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<?xml version="1.0" standalone="yes"?> <Paper uid="J89-3003"> <Title>NON-SINGULAR CONCEPTS IN NATURAL LANGUAGE DISCOURSE</Title> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 3 NON-SINGULAR TERMS IN DISCOURSE </SectionTitle> <Paragraph position="0"> The rules discussed in Section 2 cover selected cases of intersentential anaphora where the reference level in discourse does not change from one sentence to another. There exists, however, a class of intersentential dependencies whereby a reference is made across boundm'ies of different reference levels in discourse.</Paragraph> <Paragraph position="1"> For example, in My new pet is an alligator. But the alligator cannot live in our climate.</Paragraph> <Paragraph position="2"> 174 Computational Linguistics, Volume 15, Number 3, September 1989 Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse &quot;the alligator&quot; in the second sentence most likely refers to a generic object of which the alligator in the first sentence is an instance or extension. Thus we can say that the second alligator is a non-singular superobject in which the first alligator somehow participates. The extent of such participation is not clear, but in general it can be observed that certain predications true of complexes of different kind are not preserved for their parts or elements, and vice versa. To represent this new kind of intersentential dependency we introduce a multilevel model for interpreting natural language expressions, such that the levels in the model would correspond (roughly) to the levels of reference in discourse. For instance, in the example above, the resulting representation would have both alligators placed at different, though related, &quot;object levels.&quot; Because of an inherent subjectivity of such classifications, the levels in the model may have fuzzy boundaries and are only partially ordered with the &quot;lower than&quot; (i.e., &quot;more detailed than&quot;) relation with respect to some current level (corresponding to the level of reference at a present point in discourse).</Paragraph> <Paragraph position="3"> Consider now a somewhat larger fragment of text, excerpted from an article appearing in The New Yorker magazine.</Paragraph> <Paragraph position="4"> The two Ashanti kings are in somewhat different situations: the Ghanaian king is royally born, richly rewarded, divinely inspired and holds his office for life. The American Ashanti king is elected every two years from the ranks of an Ashanti social and cultural organization called the Asanteman Association of the United States of America, Inc. The first Stateside king, Kwadwo Tuffuor, was a plumber. The second, Kusi Appouh, repaired air-conditioners and refrigerators. Kwabena Oppong is the third king; he drives a cab. 2 The first two sentences in this fragment contain direct references to the higher-level entities, which are the two Ashanti kings, as if they were ordinary singular objects. The remaining three sentences directly refer to the entities that are instances of one of these kings, now seen as superobjects, at different time intervals. The discourse reference level has changed, and now we talk about lower-level entities. The superobjects can still be referred to, but only indirectly, through their instances; this is what we call the remote reference.</Paragraph> <Paragraph position="5"> The primitive notions of our theory are these of a singular object and a coordinate, a usually ordered set specifying a type of dimension that the object in question spans. A singular object is any entity to which we can directly refer using a nominal phrase of our language. The most common of the coordinates are time and space but other more abstract ones are also possible. These two basic notions are then used to define the notion of the object's instance with respect to some coordinate. Thus the pet alligator in the example above is related to the generic concept of alligator by some species coordinate that somehow ties (or enumerates?) all alligators around the world. Similarly, if we use an appropriate coordinate consisting of two-year intervals we can decompose the American Ashanti king into its elected instances. If we reverse this process we can combine objects into complexes to which we can subsequently refer using collective terms, singular or plural, such as, for example, &quot;people&quot; or &quot;the man&quot; (generic). The lower than relation between levels in the universe model derives from expanding the notion of instance over collections of objects. The relation introduces a partial ordering within the universe model and thus helps to trace changes in the reference level of discourse. The highly discrete approach taken here is favorably contrasted with other existing approaches to non-singular terms, including Quine (1960), Kripke (1972), Montague (1974), Carlson (1982), and others.</Paragraph> <Paragraph position="6"> While insights of Quine, Kripke and, perhaps even more so, Carlson are undoubtedly of great influence, they require reworking in more discrete terms. Finally, we may note that the research in artificial intelligence and computational linguistics has devoted relatively little attention to treatment of non-singular terms in natural language in general and in natural language discourse in particular; see, however, Sidner (1979) for some early attempts to recognize generics in discourse. One of the goals of the present research is to fill this gap.</Paragraph> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 4 NON-SINGULAR TERMS IN LANGUAGE </SectionTitle> <Paragraph position="0"> Many philosophers and logicians, Quine (1960), Kripke (1972), Donnellan (1971), Vendler (1971), Montague (1974), and Barwise and Perry (1983), note that the usage of the italicized nominal phrases in Example 1 has a general or generic character, except for regular singular interpretations, which are only possible in some cases.</Paragraph> <Paragraph position="1"> Example 1.</Paragraph> <Paragraph position="2"> la. The king wears a crown.</Paragraph> <Paragraph position="3"> lb. The president is elected every four years. lc. Gold is a yellow metal.</Paragraph> <Paragraph position="4"> Id. Temperature is a measure of molecular motion. Hundreds of similar examples involving such non-singular terms, and corresponding non-singular objects, as water, heat, the Pope, the number, etc. can be devised. Unfortunately, there is no generally accepted account of these non-singular terms in the philosophical literature. Some authors, for example, Vendler (1971) and Barwise and Perry (1983), cautiously called them generic, or general (for example, &quot;the king&quot;), or functional (such as &quot;the number of students,&quot; &quot;the temperature&quot;) uses of definite descriptions. Other authors, for example, Kripke (1972), were quite close to considering these kinds of non-singular terms as names (or at least some of them: heat, gold). Still other authors writing on the subject, for example, Quine (1960, 1973), advocated the notion of abstract terms as comprising of attributes, such as (being) red (further abstracted as &quot;redness&quot;), or Computational Linguistics, Volume 15, Number 3, September 1989 175 Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse (being) the man drinking the martini (which cannot be so easily nominalized), which can be predicated of &quot;concrete&quot; objects. If we consider this discussion, the so-called &quot;attributive use&quot; of singular definite descriptions as identified by Donnellan (1971), may be considered as addressing some abstract, higher-level, and therefore (in our interpretation) non-singular concepts.</Paragraph> <Paragraph position="5"> Carlson (1982) discusses the case of so-called &quot;natural kinds,&quot; a specific type among generic terms. He advocates the view in which generic terms are taken as denoting entities in the same way that singular terms do.</Paragraph> <Paragraph position="6"> If we accept Carlson's position, then our ontology of objects becomes far richer than before, and we need to impose more structure on our representation to reflect this new situation accurately. Indeed, Carlson introduces a special R relation into Montague's IL which allows him to create individual objects out of generic objects, as well as stages out of ordinary singular objects. Thus, &quot;Max believes that dogs are here&quot; receives the following translation, where m and d are individual constants denoting Max and dog kind, respectively (Carlson 1977):</Paragraph> <Paragraph position="8"> This formula says that Max believes that some stages of dog-kind are here. One problem with this representation is that we have no idea how these stages are to be identified, in other words, how do we decompose a kind-level object into stage-level individuals. If other types of non-singular terms denote in similar fashion, then we may need an even more complex model in which various stages (or levels) of object aggregation can be reflected.</Paragraph> <Paragraph position="9"> Quine (1960) presents the most comprehensive account of various categories of terms found among natural language expressions. Almost everything that one can say is made up of different kinds of terms, appropriately connected to yield meaningful utterances, which he classifies as singular, general, relative, abstract, attributive, etc. At present, we do not draw such fine distinctions in our classification, reserving the right to develop extensions along the lines of Quine as needed. We present a few examples to provide additional insight and lay the foundation for our theory of names and descriptions.</Paragraph> <Paragraph position="10"> There are numerous linguistic puzzles involving non-singular definite descriptions, among them Partee's (1972) famous temperature problem. Example 2 illustrates this phenomenon.</Paragraph> <Paragraph position="11"> Example 2 2a. The temperature is rising.</Paragraph> <Paragraph position="12"> The temperature is ninety.</Paragraph> <Paragraph position="13"> Thus, Ninety is rising. 3 2b. The president is elected every four years.</Paragraph> <Paragraph position="14"> The president is Reagan.</Paragraph> <Paragraph position="15"> Thus, Reagan is elected every four years.</Paragraph> <Paragraph position="16"> 2c. The tiger lives in the jungle.</Paragraph> <Paragraph position="17"> My pet is a tiger.</Paragraph> <Paragraph position="18"> Thus, My pet lives in the jungle.</Paragraph> <Paragraph position="19"> 2d. Americans drive big cars.</Paragraph> <Paragraph position="20"> John is an American.</Paragraph> <Paragraph position="21"> Thus, John drives big cars/a big car.</Paragraph> <Paragraph position="22"> The arguments in (2a-d) are normally considered invalid. Various researchers agree that the definite descriptions &quot;the temperature,&quot; &quot;the president,&quot; and &quot;the tiger&quot; in the first sentences of (2a-c) should be interpreted functionally, that is, as intensions (Montague 1974d), or functions over situations (Barwise and Perry 1983). Note that if the descriptions were to be interpreted singularly or as enumerating all instances of a non-singular object (that is, statements containing them were understood as making claims about each instance), the reasoning would be valid. In (2d) the situation is somewhat more complicated, because every extension of tile entity in the denotation of Americans is itself a compound entity, namely a generic entity. Thus, intensionality alone cannot explain why the first two sentences in (2d) appear connected. As a matter of fact, the reasoning displayed in (2d) is far more acceptable that those in (2a) to (2c). The reason, it seems, lies in our reluctance to apply singular interpretations to non-singular terms in first sentences of (2a-c), while this same move is more likely in (2d). In other words, while (2b') is an unlikely interpretation of (2b), (2d') can occasionally be accepted as a simplified version of (2d). 2b'. Every president is elected every four years.</Paragraph> <Paragraph position="23"> \]Reagan is the president.</Paragraph> <Paragraph position="24"> Thus, Reagan is elected every four years.</Paragraph> <Paragraph position="25"> 2d'. \]Every American drives big cars/a big car.</Paragraph> <Paragraph position="26"> John is an American.</Paragraph> <Paragraph position="27"> Thus, John drives big cars/a big car.</Paragraph> <Paragraph position="28"> Of course, (2a) and (2b), but not (2c) or (2d), could be valid if the definite noun phrases in the second sentences (the temperature, the president) were understood as co-referential with appropriate noun phrases in the first sentences. This avenue is also closed, however, because we do not regard these phrases as co-referential, that is, they cannot be substituted for one another, nor cant be their denotations. Compare this with (2e) below, where the co-reference between &quot;carnivorous animals&quot; and &quot;these beasts&quot; is easily made. 2e. Carnivorous animals live in these forests.</Paragraph> <Paragraph position="29"> These beasts are tigers.</Paragraph> <Paragraph position="30"> Thus, tigers live in these forests.</Paragraph> <Paragraph position="31"> We claim here that no two descriptions can be considered co-referential unless they are used to refer to the objects that are at the comparable stages of aggregation. We call such object relatively singular and place them within the same level in our model. It is clear that the objects referred to by the corresponding descriptions in the firs~t two sentences of (2a) to (2d) are not relatively singular, and thus the conclusions in the third sentences are not forthcoming. Another type of co-reference, which we call a remote co-reference, can still occur, and we put this view forward in this paper.</Paragraph> <Paragraph position="32"> 176 Computational Linguistics, Volume 15, Number 3, September 1989 Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse</Paragraph> </Section> <Section position="6" start_page="0" end_page="0" type="metho"> <SectionTitle> 5 A MULTILEVEL MODEL FOR INTERPRETING NATURAL LANGUAGE TERMS </SectionTitle> <Paragraph position="0"> Initially, we note that our language tends to deal with singular objects only, no matter how complex their structure happens to be. A singular object is any entity that can be taken as a coherent whole, in other words, it can be referred to directly using a referring expression of language: a name, a definite description, a pronoun.</Paragraph> <Paragraph position="1"> Thus, at least as far as our ability to refer is concerned, all objects appear singular. Still, it is not the case that all objects are singular in the same way. Take, for example, two persons John and Mary. They are singular objects and they seem singular in the same way, in other words, singular relative to one another. Next take alligator, the species, and the alligator John owns. Although both are singular in their own right, they are not compatible when related to one another: the alligator John owns appears only a manifestation, or extension, of alligator the species at a certain space-time location. The individual alligator, which at some period of its life is owned by John is, therefore, singular in the same way John is, but this will not be the case when we consider the time slice of this alligator (an alligator stage, in Carlson's words) while it was owned by John. This latter appears only an instance of the individual alligator at some time interval. By the same token, if John owns many different alligators at different times (but, arguably, never more than one at a time) then we may risk to refer to John's alligator, an abstract object that generalizes over all alligator stages of all individual alligators ever owned by John (for example, &quot;John always walks his alligator in the morning&quot;). The new object, again, seems to belong in the same class of objects as John and Mary.</Paragraph> <Paragraph position="2"> Let us introduce, only intuitively at first, the relation Of relative singularity among objects. As suggested above, this relation will help us to break down the universe of objects into classes of relatively singular objects, which we call levels. The levels can be subsequently partially ordered with lower than relation, i.e., L 1 < L2, indicating that level L~ consists of manifestations (extensions, instances) of objects at level L 2. Let L 0 be an arbitrary level we select as our reference point; if our discourse operates at this level then L 0 defines the current level of reference of the discourse. Let L+ 1 and L_1 be two other levels different than Lo and such that L-I < Lo < L+~. At level L+I we place the objects we consider to be generalizations (or abstractions) of some measurable amount of objects from Lo. It is only from the perspective of L+~ that we are able interpret &quot;The tiger lives in the jungle,&quot; or &quot;The president is elected every four years,&quot; or &quot;Birds can fly,&quot; or &quot;Tourists start forest fires.&quot; The objects at L+~ are singular but only when related to one another within the same level; when viewed from L o they appear generic or functional or the like, in other words, non-singular. When we attempt to find a denotation for a nominal, such as &quot;the tiger&quot; or &quot;tourists,&quot; within Lo, we attempt to give it either a singular or measurably singular interpretation.</Paragraph> <Paragraph position="3"> In a singular interpretation (if possible at all), we have a nominal refer to a specific object within the level (John's pet tiger), while in a measurably singular interpretation we use a quantification over a finite set of singular objects, also within the same level (every tiger, some tourists). Nominals denoting objects which are non-singular with respect to Lo, on the other hand, may have neither singular nor measurably singular interpretations within this level. In order to find proper denotations for them we must change the reference level from Lo to L+I. Thus, while the statement of &quot;The President lives in the White House&quot; when interpreted at level L/~ can be argued to be equivalent to the statement &quot;Every president lives in the White House&quot; interpreted at L0, the same cannot be said of &quot;The tiger lives in the jungle&quot; and &quot;Every tiger lives in the jungle.&quot; We must note that some objects found at L+~ could have been placed there by design rather than as a result of generalizing from Lo; an example of such higher-level object may be The President.</Paragraph> <Paragraph position="4"> If level L/~ contains generalizations of objects from Lo, then level L_ l will contain their specializations or extensions. Descending upon L_~ we can see that what we previously considered to be the atom actually denotes many different kinds of atoms (H, O, Ca, Fe, etc.), or that the mail is not the same every morning, or that Nicolas Bourbaki is the name of a group of mathematicians. 4 A few definitions will help to put the above intuitions into a more formal setting.</Paragraph> <Paragraph position="5"> Def. 1. A use of a description is called singular if it refers to a singular object. A use of a description will be called measurably singular if it refers to some measurable quantity of a singular object. Otherwise we shall talk of non-singular use.</Paragraph> <Paragraph position="6"> Def. 2. An object level, or simply a level, is an arbitrary collection of relatively singular objects. On the language side, the corresponding reference level encompasses those singular and measurably singular uses of descriptions that refer to the level's objects.</Paragraph> <Paragraph position="7"> Def. 3. For any level L, there are at least two distinct levels L_ I and L+t such that L+t contains these objects which are non-singular from the perspective of L, and L-I contains the objects for which the objects at L are non-singular.</Paragraph> <Paragraph position="8"> Def. 4. The level L o is an arbitrarily chosen level serving as a reference point.</Paragraph> <Paragraph position="9"> As described, the structure of levels is not yet adequate to capture the full complexity of the reference structure of discourse. A notion of coordinate has to be introduced along the following lines. We shall call T a coordinate, if T is a set of points or locations at which certain general (or abstract) objects, for example the president or the atom, are assigned more specific extensions or instances, such as President Reagan or H, Fe, the ordering may be partial only. Almost any object we can think of appears an instance of a more general concept, and often there will be more such concepts available, if we consider different coordinates. Water in a glass is an instance of some totality of water in the universe (space' coordinate), and also an instance of a concept of water as in &quot;Water boils at I00 degrees Celsius.&quot; These examples suggest that a coordinate is usually a large set, often an infinite set, though perhaps no more than recursively enumerable. A non-singular object can be decomposed into instances in more than one way, depending which coordinate is used. An important observation is that when a higher-level object is decomposed with two different coordinates the resulting sets of instances need not belong to the same level. The concept of the American president in the twentieth century is an instance of The President, but is still non-singular when compared with President Reagan, also an instance of The President, though with respect to a different coordinate. A somewhat finer structure of levels is required.</Paragraph> <Paragraph position="10"> Let L~'I T be the level where we place the instances of object N decomposed with coordinate T. By analogy, we define L~'I r to be the level such that for any object M, M ~ LN+'I r if N ~ L_ml r. In other words, L~'l r contains the superobject M generalizing over object N with the use of coordinate T. Suppose that we have an object N at level L0, to which we refer using a description N. 5 Suppose further that coordinate T is selected so that for any x, y E Twe have that N-at-x :/: N-at-y. Let us use Nx to stand for N-at-x, where x is an element of T, and let (N x) be an expression (as translated into our meaning representation language) that refers to the object Nx, whenever the expression N refers to N, We obtain therefore that F1. Vx,y E T \[x --/: y ~ (N x) ~ (Ny)\] The new objects Nx'S cannot be placed at L 0 because, being instances of N, they are not singular relative to N (see Def. 2). Instead, we move them onto a new level LN_'I r leaving the original object N at L o. We say that the TN,T level rN,r is lower than the level L o, and write -~-1 L.~ \] L 0. Often we drop the superscripts N and T over the level symbol, assuming some lower level L_ 1, whenever it does not lead to ambiguity. Example 3 helps to illustrate the phenomenon just discussed.</Paragraph> <Paragraph position="11"> Example 3. Let us consider a rather naive concept of bird, as that of a winged creature that lay eggs and can fly. Let B be extension of this concept in our model and let Lo be set so that B ~ Lo. Using a genus coordinate, G, we can construct a level La_'~ containing such objects as eagle, hawk, and goose. Let's suppose that, initially, penguin can also be found at level La_'I c. Upon discovery that penguins cannot fly, however, one would wish to relax the characteristics of the concept B from Lo to contain both flying and non-flying birds. Nonetheless, two distinct concepts emerge, that of flying birds and that of non-flying birds, both of which become subordinates of the (now) more general concept B. These new concepts are placed at a new level La_'~ x, where K={kl, k2} is a class coordinate such that Bkl is the concept of flying bird, FB, and Ba~ is the concept of non-flying bird, NFB. The old level LB_'~ remains intact, though it is different from ra,x Moreover, rB,G < rB,X because Jr.,_ I . L,_ 1 z-,_ 1 B,G ~a',cB, (- L_~ , where B' is either FB or NFB, and GB,C_G. There is another way of interpreting concept B as well: we introduce a specimen coordinate S that allows us to pick up specific birds, such as Opus, the penguin, at level ,-.-1 .rB's Note that this level is lower than LB'I c_ because it contains all levels &quot;-.-lrx'sx, where X ranges za,o and SxC_S. Note that the structure over objects at ---1 , of levels has been created as described because of the following set of conditions.</Paragraph> <Paragraph position="13"> Figure 1 further illustrates the concept of levels and coordinates. 6 For an easy interpretation of this drawing, note that gl ..... g5 E G, GFn O GNvB = G and gl, g2, Now we can attempt to represent meanings of some simple statements about birds. For example, Birds can fly is represented at L0 as can-fly(B), while Opus is a bird would translate as 3sES \[(B s) = Opus\]. We cannot infer from these statements that Opus can fly because this would require the formula Vs \[can-fly((B s))\] to be true, which does not have to be the case considering that B may now contain instances of non-flying birds.</Paragraph> <Paragraph position="14"> Indeed, Opus cannot fly, which translates to --xan-fly(Opus), is not necessarily inconsistent with the above two. 7 The only thing we could say about each 178 Computational Linguistics, Volume 15, Number 3, September 1989 Tomek Strzalkowskiand Nick Cercone Non-Singular Concepts in Natural Language Discourse instance of B is, perhaps, that it is a bird and that every /'B,S bird is an instance of B. In other words, if x is a ,_,_ l variable then Vx \[bird(x) -= 3sES\[x=(B s)\]\] is true. This would lead to an equivalent L_ l translation of Opus is a bird as bird(Opus). Later, we will see that this equivalence is not generally valid.</Paragraph> <Paragraph position="15"> Let us now resume our general discussion on the characteristics of the structure of levels. At level L_ ~, we have an enumerable collection of different objects Nx's. Extending the description used for N (at Lo) over Nx's we refer to them as the N, a N, some N(s), every N, etc. It is possible, of course, that some other object N' found at L 0 is now disclosed to be N~, for some x E T. What that means is that we have placed N' incorrectly at Lo, because it actually belonged to L_~. Consider the case of an ancient astronomer who believed that the Morning Star and the Evening Star were not only two different heavenly bodies, but also of the same genre as other stars. In our conventions, both the Morning Star and the Evening Star were placed at the level to which the planet Venus now belongs. When correcting this ancient misconception, we faced the problem of an instance of some object and the object itself were mistakenly assigned the same singularity level; that is, we had both Nx and N at Lo. Nevertheless, this situation represented the state of our knowledge of the world at the time.</Paragraph> <Paragraph position="16"> We may now give names to some Nx'S and N may very well happen to be among them. This time, however, N will not denote the old object from L0; this will be quite a different name referring the selected N~ and can be replaced by a definite description (N x). To illustrate this phenomenon, we may compare the concept of a programming language, such as Lisp, with its various implementations or versions available at different sites or times, and which are locally called Lisps as well. A more common transition of a name is, however, from an instantiation to a concept, which, ultimately, can create a similar effect. Consider for the moment the concept of a sun as a center (or one of several centers) of a solar system, and our own Sun as a specific instantiation of this concept. A schematic illustration of the descent process is shown in Figure 2.</Paragraph> <Paragraph position="17"> A process reverse to decomposition is that of ascending to a higher level within the level hierarchy. Suppose that for some objects N ~, N 2, * *., considered distinct at L 0, we discover they share a certain property, such as being an N, so that we need a generalizing concept to talk about them. We pick up a coordinate T, and climb onto some higher level L~, that is, Lo N,r = t_, < = L 1, and establish a new object N there, a &quot;superobject.&quot; Now, as viewed from L\], all Ni's are just the occurrences of N at different values of coordinate T. In other words, the following equation holds: T for simplicity. No matter how we name N at L+~, the following Formula of Discovery summarizes our action: F3. Yx,y E T \[(N x) = (N y)\] It must be noted that (F3) is valid only when stated from the perspective of L+~. At Lo, Ni's remain distinct traditionally, so they remain distinguished in the language as well; cf. Formula F1. The generalization of other objects from L o onto L+~ may follow but, as in the case of decomposition discussed earlier, the process will largely remain implicit. These observations are illustrated in Example 4.</Paragraph> <Paragraph position="18"> Example 4. At level L0, we have object TP, named The President. Let T be the time coordinate (which is different than T in the last two examples). At Lo, we have, according to (F3), that Vx,y ~ T \[(TP x) = (TP y)\]. Later, we may discover that for some tl, tz E T, (TP t0=N and (TP t2)=R, and that at some level rTP,r ~L-J__ l where N and R belong, they are considered distinct and named Nixon and Reagan, respectively. But at Lo, R=N is true. The last observation can be made clearer if one imagines that TP is some abstract individual who, when observed in the early 1970s, is named Nixon, and who, when observed in the 1980s, is named Reagan.</Paragraph> <Paragraph position="19"> As described, the case of generalization (or abstraction) is extremely common in natural language, and the best tangible manifestation of this phenomenon is common nouns. Common nouns should be considered names of generalized concepts, which may be either of a physical or otherwise measurable nature (mass concepts such as water, snow, gold, temperature .... ) or of a non-physical, abstract nature (usually based on enumerable quantities of specific instances: tree, man, president .... ), or both (like fish, people .... ).</Paragraph> <Paragraph position="20"> Once the superobject N has been created, it begins to live a life of its own. Some new objects from Lo, different than Ni's, may now become instances of N at some, as yet, unutilized values of coordinate T. Also, we may use descriptions (N x) without caring whether they actually refer to any objects at Lo. In other words, the fact that a superobject N has no instance at certain location x ET at level L~'~ T does not preclude the use of Computational Linguistics, Volume 15, Number 3, September 1989 179 Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse the description (N x) in the language. The resulting expression may not always be well defined, though, as is in the case of&quot;the present king of France,&quot; where the adjective &quot;present&quot; specifies the value of time coordinate decomposing the superobject the king of France.</Paragraph> <Paragraph position="21"> This problem of non-denoting expressions created out of denoting general terms, which is widely discussed by Quine (1960, 1973), receives an elegant explanation in our theory. One remaining problem is the relationship between a superobject and its instances in some given decomposition. It is important that we do not equate a superobject at L+I with the set S of its instances at Lo, even if they may have actually given birth to this superobject. A superobject cannot be understood as a set of appropriate lower-level instances, since we would obtain only a measurable collection of singular objects.</Paragraph> <Paragraph position="22"> Instead, a superobject N can be identified with a family of functions {qb~ IT is a coordinate} such that each ~ is a function from coordinate T into an appropriate lower level, L~'I ~r. In particular, a superobject N at LP+'~, where P E L 0, can be considered (from Lo perspective) as a function ~ from T into L0 such that, whenever s ESC_Lo, then there is t E T such that ~ (t) = N t = S.</Paragraph> <Paragraph position="23"> The function qb~. is then arbitrarily extended beyond the set S. The following definition may be suggested.</Paragraph> <Paragraph position="24"> Def. 5. Let L and M be any two distinct levels of relatively singular objects. We say that level L is lower than level M, L < M, if there exists an object P at level M and a coordinate T such that L D ~'te'r ~vJt_ l . In order to avoid any misunderstanding, we add the following remark. It is possible that some Nj from among the Ni's was already recognized properly at Lo as our goal object N from L/l, although its other occurrences Ni for i :k j were not identified with it (cf. the Morning Star, the Evening Star, and Venus). In some sense, therefore, the previous concept of N was incomplete, since it did not contain these other instances which, in turn, allowed this former concept to coexist with some of its would-be instances at the same level.</Paragraph> <Paragraph position="25"> This fact can be further reinforced in our language when we choose to name N after N j. This should not suggest that N and N j are one and the same object. The former is in a sense more mature, although, when referenced by name, one can hardly tell which one of the two concepts is being referred to, unless, of course, some additional clarifying context is present. Examples 5 and 6 below further illustrate this point.</Paragraph> <Paragraph position="26"> Example 5 We have the following distinct objects at level Lo: V, called Venus; MS, called the Morning Star; and ES, called the Evening Star. Upon discovery that they all represent occurrences of the same planet, we create a new object V', named Venus, at the level LI = LV'l T, and such that for some x,y,z E T, where Tis a time coordinate, (V'x) = V, (V'y) = MS, (V'z) = ES, where V', V, MS and ES are individual constants denoting V', V, MS and ES, respectively. Using the formula (F3), we conclude, from the perspective of L~, that V = MS = ES, while the same conclusion made at L o is false.</Paragraph> <Paragraph position="27"> Example 6. Let the level L o be as in Example 5, except that the object V is discovered not to be uniform. In fact, it contains occurrences of three different objects: planet Venus and some two heavenly bodies assumed to be Venus :in the mornings and the evenings. Now we cannot use our time coordinate T from the previous example to get the desired result of the object V' at L+ 1. Instead, we first descend to LV'l s over a coordinate S to differentiate the objects Vs,, Vs 2, V~ 3 for some st, $2, $3 E S. Let the V~, be a part of the ultimate object V'. In the same way, we create instances of MS and ES over the coordinate S at levels tMS,S and fES,S respectively. z.~_ 1 z.-_ I , Because V, MS and ES are relatively singular, their instances in decomposition with respect to the same coordinate yield objects that are also relatively singular. ?V,S In other words, there is a level L 2 such that L 2 ~ ,-,-i t..J LMS,S tES,S Let MS~ be called the Morning Star and --1 \[J 1&quot;--1 &quot; ES~3 be called the Evening Star at L2. Now we can construct the ultimate object V' out of Vsl, MS~2, and ES~3 using a coordinate T such that {s~,sE,s3} C_ T and V&quot; I = V~t, V&quot; 2 = MS~2, and V's3 = ES~. We place V' at a new level L 3 L v d I&quot;T. Note that L 3 C_L o, and thus that V' is singular at L0.</Paragraph> <Paragraph position="28"> Example 6 is more &quot;realistic&quot; than Example 5, but both examples have equal linguistic significance. Note, however, that having L 2 as L 0, and L 3 as L t, we could reconstruct Example 5.</Paragraph> </Section> <Section position="7" start_page="0" end_page="182" type="metho"> <SectionTitle> 6 REMOTE REFERENCES, SUPERCONTEXTS~ AND SUBCONTEXTS </SectionTitle> <Paragraph position="0"> Let us examine how the foregoing theory of non-singular terms could be utilized in assigning meaning representation to natural language discourse. In particular, we, are interested in the problem of computing extrasentential dependencies in text.</Paragraph> <Paragraph position="1"> The translation rules described briefly in Section 2 are applicable only when anaphoric references arise between relatively singular descriptions or, as we would say now, between descriptions denoting objects classified within the same single level. We show that a similar technique can be used to compute another type of extrasentential dependencies, which we call the remote co-reference. Let us start with an example. The arrow symbol-~ is used here, and elsewhere, to mean &quot;translates to&quot;.</Paragraph> <Paragraph position="2"> ExamplE; 7. Consider the following discourse fragment.</Paragraph> <Paragraph position="3"> 7a. The president 1 is elected every four years.</Paragraph> <Paragraph position="4"> 7b. The president 2 is Reagan.</Paragraph> <Paragraph position="5"> There i,; nothing to prevent us from interpreting the president I and the president 2 at levels L t and L2, respectively, so that one of the following takes place.</Paragraph> <Paragraph position="6"> Either Lq = L 2, or L 1 < L 2, or L 2 < L 1, or simply L1 L2, where < stands for the extended lower-level relation introduced in definition 5. The latter case does not interest us, since, in such an interpretation, both sentences were uttered at different occasions with no connection between them.</Paragraph> <Paragraph position="7"> 180 Computational Linguistics, Volume 15, Number 3, September 1989 Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse Consider first that L~ = L 2 = L0. If the two definite descriptions were to co-refer then we would be talking of the same object (individual) in both sentences. That interpretation, although possible, does not agree with our intuition. In this case the conclusion of 7c. Reagan is elected every four years.</Paragraph> <Paragraph position="8"> follows immediately.</Paragraph> <Paragraph position="9"> Let us assume next that L 2 TP, T DL_ 1 < L 1 = L o, where TP is the object at LI referred to by the presidenh, and T is a time-based coordinate. If the presiden h is used as a name, we can expect to obtain the following translations: null</Paragraph> <Paragraph position="11"> with &quot;elected every four years&quot; ---> eefy, &quot;the president&quot; ~ AP\[P(TP)\] where t E T and SL is a selector over T provided by the discourse situation (for example now, here, etc.). This selector could be obtained if we consider a possible translation of (7b) at L2 as pres(R) & loc(R, now), from which it follows that 3tET\[((TP t)=R & loc((TP t), now)\], with SL = At\[loc((TP t), now)\].</Paragraph> <Paragraph position="12"> In a more general case, we would take the phrase the presidenh as an ordinary definite description. Assuming some external context C which allows for the use of the definite article, the following translation can be derived: 8</Paragraph> <Paragraph position="14"> where &quot;president&quot; ---> p.</Paragraph> <Paragraph position="15"> The first of these formulas can be interpreted, with respect to some implicit context C (which may be reasonably assumed to limit our attention to the U.S. government), as: there is exactly one object such that it is a president and has a property of being elected every four years. Clearly, the object we refer to in (7a) is not any particular person, but the office of the President of the U.S. The other formula says, in turn, that there is a value t of the time coordinate T at which the instance of the general object referred to in (7a) (the President of the U.S.) is identical with the object R (which stands for Reagan). Observe that R refers to the individual Reagan as he appears at t, but not necessarily beyond that. The latter formula gives the full translation of (7b) when the remote reference to level L~ has been resolved. Before this interlevel connection is established, however, we can assign (7b) only the a literal translation at L2, as shown below, where C2 is a local context at level L2.</Paragraph> <Paragraph position="17"> A point that is worth discussion is the role of the selector SL in the full translation of (7b). This selector may be interpreted as a local context at L z determined by a discourse situation, as in Barwise and Perry (1983).</Paragraph> <Paragraph position="18"> To understand its influence on the form of the reference-making sentence, compare Example 7 with Example 8 below.</Paragraph> <Paragraph position="19"> Example 8 8a. The tiger lives in the jungle.</Paragraph> <Paragraph position="20"> 8b. My pet is a tiger.</Paragraph> <Paragraph position="21"> Unlike (8b), in the &quot;story&quot; of the president the local context allows (and requires) us to use &quot;the&quot; in (7b) because there may be at most one instance of the general term at a given discourse situation (in this case, the President of the U.S.). This is not the case when more than one value of the coordinate T satisfied the selector condition. The use of a definite description in such a context suggests, therefore, that a single instance of a general term is being picked up by the selector. This means, in turn, that except for a main &quot;remote reference&quot; in cross-level referencing, another local reference is being made at these levels where the definite description is used to denote some object. The remote reference itself does not need a definite description to be used for establishing a connection between an instance and the general term. The fact that we use a definite article when referring to an instance of a general concept, as in (7b), implies only that the local context SL contains (or is expected to contain) exactly one instance of the general object. Notice that the temporal aspect of this local context is extremely influential. 9 When we decompose a general object with respect to a time coordinate, we are more likely to obtain a unique instantiation in a local context. However, when a coordinate does not contain a time element, as in (8a,b), we cannot, in general, exclude the possibility that instances other than the one we intend to refer to in subsequent statements may be present in a local context. Thus a definite description is not used until the local context is properly narrowed. In the case of (8a,b) above, for example, we may continue the discourse at the lower level, saying &quot;The animal is quite friendly,&quot; and meaning &quot;The animal which is my pet tiger is quite friendly.&quot; This discussion strongly supports our theory of names and descriptions, particularly the existence of levels and coordinates. Having established a higher-level object (or superobject), we can freely discuss its instances across various coordinates. The local references can be accounted for by singular translation rules, which we discuss in Strzalkowski and Cercone (1986).</Paragraph> <Paragraph position="22"> Finally, let us define the notion of remote reference for objects classified into different naming levels.</Paragraph> <Paragraph position="23"> Def. 6. An object N at a level L n is said to be remotely referenced by a description M if M refers to an object M at some level L m such that either L m D_ L~ 'r, or L m D_L~f, for some coordinate T.</Paragraph> <Paragraph position="24"> Let us summarize this discussion briefly now. In some part of a discourse, a certain (general) object X is addressed; that is, there is some part, $1, of the discourse (presented as a single sentence in our examples, Computational Linguistics, Volume 15, Number 3, September 1989 181 Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse for simplicity), such that S 1 predicates something of X--that is SI(X), where X is a description that refers to X. In a subsequent part of the discourse, however, the discourse changes the level of reference and only some instance(s) of X with respect to some coordinate T is addressed; that is, there is some t E T such that S2((X t)), where S 2 is this new part of the discourse. Apparently, the discourse internal cohesion would be compromised if we did not allow the higher level object X be a target of a remote reference by a description (X t) denoting one of its instances. In such a case we say that SI(X) creates a supercontext for (X t). We can further say that X and (X t) are remotely co-referential. By analogy, if S! contains a reference to an instance of an object X, that is, we have SI((X t)) for some coordinate Twhere t E T, and $2 contains a reference to X, that is, S2(X), then SI((X t)) is a subeontext for X. In this case too, X and (X t) are remotely co-referential.</Paragraph> <Paragraph position="25"> We have just arrived at a very important property of a natural language discourse, which influences both coherence and cohesion, and is absolutely essential in proper representation of discourse meaning content.</Paragraph> <Paragraph position="26"> The following translation rule specifies how such a representation should be derived.Ideg The approach presented here is entirely original, although the problem of the &quot;general term/an instance&quot; connection has been known for a long time and other, less general, solutions have been suggested (see Partee 1972 and Montague 1974).</Paragraph> <Paragraph position="27"> Rule 11 (Supercontext Translation Rule): If the context-setting sentence St with the translation L+1, where 3x \[Pl(x) & Fl(x)\] is interpreted at level g.r is an object satisfying sentence S 2 when interpreted at level L o, and $2 contains an unresolved remote reference P2, that is,</Paragraph> <Paragraph position="29"> then the full translation of S 2 is obtained as AQ\[AC\[MQ,c\](C1)\](Au\[3t\[AY\[P2(y) & F2(Y)\]((u t))\]\]), where the supercontext C1 is hx\[Pi(x) & Fl(x)\], and MQ, c abbreviates the following expression</Paragraph> <Paragraph position="31"> Thus far we have considered two types of cross-level references: the trivial one where L~ = L2 = Lo, and the remote reference in supercontext, that is, with L~ < L2 = L o. Let us now turn to the remaining type, for which L 0 = L 1 < L2. We call such situations remote references in subeontext.</Paragraph> <Paragraph position="32"> Example 9. Let TP and T be as in Example 7. This time suppose that we have the following situation: L 0 = L I < L 2 ~_Lt+p |r, where tp is TPt for some t E T. Our discourse may now look as follows.</Paragraph> <Paragraph position="33"> 9a. A president1 sits in the first row.</Paragraph> <Paragraph position="34"> 9b. The president 2 is elected every four years.</Paragraph> <Paragraph position="35"> Sentence; 9b has the following translation with a remote reference to &quot;the president&quot; in (9a).</Paragraph> <Paragraph position="37"> where &quot;sits in the first row&quot; ---> sfr.</Paragraph> <Paragraph position="38"> The formula translating (9b) says that the president, a unique superobject denoted by x, is elected every four years. The uniqueness of the president superobject is determined by the fact that one of its instances with respect to some space-time coordinate, identified in the formula by the term (x t) where t is an element of this coordinate, has the property of sitting in the first row.</Paragraph> <Paragraph position="39"> One may wonder whether the use of the definite description in (9b) is always necessary to maintain the remote reference, since we rejected such a necessity in supercontexts. It should be clear enough, considering the level structure introduced earlier, that we can and have to use definite descriptions in subcontextual references. We illustrate this point with another example.</Paragraph> <Paragraph position="40"> The problem is discussed at a great length in Strzalkowski (1986).</Paragraph> <Paragraph position="41"> Example 10. In the following discourse, the descriptions a presidentl and a president 3 refer to objects which are not relatively singular and therefore do not belong to the same level. Although the discourse seems connected and coherent, there is no straightforward correspondence between the two sentences.</Paragraph> <Paragraph position="42"> 10a. ,lohn wants to become a king rather than a presidenti.</Paragraph> <Paragraph position="43"> 10b. That is because a president3 is elected every several years, while the king rules for a lifetime.</Paragraph> <Paragraph position="44"> Here a presidentl refers to an L0-instance PI of an rv,,r A president 3 L+ l-level object Pz specifically Pz E ~+1 * refers to some object P3, which is still non-singular from the perspective of level Lo. P3 is an instance of Pz but with respect to a coordinate T' different than T. In other words, P3 is placed on the level L~:i r, which is not a part of L0. There is no direct correspondence between a president~ and a president3 beyond the fact that P2 E f P3,r' It may well happen, however, that for LP'\[ r n ~+1 * some coordinate U and some u ~ U, we shall have (P3 u) denoting Pv This is why John would rather be a king .... What relates these two &quot;presidents&quot; is a composition of two remote references: the one made by a president I to the object P2 (we may wish to call this object the president2), and the other made by a president 3 to P2. This same effect will be produced if we use other measurably singular descriptions like &quot;every president,&quot; &quot;some presidents,&quot; or &quot;most presidents&quot; in (10b). If we had the defini~te &quot;the president&quot; instead of indefinite &quot;a president&quot; in (10b), however, we would obtain a clear instance of a remote reference in subcontext, that is, with P2 = P3. The objects and their respective levels in the presidents case are illustrated in The case of the remote reference in subcontext is summarized with the translation Rule 12.</Paragraph> <Paragraph position="45"> Rule 12 (Subcontext Translation Rule): If the context-setting sentence S t with the translation ~,T 3x \[Pn(x) & Fl(x)\] is interpreted at some level L_ 1, where ~ is an object satisfying the current sentence S 2 when interpreted at L o, and S 2 contains an unresolved remote reference P2, that is, where the subcontext Cn = Ax \[Pn(x) & Fn(x)\] is derived from $1.</Paragraph> </Section> <Section position="8" start_page="182" end_page="183" type="metho"> <SectionTitle> 7 SUPEROBJECTS VS. PLURAL TERMS </SectionTitle> <Paragraph position="0"> We now examine the nature of superobjects, that is, the objects placed at level L/ 1. In particular, we are interested in what sets them apart from their instances. Let us again consider the term &quot;The President&quot; as referring to the superobject TP at some level L+~ (of. Example 7).</Paragraph> <Paragraph position="1"> We may say that such superterm has the property of collective referring to all of its instances at once, but without necessarily making a reference to any of these instances in particular. At level L0, if we want to refer to a set of objects, we use one of the enumerating quantitiers &quot;every&quot; or &quot;each,&quot; or we use some sort of plural, such as in &quot;These presidents were married&quot; (each). We note, however, that most plurals, especially the so-called &quot;bare&quot; plurals, like presidents, tigers, or meetings can actually refer collectively, which makes them akin of superterms, though many of these plurals can be ambiguous between the collective and a non-collective interpretations. Just like in the case of singular superobjects, the objects denoted by plural terms, such as presidents, tigers, Americans, etc., cannot be always identified with the corresponding sets of lower-level instances. It turns out that the plural terms actually denote superobjects, 11 and therefore they should be interpreted at the same level as respective singular superterms. We will see that the generalization leads naturally to plural terms that may or may not induce equivalent singular superterms. Conversely, a plural equivalent to a singular term may suggest the most natural coordinate to decompose the superobject in its denotation into lower-level instances. When a singular term lacks a plural equivalent, however, we may admit that the object in its denotation is not naturally decomposable and that we are now looking at the bottom-most level in some decomposition hierarchy. A further decomposition may be still possible, but it can only produce objects that will never assume an independent status and will remain recognized only as instances of some more general superobject scattered over that or another coordinate. This phenomenon is characteristic of so-called mass objects and their corresponding mass terms. As an example, consider such nouns as water, gold, or heat. They lack plural equivalents, and there is no obvious way to decompose them into lower-level instances, except with 4-dimensional space-time coordinate. Note that although we can occasionally use a morphologically plural term, like waters, these usually will not be equivalent to singular superterms, and they will often denote other mass objects as well, such as in &quot;waters of the Nile.&quot; Space-time coordinates can also be used to obtain instances of otherwise non-decomposable entities, such as John, into space-time slices, called &quot;stages&quot; (Carlson 1982). Quite naturally, the question of where one level ends and another begins arises. The following two examples provide some insight into the level-boundary problem.</Paragraph> <Paragraph position="2"> Example 11. Consider the following sentences.</Paragraph> <Paragraph position="3"> Let &quot;water&quot; in (1 la) be the name of some superobject W at level L+I. Presumably, Mary brings only a part of W, but we can say that W is being brought by Mary every day. This is the same W every day, although each time possibly a different part of it is in transit, which leads to the obvious translation (at L+t), i. 1 la ---> brings-every-day(M,W) where M and W are individual constants denoting L/I objects M (Mary) and W (water). Alternatively, if we evaluate (lla) at level L o = ,-,-lrw'r with a space-time coordinate T, in which case (lla) should read &quot;Mary brings some water every day,&quot; we obtain the following interpretation: ii. 1 la---> Vx \[day(x) D 3t \[brings(x, M, (W t))\]\] where W is as before, M is an individual constant denoting a space-time stage of Mary,12 and brings(x,y,z) should be read at x y brings z. Now (W t) denotes some instance of the superobject W at level L 0' of which we Computational Linguistics, Volume 15, Number 3, September 1989 m _ Tomek Strzalkowski and Nick Cercone Non-Singular Concepts in Natural Language Discourse can say that it is water as well; that is, water((W t)). At L 0' we replace (W t) by a singular variable u, obtaining ii'. 1 la---~ Vx \[d(x) D 3u \[water(u) & brings(x, M, u)\]\] The translations featured in (i), (ii), and (ii') may appear to be equivalent statements made from different perspectives and at different levels of detail. This is not the case, though. In particular, (ii) and (ii') are not equivalent. This point is further illustrated with (1 lb). Let m be the mail that John is picking up at any one particular occasion. There is a space-time coordinate T such that the level L~I r contains a unique superobject for which the following holds.</Paragraph> <Paragraph position="4"> iii. 1 lb ~ 3x \[mail(x) & C(x) & Vy \[(mail(y) & C(y))</Paragraph> <Paragraph position="6"> The context C is used to determine the uniqueness of John's mail superobject. Note that the quantification m,T ranges over objects at L+~ . Because John's mail superobject is decomposable with T we obtain the following equivalent cross-level interpretation.</Paragraph> <Paragraph position="7"> iv. 1 lb ~ 3x \[mail(x) & C(x) & Vy \[(mail(y) & C(y)) 3 (x = y)\] & Vu \[morn(u) D 3t \[picks(u, J, (x t))\]\]\] Here, t ranges over elements of coordinate T, and (x t) refers to an L 0 instance of John's mail superobject. The three arguments of picks(x,y,z) are understood as at x y picks up z. As it stands, (iv) still needs to be interpreted m,T at L+~ . To reduce this translation to the form that would be interpretable at Lo, we have to replace the definite reference in the first line of (iv), that is,</Paragraph> <Paragraph position="9"> with a predicate, interpretable at L o, that would uniquely indicate John's mail. Then replacing (x t) by a variable z, we obtain the following formula.</Paragraph> <Paragraph position="10"> iv'. Vu \[morn(u) D 3z \[John's-mail(z) & picks(u, J, z)\]\] Note that equivalence to (iv) requires the John's-mail predicate to be true of an empty z, which can happen if, at some occasion, John receives no mail at all. Otherwise (iv) and (iv') are not equivalent. In (iv) we can maintain truth of picks (u,J,(x t)) because we do not require (x t) to denote anything at Lo; we merely say that a t exists. This example actually shows the power of the multilevel representation. Observe that we have just delivered a very strong argument supporting the claim that superobjects are not merely sets of their instances. It must be noted here again (cf. footnote 7) that some limit needs to be set up for the amount of exceptions to a general statement like (iv) that we are willing to tolerate.</Paragraph> <Paragraph position="11"> There remains one more reading of (lib) that does not seem to require any reference to higher-level objects. This reading could be paraphrased as: Every morning there exists the mail such that John picks it up, or more formally</Paragraph> <Paragraph position="13"> This time the variable u is bound at Lo. Is this translation feasible? We can answer both &quot;yes&quot; and &quot;no.&quot; A &quot;yes&quot; answer indicates that the transformation gives us a singu\]lar interpretation of (11b) at L o. Note that because of the uniqueness clause in it, (v) says no more than that John keeps picking up the same thing every morning, since the context C does not depend on x and is the same each time. A &quot;no&quot; answer indicates that the latter interpretation most probably does not express our intention. The translation of (v), although possible, is not equivalent to either (iii), (iv), or (iv').</Paragraph> <Paragraph position="14"> At the beginning of this section we stated that bare plurals, like presidents or meetings, behave much like singular superobjects. Let us look somewhat closer at this issue now. Consider the pairs of sentences (12a,b) and (12c,d) below.</Paragraph> <Paragraph position="15"> Example 12.</Paragraph> <Paragraph position="16"> It seems to us that the preferred interpretation of sentences in (12b) and (12d) is such that the plural noun phrase ,(faculty meetings, rats) is understood as denoting a higher-level concept. For the same reason as (12a), and (lib) before it, (12b) will remain a truthful L/I statement in spite of the fact that, at some occasions, meetings can be canceled (due to holidays, for example). In the case of (12c) and (12d) the point is somewhat finer. &quot;\['his time we may be less reluctant to say that (12d) means, in fact, that every rat can live in most countries, because we talk about possibilities. There is a catch in here, however; though every rat can live at many different places, not every one, or perhaps even none, car~ live at most of these places.</Paragraph> <Paragraph position="17"> One of this section's key observations is that plural terms are, in many respects, equivalent to singular superterms. We can assume, for now, that plural terms, cautiously named prototypes of superterms, actually denote superobjects as well. For some plural terms, we can find equivalent singular versions like tigersmtiger, presidentsmpresident, etc. Others do not have this property. Alternatively, mass terms will usually lack plural equivalents, for example, &quot;water&quot; cannot be identified with &quot;waters&quot; in general. Still others may expose a surprising mixture of the properties, like &quot;people,&quot; which is a plural term with morphologically singular form and which may also be used as mass superterm.</Paragraph> </Section> class="xml-element"></Paper>