes much simpler.</Paragraph>
      <Paragraph position="32">  Agentless passive constructions of verbs like to admire, to hate, to neglect or ~o laugth at, in some contexts lend very well themselves to a representation with defaults. For example, a sentence like: Mary is admired.</Paragraph>
      <Paragraph position="33"> can be paraphrased by: Most persons who know Mary admire her.</Paragraph>
      <Paragraph position="34">  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.</Paragraph>
      <Paragraph position="35"> 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 intensionality. Using possible world semantics could be a mean to solve this problem, but this is still unclear to us.</Paragraph>
    </Section>
  </Section>
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