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E@ qbjbj{rVGVGVGpHIr^JKJKTLLLLLLPPPP;PVW\$,`R~bL{\LLLLL{\LL@^JNJNJNLLLPJNLPJNJNbNbNL>KVGtMbNNV^0^bNbJNbbNrrbbN$JNLLL{\{\rrDVGJNrrVGIII Defaults and the Meaning of Generic Sentences
Abstract
After giving a short introduction to nonmonotonic logic, several approaches to the analysis of generic sentences are considered and their shortcomings are discussed. A metalinguistic model of the semantics of distributive generics is proposed that relates generic sentences and default expressions in a less direct way than the so-called quantified defaults approach. The metalinguistic conception profits from the notion of markedness as developed within intrasegmental phonology and extends it to the conceptual domain. The new approach maintains the main advantages of earlier approaches to generics but avoids their shortcomings.
1. General introduction
Nonmonotonic logic (Reiter 1980, McDermott & Doyle 1980) 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 nonmonotonic logic was to give a more accurate semantic representation to generic sentences like Ravens are (normally) black, Birds fly, and A dog barks. The idea was to define a representation that permits exceptions to be compatible with a general statement or law. Very soon the linguistic motivation became somewhat neglected, yielding the way to theoretical investigations in nonmonotonic reasoning, to the development of model theories and proof procedures.
In this paper, we come back to the very first motivation of nonmonotonic logic and explore its integration into the semantics of generic sentences.
Generic is not an uniform phenomenon. For example, the distinction between distributive generics (e. g. A dog barks, Ravens are black) and non-distributive generics (e. g. The dodo is extinct) is well-founded (cf. Geurts 1985, Heyer 1987, Krifka 1987). In this paper we will consider only the semantic of distributive generics.
It seems that distributive generics are paraphrasable as quantified expressions. For example, the sense of Ravens are black can be expressed equally well by saying that all "normal" (or "typical", or "unmarked") ravens are black. Nonmonotonic logic suggests quantified default expressions to represent the meaning of such sentences. In the system of McDermott & Doyle (1980) this may explicitly be stated as follows for the given example:
(1) (x(RAVEN(x)(M(BLACK(x)) ( BLACK(x))
This formula can be read informally as "For all x, if x is a raven and it is consistent to assert that x is black, then x is black". Let us call this approach the quantified default approach to distributive generics. Clearly, it gives a direct translation of generic statements into nonmonotonic logic.
After giving a short introduction to nonmonotonic logic (Sections 2 & 3), it is the first aim of this paper to argue that the quantified default approach fails: the meaning of (distributive) generic sentences cannot be represented by quantified defaults adequately (Section 4). Moreover, I will consider two further approaches to generic statements in order to demonstrate their shortcomings. One is Carlson's (1980) analysis of generic terms treating them as proper names of kinds (Section 5). The other is an analysis put forward by Delgrande (1987) which treats distributive generics as variable conditionals within an improved framework of possible-worlds semantic (Section 6). Finally, I will propose a metalinguistic model of the semantics of distributive generics (Section 8). This approach relates generic sentences and default expressions in a less direct way than the quantified default approach. Generic sentences are associated with certain meta-theoretic statements about a modal base that is realized by a default theory. The metalinguistic conception profits from the notion of markedness as developed within intrasegmental phonology (Section 7) and extends it to the conceptual domain. I hope to show that the new approach maintains the main advantages of both Carlson's and Delgrande's analyses but avoids their shortcomings.
2. An introduction to nonmonotonic logic
Let us first take a closer look at a formula like (1) and the inferences it sanctions. Let us assume we have a theory whose only two axioms are
(1) (x(RAVEN(x)(M(BLACK(x)) ( BLACK(x))
(2) RAVEN(ALBIN)
Here M(p) does not merely mean that p is consistent - it means that p is consistent with the (nonmonotonic) theory that contains those two axions. We would expect BLACK(ALBIN) to be a theorem of this theory. If we change the theory by adding
(3) (BLACK(ALBIN)
as an axiom (Albin might be an albino raven), we then change the meaning of M(p) to be that p is consistent with the nonmonotonic theory that contains the axioms (1) - (3) and we would not expect BLACK(ALBIN) to be a theorem. The operator M changes its meaning with context just like indexical words do in natural language.
This type of reasoning is an important form of nonmonotonic commonsense reasoning. It is not a form of default reasoning as described, e. g., by Reiter (1980); it rather seems to be more like reasoning about one's own knowledge or belief. Moore (1985) referred to it as autoepistemic reasoning. Autoepistemic reasoning is nonmonotonic because the meaning of an autoepistemic statement is contextsensitive. (In contrast, default reasoning in the spirit of Reiter is nonmonotonic because, to use a term of Moore, it is defeasible: its conclusions are tentative, given better information, they may be withdrawn.)
In order to give a more formal account of autoepistemic reasoning let us introduce a language L for expressing self-belief. Our treatment generally follows Moore (1985) and Konolige (1987). The logic will be based on the notion of belief, so we will take L to mean "is believed", treat it as primitive, and define M as (L(. This gives us the same notion of consistency as the one of McDermott & Doyle (1980): a formula is consistent if its negation is not believed.
Let Lo be an ordinary first-order language (i. a., the normal formation rules for formulas of first-order logic hold). A sentence of Lo is a formula with no free variables; an atom is a sentence of the form P(a1,...,an) where P designates a n-ary predicate letter and a1,...,an designate constant letters. We extend Lo by adding the unary modal operator L (and its dual M = (L( if required). L can be defined recursively as containing all the formation rules of Lo, plus the following:
(4) If p is a formula of L, then so is L(p).
The primary focus of autoepistemic logic is a normative one: given a set T of sentences of L (T may be interpreted as an initial (or base) set of beliefs about the world), what final set S should an ideal introspective agent settle on? If we restrict ourselves for the moment to Lo (a language without the self-belief operator), then clearly an ideal agent should believe all of the logical consequences of her base beliefs, a condition sometimes referred to as logical omniscience. The presence of a self-belief operator complicates matters. Because the intended meaning of L(p) depends on the final belief set S of the agent, the definition of S itself becomes circular, which necessiates the use of a fixed-point equation to define S.
Before we present this definition it is useful to specify the closure conditions that we would expect the beliefs of an ideally rational agent to possess. Viewed informally, the beliefs should include whatever the agent could infer either by ordinary logic or by reflecting in what he believes. Following Stalnaker (1980) we call a belief set S stable if it satisfies the following three properties:
(5) Th(S) ( S (Th is the ordinary first-order closure operator; p ( Th(X) iff p is a first-order consequence of X)1
(6) If p ( S, then L(p) ( S.
(7) If p ( S, then (L(p) ( S.
Clearly, the states of belief characterized by these conditions are stable in the sense that no further conclusion could be drawn by an ideally rational agent in such a state.
Now we are ready to give a formal definition for the possible sets of beliefs that a rational agent might hold, given T as his premises. Moore (1985) calls these sets the stable expansions of T.
(8) S is a stable expansion of T just in case S is the set of first-order consequences of
T ( {L(p): p ( S} ( {(L(p): p ( S}; in other words:
S = Th(T ( {L(p): p ( S} ( {(L(p): p ( S}).2
It can be shown that a stable expansion of T fulfills conditions (5) - (7) (thus justifying the term stable expansion). Furthermore, a stable expansion of T is grounded in T in the sense that S is derived from T and assumptions about self-belief (cf. Moore 1985).
If we consider a set of premises T as axioms, what do we consider the nonmonotonic theorems of T to be?
(9) The set TH(T) of nonmonotonic theorems of T is the intersection of the set of all sentences of L with all the stable expansions of T.
This definition (borrowed from McDermott & Doyle 1980) yields TH(T) to be the set of sentences of L that are in all stable expansions of T if there is a least one, and it makes TH(T) inconsistent if there is no stable expansion of T. As Moore (1985) argues, TH(T) represents what an outside observer would know, given only knowledge of the agent's premises T and that he is ideally rational. We will use the abbreviation T (( p to indicate that p ( TH(T).
In order to give an illustration (which will be used later) of the notions developed so far let us consider a special kind of base sets. Let be a set of classical sentences (i. e., contains only sentences of Lo). We now define the set () of elementary defaults with respect to as
(10) () = {M(p)(p: p ( }
Let H be another set of classical sentences. Then we call the set H(() an elementary default system with core H.3 A set ' ( is called compatible with H iff '(H is consistent. Furthermore, we introduce the notion MCS(,H) for the set of all maximal subsets of which are compatible with H. Clearly, MCS(,H) = {(} if H is inconsistent. Now the task of calculating nonmonotonic consequences of H(() can be simplified by using the following theorem:
THEOREM 1 Let H(() be an elementary default system. For any sentence p of Lo proposition (11) holds:
(11) H(() (( p if and only if H(' (( p for all ' ( MCS(,H).
Note that the right hand side of (11) is quite similar to the definition of compatibility-restricted entailment given by Rescher (1964). Accordingly, Theorem 1 says that nonmonotonic entailment (restricted to sentences of Lo) and compatibility-restricted entailment are equivalent in the case of elementary default theories.
We omit the proof of Theorem 1. In principle, it can be taken from Konolige (1987) who proved the formal equivalence of Reiter's default logic and autoepistemic logic. The observation of Ginsberg (1984) that the fixed points of normal default theories are in natural correspondence with the maximally compatible subsets of would complete the proof. A simple consequence of Theorem 1 is that an elementary default system H(() is consistent if and only if H is consistent.
3. Elementary default systems: An example from intrasegmental phonology
Nontrivial examples of elementary default systems may be extracted from intrasegmental phonology. The fact that intrasegmental phonology is a source of inspiration for developing theories of markedness (e. g., Chomsky & Halle 1968, Kean 1975) makes the following example a valuable one in the context of genericity (viewing genericity as unmarkedness, see Section 7).
Consider the following fragment of the vowel system (adapted from Kean 1975):
(12)
(back
+back
/i/
/u/.
+high
/e/
/o/
(high/(low
//
//
/a/
+low
Formally, the segments /i/, /u/, ... may be represented as constant letters i, u, ... of a first-order language Lo. The phonological features may be represented, accordingly, as predicate letters BACK, LOW, HIGH, ROUND. The knowledge of the phonological agent concerning this fragment may be represented explicitly as in the following table:
(13)
i
u
e
o
a
BACK
(
+
(
+
(
+
+
LOW
(
(
(
(
+
+
+
HIGH
+
+
(
(
(
(
(
ROUND
(
+
(
+
(
(
+
The table corresponds to a set E of atoms of Lo:
(14) {(BACK(i), (LOW(i), HIGH(i), (ROUND(i),
BACK(u), (LOW(u), HIGH(u), ROUND(u), ...}
It is evident that this list contains strong and weak (or "probabilistic") redundancies. For example, all segments x with HIGH(x) are correlated with (LOW(x) (strong redundancy) and most segments x with BACK(x) are correlated with LOW(x) (weak redundancy).
Now let us search for an axiom set T such that (a) T represents these redundancies and (b) E is deducible from T, i. e., E ( TH(T). The task (a) can be solved approximately by adopting the following axioms:
(15) (i) LOW(x) ( (HIGH(x)
(ii) (BACK(x))
(iii) (BACK(x) ( LOW(x))
(iv) ((BACK(x) ( HIGH(x))
(v) (BACK(x) ( (LOW(x) ( ROUND(x))
Here and in the following a formula (x) with free variable x abbreviates the set of all substitution instances (ai) where ai is a constant letter of Lo. It is clear that (15) represents an elementary default system that can be written as Ho(() (Ho corresponds to (i) and () corresponds to (ii) - (v)). It also is clear that in the given example is compatible with Ho; consequently, we get MCS(,Ho) = {}. We observe further that Ho( is (classically) equivalent to
(16) {BACK(x), LOW(x), (HIGH(x), (ROUND(x)}
Using Theorem 1 we obtain that the nonmonotonic theorems of Ho(() are exactly the classical consequences of (16). Next we notice that (16) describes a vocalic segment that has been called "the unmarked case" in the vowel system. It is exemplified by the segment /a/ in table (13).
In order to solve task (b) mentioned above we must augment the core Ho (given by (15)(i)) with all the exceptions from the "general tendencies" formulated by the default expressions (15)(ii) - (v). These exceptions are underlined in table (13) and may be rewritten as follows:
(17) {(BACK(i), HIGH(u), (HIGH(e), (BACK(e),
(LOW(o), LOW(), (BACK(), ROUND()}
Now let H be the union of Ho with set (17). It is a simple task to show, using Theorem 1, that E ( TH(H(()), i. e., task (b) has also been solved. We come back to this example in connection with the discussion of unmarked and prototypical properties in Section 7.
4. Distributive generics and quantified defaults
After having introduced nonmonotonic logic let us now return to the hypothesis that the meaning of distributive generic sentences may be approximated by quantified default expressions. This hypothesis has been illustrated by pairs like
(1() Ravens are black.
(x(RAVEN(x)(M(BLACK(x)) ( BLACK(x))
The translation of distributive generics into quantified defaults has some very attractive features. First, as we have already seen, a quantified default expression represents an exception-allowing general statement, i. e., it cannot be falsified by counterexamples (reconsider the case of albino ravens). The same, of course, holds for a generic statement which allows exceptions as well. Moreover, examples like A unicorn has a horn show that a generic sentence can be true even if there is no individual in the real world which satisfies the antecedent. Formally, the same is correct for quantified default expressions.
Second, the default approach can handle examples like A bird lays eggs which should be equivalent to A female bird lays eggs. As Krifka (1987:7) argues this comes out formally if the assumed facts contain, among others, the propositions that only female animals give birth, that laying eggs is a form of giving birth, and that birds are animals.
A third important point (discussed by Krifka as well) is that the default approach (based on autoepistemic logic, cf. Section 2) can employ different "modal dimensions". This arises from the indexical nature of the autoepistemic operator M. (As we have stressed in Section 2 the operator M changes its meaning with context = base set of beliefs.) That genericity is a modal concept has already been observed by Dahl (1975). Examples like A spinster is unmarried, Two plus two equals four, A boy doesn't cry demonstrate this fact with respect to the modal dimensions of analytic necessity, mathematical necessity and deontic necessity, respectively (cf. Krifka 1987).
Unfortunately, the interpretation of distributive generics via quantified defaults has some defects and shortcomings which undermine this analysis. In conventional logic as in almost all theories of natural language semantics the meaning of a composite expression can always be defined in terms of the meanings of its parts, even if these meanings are infinite things like functions on sets of possible worlds. In nonmonotonic logic, however, this is impossible. The meaning of M(p) can never be purely a function of the meaning of p, because it depends on the available belief set as well. Unfortunately, the complete expression of which M(p) is a part is an element of this belief set itself (due to the a u t o epistemic character of M(p)). Thus, the meaning of M(p) depends on its sentential context. This "holistic" property seems inescapable whatever the precise semantic definitions are (cf. McDermott & Doyle 1980, Moore 1985). Consequently, the price we must pay when we analyze distributive generics as quantified default expressions is the abandonment of compositionality as a semantic principle.
Whereas the unorthodox semantics of nonmonotonic systems may frighten perhaps only an orthodox semanticist the following shortcomings of the quantified default approach should have a chance of broader apprehension. One of the problems is connected with negation. The negation of a statement containing a generic description as its subject attaches - just as in the case of proper names - directly to the grammatical predicate. However, the negation of quantified assertions like (1) would attach to the quantifier expression:
(18) ((x(RAVEN(x)(M(BLACK(x) ( BLACK(x))
( (x(RAVEN(x) ( M(BLACK(x) ( (BLACK(x))
The resulting formula is not only incompatible with the intended translation
(19) Ravens are not black.
(x(RAVEN(x)(M((BLACK(x)) ( (BLACK(x))
Worse, expression (18) is logically inconsistent.4
A further problem arises from the well-known fact that generic NPs have far wider coindexing possibilities than ordinary quantifier expressions. For example, generic NPs can coindex pronouns across a sentence boundary: A raveni is a bird. Iti has black feathers. Carlson (1980) and Hornstein 1984) have argued convincingly that generic NPs have further namelike properties. Thus, the analysis in terms of ordinary operator-variable structures is excluded.
We conclude that the analysis of distributive generics in terms of quantified default expressions is - despite some tempting attractiveness - a disappointment.
5. The proper name analysis
In the area of linguistic semantics the best-known work with respect to generic statements is perhaps that of Carlson (1980). In brief, Carlson argues against any quantificational treatment of generics (and with them, kinds) and proposes instead that kinds be treated as individuals. In his semantics the term ravens in Ravens are black is treated as a proper name and the statement is true (roughly) if the kind RAVEN has the property of normally being BLACK.
While this approach appears to address a wide class of recalcitrant problems (the problems with negation and coindexing just mentioned are only the simplest ones), it does not seem to address fundamental concerns of cognitive and computational linguistics. There is no indication as to what it means to be a "normal" property of kind, nor how such properties interrelate, nor how one could reason about such properties. Carlson's approach does not even explain the simplest inferences associated with distributive generics, e. g.:
(20) If ravens are (normally) black and black things are not white, then ravens are (normally) not white.
(21) If ravens are (normally) black and ravens (normally) have dark eye pigment, then ravens that are black (normally) have dark eye pigment.
(22) If ravens are (normally) black and ravens that are black (normally) have dark eye pigment, then ravens (normally) have dark eye pigment.
(23) If ravens are (normally) black, but it isn't the case that ravens are (normally) albino, then ravens that aren't albino are (normally) black.
(The examples are taken from Delgrande 1987.) Furthermore, we can say without inconsistency:
(24) Ravens are black, but albino ravens are not black.
(25) Penguins are birds and birds fly, but penguins do not fly.
Example (24) shows that strengthening the antecedent is an invalid inference in case of distributive generics, and (25) demonstrates that transitivity is invalid as well.
One of the obvious aims of a logical analysis of distributive generics is to provide a formal system that deals with the inferential patterns underlying examples like (20) - (25). Ideally, all intuitively valid inferences should be rendered valid in the formal system, and, conversely, all intuitively invalid inferences should be rendered invalid in the formalization. Carlson's approach appears not to address these concerns. The following sections are an attempt to fill in this gap.5
6. Conditional logic and the variable conditional approach to generics
Delgrande (1987) presents an approach to distributive generics that explicitely addresses the problem of reasoning with generic sentences. His analysis makes use of a modal base in terms of accessibility between possible worlds. A rather similar theory has been developed by Dahl (1975) that we should consider first.
With a little reformulation Dahl's semantic description of Ravens are black would look like
(26) (x(RAVEN(x) ( BLACK(x))
where the operator ( is the strict conditional (p ( q =def ((p(q) with the usual modal operator (). This analysis has undesirable properties. Unfortunately, it produces inferential patterns like transitivity, contraposition, and strengthening antecedents, which reflect neither the logical properties of natural language conditionals nor those of generic statements.
The advantage of Delgrande's (1987) analysis is that it overcomes these difficulties and, at the same time, captures interesting intuitions regarding the intricate inferential properties that govern distributive generics. The heart of the underlying semantic theory is a "variable conditional" operator >. The statement p>q is interpreted informally as "in the normal course of events, if p then q". Analyzing Ravens are (normally) black as
(27) (x(RAVEN(x) > BLACK(x))
this should allow a consistent assertion along with Albino ravens are (normally) not black:
(28) (x((RAVEN(x) ( ALBINO(x)) > (BLACK(x))
In extending the semantics of predicate calculus to account for >, Delgrande (1987) adopts a possible-worlds approach, where p>q is true if q is true in the "least exceptional" worlds where p is true. The resulting formal system is a normal conditional logic in the sense of Chellas (1975). For convenience, we replicate only its propositional part, called NP.
(29) The conditional logic NP is the smallest logic containing the standard propositional calculus, and closed under the following axiom schemata and rules of inference:6,7
ID p>p
CC (p>q ( p>r) ( (p>(q(r))
MOD ((p>p) ( (q>p)
CA (p>r ( q>r) ( ((p(q)>r)
CSO (p>q ( q>p) ( ((p>r) ( (q>r))
CV ((p>q) ( ((p>r) ( ((p((q)>r)
RCM From q ( r infer (p>q) ( (p>r)
The logic NP differs from conditional logics for counterfactual reasoning primarily in that these other systems generally have one or both of the following formulas as theorems:
(30) MP (p>q) ( (p ( q)
CS (p ( q) ( (p>q)
(See Lewis 1973 and Nute 1984 for taxonomies of such systems.)
If we would add (30) to the axioms of NP we would exactly obtain Lewis' (1973, 1981) system VC. Delgrande (1987) suppresses the axioms (30) (by abandoning Lewis'principle of "centering"). The exclusion of axioms (30) is important. As Delgrande (1987:29) tells us, admitting (30) would collaps a quantified conditional like (27) into entailment and that clearly would be inadequate with regard to the semantics of generic sentences.
While appreciating the advantage of this analysis - it handles some inferences associated with distributive generics in a principled way8 - we also see its shortcomings. Most crucially, Delgrande (1987) performs a quantificational analysis of generics that suffers from most problems attacked by Carlson (1980). Furthermore, Delgrande uses a rather exotic possible-worlds semantics that differs from other systems of conditional logic in a way that appears to be somehow ad hoc.
One of the motivations of Delgrande's (1987) analysis was to relate generic sentences and expressions about prototypical properties. Saying that Ravens are black, for example, should be equivalent to saying that the property of being black is a prototypical property of the species raven. This idea looks very promising as a guide for a theoretical analysis, and indeed, it may be understood as the common integrator of quite different particular models (e. g., Platteau 1980, Strigin 1985, Guerts 1985, Krifka 1987, Heyer 1987). How to explicate the notion of prototypical property? Delgrande would tell us something like
(31) P is a prototypical property of A just in case
(x(A(x)>P(x)) is true.
Intuitively, P is called a prototypical property of class/kind A just in case each normal (or
typical) representative of A has property P. The "definition" (31) expresses something quite different - something like: each member of class/kind A has P as a normal property. Accordingly, the statement that P is a prototypical property of A would be falsified already by discovering one counterexample "a" that falsifies A(a)>P(a). That clearly seems inadequate.
In the next section we will try to develop a more satisfactory definition of prototypicality. This definition profits from the notion of "markedness" as developed within intrasegmental phonology.
7. Intermezzo: Markedness and prototypicality
In Section 3 we have considered a simple fragment of the vowel system and we have given a knowledge base T that yields the intrasegmental feature specifications of any segment of the fragment. It has been pointed out that some feature specifications of a given segment can be derived via strong and weak redundancy rules and need not be marked explicitely in the knowledge base. Let us call these specifications the unmarked properties of the segment. Other specifications must be formulated explicitely in the knowledge base as exceptions from the general rules. These specifications we call the marked properties of the segment. Table (13) designates the marked properties by underlining. The vowel /a/, for example, has only unmarked properties, and the vowel /i/ contains (BACK as marked, but (LOW, HIGH, (ROUND as unmarked.
The following extensions of these notions from single segments to classes of segments are straightforward:
(32) A property P is called an (un)marked property of a class A of segments iff for each member of A that has the property P this property is its (un)marked property.
Using this definition we can extract the following particular statements from (13):
(33) (a) BACK is an unmarked property of the class of vowels
(BACK is a marked property of the class of vowels
(b) LOW is an unmarked property of BACK (the class of back vowels)9
(LOW is a marked property of BACK
(LOW is an unmarked property of (BACK
LOW is a marked property of (BACK
(c) (HIGH is an unmarked property of BACK
...
(d) ROUND is an unmarked property of BACK((LOW
...
The reader may already have seen that these statements exactly reflect what has been called markedness conventions in the generative tradition (e. g., Chomsky & Halle 1968, Kean 1975).
It is remarkable that the markedness conventions (33)(a)-(d) are in strict correspondence to the default expressions () of our knowledge base T (see Section 3). For example, the statement (33)(b) that LOW is an unmarked property of back vowels corresponds to the default expression
(15)(iii) (BACK(x) ( LOW(x))
Using the present formalization it is possible to derive the markedness conventions as metatheoretic statements about the underlying default theory (knowledge base T).
Next let us consider prototypicality and its relationship to markedness. A prototype of a class A may be introduced as the most "typical" or most "normal" member of the class. In more theoretical terms we will define the prototype(s) of class A as the least marked member(s) of the class. So /a/ is a prototype of the class of all vowels (it has only unmarked properties), /i/ is a prototype of the non-back vowels (it has only (BACK marked), /u/ is a prototype of the high vowels (it has only HIGH marked), etc.
Calling the properties of the prototype(s) of class A the prototypical properties of A let us ask now for a mechanism that determines the prototypical properties of A from the underlying knowledge base T. Since prototypes are represented in the knowledge base implicitly (only highly marked segments have an explicit representation in T; prototypes are the least marked members of a class), direct search for prototypes in the knowledge base is impossible. The best what the mechanism can do is to assume an arbitrary segment not introduced into the knowledge base so far and to assume that instantiates A. It can be shown that the derivable properties of coincide with the properties of a prototype of A. Consequently, the mechanism can calculate the prototypical properties P of A by deriving P() from T({A()}.
The intuitive correctness of the given mechanism justifies the following final definition:
(34) P is a prototypical property of A with regard to a knowledge base T just in case T({A()} (( P() (where must not occur in T).
I believe that this definition of prototypicality is intuitively more convincing then the one of Delgrande.10
8. A metalinguistic interpretation of conditionals and generics
Delgrande (1987) has supposed that the modal base that constitutes the interpretation frame of his conditional logic may be conceptualized as an accessibility relation between possible worlds. In this section we will take the underlying modal base T to be a set of sentences of L (the nonmonotonic first-order language of Section 2). This gives the semantics of conditionals a metalinguistic flower: we interpret "if p then q" to be true just in case q is (nonmonotonically) derivable from T combined with p. (For a related approach, cf. Blue 1981.)
Modal bases (another term would be "conversational background"; cf. Kratzer 1981) may be classified with respect to the notional categories of modality in natural language. We would like to draw attention to the so-called totally realistic backgrounds (Kratzer 1981). They characterize the world in a unique way and are in strict correspondence to the "centered" bases of Lewis (1981)
We can proceed now in more technical terms. Let L+ be the extension of Lo that originates from introducing a conditional operator (. Furthermore, let v be an ordinary interpretation of Lo. Then the pair (v,Tv) is called an interpretation of L+ iff Tv is a set of sentences of L. Furthermore, (v,Tv) is called a totally realistic interpretation of L+ iff Tv is a totally realistic conversational background in L, i. e., Tv is a set of sentences of L and the following condition is satisfied:
(35) For each sentence p of Lo: Tv (( p ( p is true in v.
With respect to the standard case of elementary default systems (Section 2) we define: (v,Tv) is a standard interpretation of L iff Tv is an elementary default system in Lo.
The assignment of truth-values to the formulas of the language L+ is consistent with the usual truth recursion for predicate calculus. If p, q are sentences of Lo, then the following assignment is given for the conditional p(q:
(36) p(q is true in (v,Tv) iff Tv({p} (( q (p(q is false otherwise).11
A sentence of L+ that is true in every member of a class of interpretations is called valid with respect to the class. An inference rule "from p infer q" is called valid with respect to the class iff for each member of the class where p is true q is true as well.
It is not difficult to show that transitivity, contraposition, and strengthening antecedents can be falsified by some totally realistic standard interpretation. Furthermore, we have the following theorem:
THEOREM 2: The following axioms and rules of inference are valid with respect to all totally realistic standard interpretations of L+:
ID p(p
CC (p(q ( p(r) ( (p((q(r))
MOD ((p(p) ( (q(p)
CA (p(r ( q(r) ( ((p(q)(r)
CSO (p(q ( q(p) ( ((p(r) ( (q(r))
CS (p(q) ( (p(q)
MP (p(q) ( (p(q)
RCM From q ( r infer (p(q) ( (p(r)
PROOF: Using definition (36) the validity od ID, CC, and RCM are obtained immediately from the fact that TH(Tv({p}) is closed under classical consequences. (Accordingly, ID, CC, RCM are valid with respect to all interpretations of L+).
For CS, suppose that p and q are true in v. Using (35) it follows that Tv (( p and Tv (( q. A general theorem of nonmonotonic logic says that TH(X) ( TH(X({p}) if p ( TH(X). So, q ( TH(Tv) ( TH(Tv({p}), and CS follows from it.
For MP, suppose that T (( (p. Then (35) says that p is false and p(q holds. On the other hand, if not Tv (( (p, then (35) yields Tv (( p. From Tv({p} (( q it follows that Tv (( q. So q must be true. The conclusion p(q again follows. (Note that our proof shows that CS and MP are valid with respect to all totally ralistic interpretations of L+).
Proving the remaining axioms requires standard interpretations. We refer to them by Tv = Hv((v), where Hv and v are sets of sentences of Lo (see Section 2). For simplicity we suppress the indices in the following.
For MOD, note that H(()({(p} (( p implies that H({(p} is (classically) inconsistent (a simple consequence of Theorem 1). MOD immediately follows.
For CA, consider an element ( MCS(,H({p(q}). Clearly is compatible with H({p} or H({q}. On the other hand, since is maximal subject to its compatibility with H({p(q} any ( with ( ( ( will have ( incompatible both with H({p} and with H({q}. It follows that ( ( MCS(,H({p}) or ( ( MCS(,H({q}) and CA follows from this by Theorem 1.
For CSO, we show that the truth of p(q and q(p implies that MCS(,H({q}) ( MCS(,H({p}). CSO immediately follows from this. Let ( MCS(,H({q}). The truth of q(p implies that (H({q} (( p (Theorem 1). Consequently, ( MCS(,H({p,q}). It follows that H({p}( is consistent. If were not maximal subject to its compatibility with H({p}, there ought to be some ( ( with ( ( MCS(,H({p}). From p(q we would get ( ( MCS(,H({p,q}), and no proper subset of ( could be an element of MCS(,H({p,q}). However, ( ( , ( MCS(,H({p,q}). From this contradiction we conclude that must be maximal subject to its compatibility with H({p}; it follows that ( MCS(,H({p}). Q. E. D.
The axiom system mentioned in Theorem 2 is identic with the system SS introduced by Pollock (for references, see Nute 1984). Thus we have proved the soundness of Pollock's system SS with respect to the present metalinguistic interpretation. The system SS is weaker than Lewis' VC since
CV ((p(q) ( ((p(r) ( ((p((q)(r))
is contained in VC while CV is not contained in SS. It is a standard task to impose an extra condition on interpretation frames to make this axiom valid in the metalinguistic formulations as well (cf. Ginsberg (1984) for solving this task by using a framework that is compatible with ours).
Now we are prepared to account for the notion of prototypicality in the present metalinguistic framework. We start by extending the language L+ by adding a new constant letter to the vocabulary of L+. This extension is called L+(). An interpretation of L+() is a pair (v,Tv) where v is an ordinary interpretation of Lo() (the classical language Lo augmented with a new constant letter ) and Tv is a set of sentences of L+. The reference to L+ is important - with it we are sure that does not occur in Tv. The condition for totally realistic conversational backgrounds remains that of (35). Consequently, for expressions like A() it is not demanded that A() is true if Tv (( A().
Let P and A be one-place predicates of L+.
(37) P is called a prototypical property of A with respect to an interpretation (v,Tv) just in case A()(P() is true in (v,Tv).
This definition exactly reflects the notion of prototypicality as developed in Section 7. Conceptualizing distributive generics as statements about prototypical properties, sentences like Ravens are black may be represented formally by expressions of L+():
(38) RAVEN()(BLACK()
Such expressions relate generic sentences and default expressions in a less direct way than does the quantified default approach. The reference to default expressions is given indirectly via the underlying modal base that may contain quantified default expressions like (1) in order to make (38) true.
The present approach, of course, is a non-quantificational approach to distributive generics. In this respect it is similar to Carlson (1980) but differs from Delgrande (1987). Expressions like (38) are distinct from ordinary operator-variable structures and thus avoid the difficulties with negation and coindexing that are typical for any quantificational approach. On the other hand, the present approach explicitly addresses the problem of reasoning about generies. Concretely, expressions like (38) sanction all inferences that are associated with Delgrande's (1987) conditional logic NP (see Theorem 2).12 In this way, the present nonquantificational analysis maintains the advantages of Delgrande's approach.
An important point is that our analysis allows to "paraphrase" distributive generics as quantified expressions under certain conditions. Consider, for example, a modal base Tv that contains only "classical" expressions (i. e., sentences of Lo). Let us assume forthermore that Tv is totally realistic. Then the truth of (38) implies the truth of
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OP !0^`0a$(Use the ordinary deduction theorem to prove this.) The introduction of modal bases in terms of quantified default expressions and the formulation of exceptions in the base would modify this result: the unrestricted quantification in (39) has to be replaced by a restricted one referring to the "normal" or "unmarked" ravens only. To prove this formally would be an interesting technical task.
Finally, we will make a remark concerning the peculiar status of our -objects. We have introduced these "objects" in purely syntactic terms (as "denoted" by a constant letter that does not occur in Tv). A more semantic characterization would be desirable. Possibly, data semantics (e. g. Landman 1986) gives the appropriate framework. Interestingly, Landman's "Alecs" are just like our -objects: they have no properties in isolation but they may grow into every character in appropriate contexts.
Acknowledgements
I would like to thank Ewald Lang, Manfred Krifka, and Anatoli Strigin for their very helpful comments on earlier versions of this paper.
Notes
1 We will also use the abbreviation X (( p to indicate that p ( Th(X).
2 McDermott & Doyle (1980) use the fixed-point equation
S = Th(T({M(p): (p(S})which can be rewritten as
S = Th(T({(L(p): p(S): Obviously, {L(p):p(S} is missing from the base of the fixed points. Moore (1985) comments on this: "McDermott and Doyle's agents are omniscient as to what they do not believe, but they may know nothing as to what they do believe" (p. 86).
3 Note that in Reiter's (1980) system H(() would be a special case of a normal default system.
4 Generally, M(p)((p is inconsistent because each stable set containing (p should also contain L((p) (cf. condition (6)). L((p) is equivalent with (M(p). Now we can infer M(p)((p((M(p) which clearly is inconsistent.
5 The analysis of non-distributive generics is not addressed in the present paper. Perhaps Carlson's theory is right in this case (cf. Heyer (1987, Krifka 1987).
6 For naming the axioms and rules of inference, cf. Nute (1984) and Chellas (1975).
7 We have modified Delgrande's original system a little in order to simplify comparisons with other systems. It is nearly trivial to show the equivalence between Delgrande's formulation and ours.
8 It is a simple exercise to reproduce the inferences underlying examples (20)-(25) in the formal framework.
9 Here and in the following we identify classes with their defining properties.
10 What about the relationship between prototypical properties and unmarked properties? Intuitively, the unmarked properties of a class are those that "most probably" reflect the redundancy structure of the class as a whole. On the other hand, the prototype of a class is the most typical representative of the class. Thus, its properties seem to reflect the (probabilistic) redundancy structure of the whole class as well. Not surprisingly, in case of our knowledge base T (Section 3) it can be shown that the prototypical properties and the unmarked properties of any class coincide (of course, this proposition depends on the structure of the base T and need not generally be true).
11 It is a shortcoming of the metalinguistic approach that it doesn't give truth assignments to embeddings like (p(q)(r. Obviously, the present approach concerns the "horizontal" dimension of the logical system only.
12 Note that CS and MP are proved valid for ordinary conditionals (not containing ) in our system too. However, the validity of CS and MP is excluded in the totally realistic domain (reconsider the proof of Theorem 2). As it was pointed out in Section 6, Delgrande (1987) must exclude CS and MP generally. We believe that this is unsatisfactory.
References
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III Defaults and the meaning of generic sentences
PAGE
PAGE 55
PAGE 56
III Defaults and the meaning of generic sentences
PAGE 57
1. General introduction
PAGE 58
III Defaults and the meaning of generic sentences
PAGE 59
2. An introduction to nonmonotonic logic
PAGE 60
III Defaults and the meaning of generic sentences
PAGE
3. An example from intrasegmental phonology
PAGE
III Defaults and the meaning of generic sentences
PAGE 61
4. Distributive generics and quantified defauls
PAGE 62
III Defaults and the meaning of generic sentences
PAGE 63
5. The proper name analysis
PAGE 64
III Defaults and the meaning of generic sentences
PAGE
6. Conditional logic and the variable conditional approach to generics
PAGE 66
III Defaults and the meaning of generic sentences
PAGE 65
7. Intermezzo: Markedness and prototypicality
PAGE 68
III Defaults and the meaning of generic sentences
PAGE 67
8. A metalinguistic interpretation of conditionals and generics
PAGE 70
III Defaults and the meaning of generic sentences
PAGE 69
Notes and references
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