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<?xml version="1.0" standalone="yes"?> <Paper uid="C88-2117"> <Title>Default Logic, Natural Language and Generalized Quantifiers</Title> <Section position="3" start_page="556" end_page="556" type="metho"> <SectionTitle> 3 Integrating default logic into Gen- </SectionTitle> <Paragraph position="0"> 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: important restriction is that universes are supposed to be finite at each time t, in order to make our notations below computationally tractable.</Paragraph> <Section position="1" start_page="556" end_page="556" type="sub_section"> <SectionTitle> 3.1 Generalized quantifiers </SectionTitle> <Paragraph position="0"> 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 generalized quantifier Q denotes a relation among sets of entities in a world W. It is noted as:</Paragraph> </Section> </Section> <Section position="4" start_page="556" end_page="557" type="metho"> <SectionTitle> QAB </SectionTitle> <Paragraph position="0"> 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).</Paragraph> <Paragraph position="1"> 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).</Paragraph> <Paragraph position="2"> 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 \[Van Benthem 86\].</Paragraph> <Paragraph position="3"> 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 characterize 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.</Paragraph> <Paragraph position="4"> or, simply as a binary relation since, in our context, R and C are identical: by-def P C.</Paragraph> <Paragraph position="5"> 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:</Paragraph> <Paragraph position="7"> which introduces another interesting type of relation, out of the scope of the present contribution.</Paragraph> <Section position="1" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 3.2 Some examples </SectionTitle> <Paragraph position="0"> As an illustration, we now present possible representations for some of the examples given above in the previous section.</Paragraph> <Paragraph position="1"> Those representations have a knowhdge-base orientation rather than a pure formal semantics one.</Paragraph> <Paragraph position="2"> (a) Many workers have a car.</Paragraph> <Paragraph position="3"> is represented by: by-def(worker(X),car(Y) A to-have(X,Y)).</Paragraph> <Paragraph position="4"> (b) John knows few bird names.</Paragraph> <Paragraph position="5"> is represented by: by-def(bird(Y) A name-of(X,Y), -1 to-know(john, name-of(X,Y)).</Paragraph> <Paragraph position="6"> (c) John sings rarely.</Paragraph> <Paragraph position="7"> 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.</Paragraph> <Paragraph position="8"> is represented as follows:</Paragraph> <Paragraph position="10"> 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).</Paragraph> <Paragraph position="11"> (f) The Mont-Blane is one Of the highest mountains.</Paragraph> <Paragraph position="12"> is represented as follows:</Paragraph> <Paragraph position="14"> is represented by: by-def(to-know (X,mary),to-admire(X,mary)). We could also add that X has the ability to admire.</Paragraph> </Section> </Section> <Section position="5" start_page="557" end_page="557" type="metho"> <SectionTitle> 4 Stability of statements represented </SectionTitle> <Paragraph position="0"> by default rules By stability of a statement, we mean the characterization of conditions under which a statement remains true when the current world is updated. In our framework, stability means the characterization of the conditions under which the set of elements x that satisfy by-def A B remains unchanged, i.e. any deduction made from that default rule for any individual x before the updating remains true after the updating.</Paragraph> <Paragraph position="1"> Representations with defaults appear to have slightly different 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 properties listed below are central to the field of Generalized Quantifiers. null</Paragraph> <Section position="1" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 4.1 Conservativity </SectionTitle> <Paragraph position="0"> 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 nature of the prerequisite A.</Paragraph> </Section> <Section position="2" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 4.2 Extension </SectionTitle> <Paragraph position="0"> 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&quot; be worlds defined on universe U.</Paragraph> <Paragraph position="1"> 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'.</Paragraph> <Paragraph position="2"> - right-irrelevance: Let W&quot; = W U {C'}, then C' is right-irrelevant iff:</Paragraph> <Paragraph position="4"> consistentw(S) is a predicate which is true if in world W the statement S is consistent.</Paragraph> <Paragraph position="5"> If W contains disjunctions of formulae, then it is necessary to consider all maximal extensions E of W to define rightirrelevance. The following condition has to be true: For all E, maximal extension of W such that: (1) E'=EU {C }.</Paragraph> <Paragraph position="6"> (2) E is consistent in W.</Paragraph> <Paragraph position="7"> (3) { x c E lco~siae~t~(n(..~..)) } = { y ~ E'I consistentw,BC..tl..)) }.</Paragraph> <Paragraph position="8"> 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).</Paragraph> </Section> <Section position="3" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 4.3 Monotonicity </SectionTitle> <Paragraph position="0"> 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 :=C/. by- defw A B (downward leftmonotonicity, noted ~MON ).</Paragraph> <Paragraph position="1"> (b) Let W&quot; 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 rightmonotonicity, noted MON.L ).</Paragraph> <Paragraph position="2"> Not surprisingly (default logic is a non-monotonic logic), upward monotonicity does not hold in general.</Paragraph> </Section> </Section> <Section position="6" start_page="557" end_page="559" type="metho"> <SectionTitle> 5 Inferential patterns </SectionTitle> <Paragraph position="0"> 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:</Paragraph> <Section position="1" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 5.1 Restricted transitivity </SectionTitle> <Paragraph position="0"> 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 corresponding to well-formed linguistic expressions explicitly containing 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.</Paragraph> <Paragraph position="1"> (2) Most animals feed their babies.</Paragraph> <Paragraph position="2"> 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)).</Paragraph> <Paragraph position="3"> Notice that A' is bound to A in the consequent, because the two formulae are merged.</Paragraph> <Paragraph position="4"> 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.</Paragraph> <Paragraph position="5"> 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 vegetarians 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 nothing 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 corresponding paraphrase cannot be directly derived. For example, for (6} we could have: The determiner some is more neutral and will be prefered in this type of situation.</Paragraph> </Section> <Section position="2" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 5.2 Distrlbutivity </SectionTitle> <Paragraph position="0"> 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.</Paragraph> </Section> <Section position="3" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 5.3 Contraposition </SectionTitle> <Paragraph position="0"> 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.</Paragraph> <Paragraph position="1"> 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'.</Paragraph> <Paragraph position="2"> 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.</Paragraph> </Section> <Section position="4" start_page="557" end_page="557" type="sub_section"> <SectionTitle> 5.4 Cosymmetry </SectionTitle> <Paragraph position="0"> 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 C/=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 cosymmetry, we have: by-def A -~ B ~ by-def B -~ A.</Paragraph> <Paragraph position="1"> Notice that, due to symmetry of formulae, there is an equivalence 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.</Paragraph> </Section> <Section position="5" start_page="557" end_page="559" type="sub_section"> <SectionTitle> 5.5 Subalternacy </SectionTitle> <Paragraph position="0"> From complex relations holding between different classes of determiners, the property of subalternaey has emerged and turns out to be relevant for statements represented by default logic.</Paragraph> <Paragraph position="1"> This property states that: For all A, B, by-def A B ==~ ~ by-def A (A A -1 B).</Paragraph> <Paragraph position="2"> For the same reasons as above, this expression can be simplified and becomes: by-def A B ~ -~ by-def A -~ B.</Paragraph> <Paragraph position="3"> (10) Most birds fly entails (11) It is false that most birds cannot fly.</Paragraph> <Paragraph position="4"> or, using eontrapasition and if few is the opposite of most: (12) Few birds does not fly.</Paragraph> </Section> </Section> class="xml-element"></Paper>