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<Paper uid="C86-1086">
  <Title>An Approach to Non-Singular Terms in Discourse</Title>
  <Section position="3" start_page="0" end_page="363" type="metho">
    <SectionTitle>
2. Non-singular terms in language
</SectionTitle>
    <Paragraph position="0"> Many philosophers and logicians, see \[1\]-\[4\], \[6\], \[10\]. appreciate that the u~age of the underlined nominal phrases in the following sentences has a &amp;quot;general&amp;quot; or &amp;quot;generic&amp;quot; character, except for &amp;quot;regular&amp;quot; singular interpretations which are possible only in some cases.</Paragraph>
    <Paragraph position="1"> E~ample j (~,d rh~ wears a crown.</Paragraph>
    <Paragraph position="2"> (lb) T~resident rs elected every four years.</Paragraph>
    <Paragraph position="3"> (lc) Gold is a yellow metal (ld) The temperature is a measure of molecular motion.</Paragraph>
    <Paragraph position="4"> :One can imagine hundreds of similar examples involving such non-singular objects as water, beat, the Pope, the number etc Unfortunately, there is no commonly accepted account of these species in  philosophical literature Some authors, see \[J\] and \[I0\], cautiously called them generic, or general (for example the king), or func tional (such as the number of students, ttre temperature) uses of (definite) descriptions. Others, like Kripke \[3\]. were quite close to consider them names (or at least some of them: heat gold). Yet others, see Quine \[6, 7\]. advocate the notion of abstract terms as being made of attributes, such as /being\] red (further abstracted as redness), or /being\] the man drinking tile martini (which cannot be so easily uominalized) which can predicate about &amp;quot;concrete&amp;quot; objects.</Paragraph>
    <Paragraph position="5"> There are numerous striking linguistic puzzles involving non-singular definite descriptions see \[I\]. \[4\], \[5\]. The following example illustrates the phenomenon  Consider the following inferences: (2a) The temperature is rising The temperature is ninety.</Paragraph>
    <Paragraph position="6"> so, Ninety is rising.</Paragraph>
    <Paragraph position="7"> (2b) The president met the Soviet leader many times The president is Reagan so. Reagan met the 5oviet leader many times (2c) The tiger lives in thejungle.</Paragraph>
    <Paragraph position="8"> My pet is a tiger.</Paragraph>
    <Paragraph position="9"> so My pet lives in the j ungle The conclusions in (2a) to (2c) are wrong in general case The explanation given by numerous researchers chiefly amounted to the corroboration that the definite descriptions the temperature, the president and the tiger in the first sentences of (2a). (2b) and (2c) respectively should be interpreted functionally, i.e., as iutensions \[4\], or functions over situations \[1\] Observe that if the descriptions were to be interpreted singularly or as enumerating all instances of a non-singular object (i.e., statements containing them were understood as making claims about each instance), the reasoning would be sound. We claim that unless some two descriptions (or names) are used singularly or measurably singularly at the same level no simple ,eference can be made between them. In fact. another type of reference that we call remote reference can still take place and we shall put this view forward in this paper 3. Tire Theory of Names and Descriptions Initially let us observe that our language deals with singular objects only, no matter how complex their structure happens to be Suppose somebody is being put into posihon of the Observer who perceives all these objects and has to use his language to describe them Some objects are sharply distinguished from others so he chooses to give them names as John, Mary, Fatsy. Sun, .. The others have no clearly perceivable boundaries but he still may name them: tea. water, grass, snow ..... and then refer to some measurable quantities of them as some tea, little snow. etc, Yet others appear to be numerous, though enumerable, displaying strong similarities to one another. It would be pointless for Observer to give them each a name Instead. he decides to refer to them as a cow, the man. this tree etc Still. he prefers to say the sun or the lake rather than to invent new names if he is not sure how many of them are there, even if he is aware of just one specimen LaLer tie may find out that some objects were gwen identical names, so having encountered them together he must refer to one as the John. the Sun, or a Fatsy. Having completed his job Observer, who is also a part of this world, may name h~mself Observer or the Observer, and happily sit down under a tree on the grass Let us call the whole collection of objects he has just described as the Observer level and use the symbol L 0 for it Suppose then we ask Observer to tell us as much as he can about L 0 Soon he finds out that his naming has its limits As he discovers new facts about his world it becomes more and more cumbersome for him to communicate in terms of every man. some cats. several trees, each president, etc He discovers that some things be originally considered distinct appear to be instances of some single object. Also he must admit that the identity of some other objects has to be put into question Being smart enough. Observer invents two new levels. L+1 and L_j. which augment his world. At level L+I he places the new objects he discovered to be generalisations (or abstractions, if you like) of some measurable amount of objects from L 0 which displayed a striking similarity or even identity. From the perspective of L+l he is able to tell us that Tire tiger lives in the .jungle. that The president is elected every four years, and that The Morning Star and The Evening Star are actually two appearances of tire plar~et Venus. The objects at L+I are singular there, bnt they appear generic&amp;quot; or &amp;quot;functional&amp;quot; or whatever of thai sort as seen from L 0. Observe that these objects may not have straightforward measurably singular descriptions at L 0 (like every tiger, some president, etc). and often it will not be possible to refer to them in the terms of the language available at L(~. In either case one may expect that some undescribable aspect of an L+I object can emerge at L n. even if they all have been derived from L 0 (which does not have to be the case). Next Observer invents a new generation of names at L+l. the president and the tiber may be among those names. On the other hand Observer might prefer to use definite descriptions here. for the similar reason he frequently decided so at L\[). In fact. we have no means to distinguish between names and definite descriptions in discourse We (.an only stick to linguistic conveutions.</Paragraph>
    <Paragraph position="10"> It probably would not take a long time before a new augmentation for L+( becomes necessary. Two new levels L+1+1 and L+I_ t carl be added in a much the same fashion The level L+~_ 1 does not necessarily have to be L0 although it probably will. More or less the same happens at the level /._1 where Observer can now say that what he previously considered to be the atom actually denotes marry different kinds of atoms (H. O Ca. Fe etc.), that tea is not so uniform and many different teas can be found, and that under the name Joe Smith was actually hidden a group of crime story writers. Subsequently the level L-1 will expand by L_I+ ( and L-a-( with the former often different than L 0. Let us now formalize our intuition Definition :1 A use of a description will be called singular if it denotes or refers to a singular object. A use of a description will be called measurably singular if it denotes or refers to some measurable quantity of a singular object. Otherwise we shall talk of non-singular use.</Paragraph>
    <Paragraph position="11"> Definition 2 A level will be an arbitrary collection of singular objects. A level language will contain these and only singular and measurably singular uses of descriptions communicating of the level objects.</Paragraph>
    <Paragraph position="12"> Definition 3 For any level L~. all names appearing in tbe L, language have singular interpretations.</Paragraph>
    <Paragraph position="13"> Definition 4 For any level L n there will be at least two distinct levels L~ l and L,+ 1 such that L,,+l contains the non-singular objects as seen from L,. and L,-t contains the objects for which the objects at L, are non-singular Definition 5 Tbe Observer level L o is an arbitrary chosen level serving as a reference point Suppose that we have an object N called N at level L 0. Let T be an arbitrary set we shall refer to as a coordinate. Suppose further that. for the coordinate T. the Observer discovers that the identity of N along that dimension can no longer be accepted. That is. there are at least two x. y E T such that N at x ~: N at y. Without losing generality we can assume that the coordinate T has been chosen so that the following non-equation holds:  moves them onto a new level L_Nt 7 leaving the original object N at L0. N may be no longer a &amp;quot;real&amp;quot; object but the concept remains in language LNt 7 can be attached to any existing level provided that the definitions 1 to 4 will never be violated It can also give a beginning to a new level. Note that the distribution of N over the coordinate T forces otber objects from L 0 to be distributed over T as well. and their instances placed at LN_I T This process may remain mostly implicit until we mm(e an utterance relating (N x) to other objects at LN_i T. In general we shall say that the level L_Ni T is lower than the level L 0, and write L~_lf&lt;Lo Often we shall drop the superscripts N and T over the level symbol assuming some lower level L_~ whenever it does riot lead to ambiguity Observe that with the above account the level structure of objects has a dynamic, ever-changing character. Ally new empirical fact to be added to our world knowledge bears a potential reveberation in the level structure involving creation of new levels and moving objects between levels. At probably non-frequent idle states the definitions 1 to 4 assure the structure balance Moving at level L_ 1 the Observer is aware of an enumerable collection of different objects N,'s Extending the description used for N over N,'s the Observer refers to them as the N. a N. some N(s).</Paragraph>
    <Paragraph position="14"> every N. etc. It is possible, of course, that some other object N' found at L 0 is now disclosed to be an Nx for some x (~ T. What that means is that we have wrongly placed N&amp;quot; at L 0. because it actually belonged to L_ l 13ut this was right at the time N&amp;quot; was placed there. ie. it mirrored tbe state of onr knowledge of the world at the time. We may now give names to some Nx's and N can very well happen among them. This time however N will not denote the old object from L0: this will be actually quite a different name referring to one selected N,. and which may be replaced by a definite description of (N~) On the other hand suppose we tlave some objects N l, N 2, considered distim:t at L 0 Suppose then that we discover some resemblance between them along some dimension (coordinate) T. so that we need a generalizing concept to talk about them. We climb to some higher level L~I 7-. i.e L0&lt;LNI 7. and establish a new object a superobject N there Now as seen from LN+j T all N/s are just the occurences of N at L 0 at different values of coordinate T In other words, the following equation holds:</Paragraph>
    <Paragraph position="16"> Observe also that all Ni's now belong to the level N T L+l--1 which is a part of Lo As before we shall drop superscripts N and T for simplicity. No matter how we name N at L+I the following Formula of Discovery summarizes our action: (FD) VxVy. x.y e T.(Nx) =(Ny) Remember that the formula FD is valid only when observed from L+I At L 0. Ni's remain distinct traditionally - so they remain distinct in the language as well. The generalisation of other objects from Lo onto LN+\] T may follow but. as in tbe case of decomposition discussed above, the process will remain largely implicit. Once the superobject N has been created it begins to live its own life. Some new objects from L 0 different than N/s. may now become instances of N at some not yet utilized values of coordinate T. Also. we may use descriptions (N x) without caring whether they actually refer to any objects at L 0. The latter property of general terms which is widely discussed by Qnine \[6 7\] gets a formal explanation in our theory. It is important not to confuse a superobject with a set S of lower level instances over some coordinate T as we would obtain a measurably singular concept only. Instead. a superobject can be identified with the function N from Tinto L 0such that whenever sPSSthen there is a teT such that (N t)=s. and then extended arbitrarily beyond the set S Exam Die 3 We have the following distinct object at some level L0: V called Venus. MS called Morning Star. and ES called Evening Star. Upon discovery that they all are just occnrences of the same planet we create a new object V&amp;quot; named Venus at some level LVl &amp;quot;T and such that for some ;~.y.z C/ T. where T is a time coordinate.</Paragraph>
    <Paragraph position="17"> (V';~) = V (V&amp;quot; v) =MS.(V'z) = ES. According to the FD formula we conclude from L+1 that V=MS=ES. while the same conclusion made at L n is false Examp~ At level Lo the OI)server is aware of the object TP named The President. Let T be the time coordinate (different than in the last  example). At L 0we have according to the FD formula that deg VxVy. x,y E T (TPx) =(TPy) Later the Observer may dicover that for some tl. t2 E T, (TP tl)=N and (TP t2)=R, aud that at some level LT_~ 'T where N and R belong, they are considered distinct and named Nixon and Reagan respectively. But at L o. R=N is true. The last observation can be made clearer if one imagines that TP is some abstract individual which (like Venus) when observed in early 70's is named Nixon, while when observed in 80's is called Reagan, \[\]</Paragraph>
  </Section>
  <Section position="4" start_page="363" end_page="363" type="metho">
    <SectionTitle>
4. Superobjects
</SectionTitle>
    <Paragraph position="0"> Let us now examine the nature of superobjects i.e.. the objects placed at level L+l. It turns out that the plural terms, e.g.</Paragraph>
    <Paragraph position="1"> presidents, tigers, etc., are actually prototypes of superobJects, see \[6\]. and they should therefore be placed at the same level as respective superobjects. We will see that the generalization leads naturally to plural terms which may or may not induce equivalent singular superobjects. Conversely. a plural equivalent to a superobject may suggest the most natural coordinate to decompose the latter onto some lower level. When a superobject lacks a plural equivalent, however, we may admit that this object's origin has been traced down. A further decomposition is still possible but this process may often produce objects that will never assume an independent status and will remain recognized only as instances of- this general concept scattered over that or another coordinate This phenomenon is characteristic of the so-called mass objects and their corresponding mass terms Quite naturally the question of- where one level ends and another begins arises The following example gives some insight into the problem of level boundaries.</Paragraph>
    <Paragraph position="2"> ExamplPS5 Consider the following sentences.</Paragraph>
    <Paragraph position="3"> (Sa) Mary brings (some) water every day.</Paragraph>
    <Paragraph position="4"> Let water in (5a) be the name of some superobject w at the 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+I ) (i) 5a --, (br-e-d M w) where br-e-d stands for brings every day.</Paragraph>
    <Paragraph position="5"> 'On the other hand. suppose that Mary brings some water every day. Except for the above interpretation, we also have the measurably singular reading at L0 where w is scattered over some coordinate T so that ~tE T such that (W t) is being brought by Mary. i.e..</Paragraph>
    <Paragraph position="6"> (~t (br M (w t))). This clause is, of course, relative to every day so at L 0 we could have (ii) 5a -~ (Vx(dx) D (~\]t(brM(wt))))t where brines -~ hr. day ~d Both translations are essentially equivalent, and this equivalence is by no means accidental. It lends a strong support for our Theory of Names and Descriptions. and explains the intuition underlying its formulation. null</Paragraph>
  </Section>
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