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<?xml version="1.0" standalone="yes"?> <Paper uid="J87-3004"> <Title>COMMONSENSE METAPHYSICS AND LEXICAL SEMANTICS</Title> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 2 REQUIREMENTS ON ARGUMENTS OF PREDICATES </SectionTitle> <Paragraph position="0"> There is a notational convention used below that deserves some explanation. It has frequently been noted that relational words in natural language can take only certain types of words as their arguments. These are usually described as selectional constraints. The same is true of predicates in our knowledge base. The constraints are expressed below by rules of the form p(x,y) : r(x,y) This means that for p even to make sense applied to x and y, it must be the case that r is true of x and y. The logical import of this rule is that wherever there is an axiom of the form</Paragraph> <Paragraph position="2"> this is really to be read as</Paragraph> <Paragraph position="4"> The checking of selectional constraints, therefore, emerges as a by-product of other logical operations: the constraint r(x,y) must be verified if anything else is to be proved from p(x,y).</Paragraph> <Paragraph position="5"> The simplest example of such an r(x,y) is a conjunction of sort constraints rl(x)/% r2(y). Our approach is a generalization of this, because much more complex requirements can be placed on the arguments. Consider, for example, the verb &quot;range&quot;. Ifx ranges from y to z, there must be a scale s that includes y and z, and x must be a set of entities that are located at various places on the scale. This can be represented as follows: range(x,y,z) : (3 s) \[scale(s)/% y E s/% z E s/% set(x) /% (V u)\[u ~ x D (3 v) v E s/% at(u,v)\]\]</Paragraph> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 3 THE KNOWLEDGE BASE </SectionTitle> <Paragraph position="0"/> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.1 SETS AND GRANULARITY </SectionTitle> <Paragraph position="0"> At the foundation of the knowledge base is an axiomatization of set theory. It follows the standard Zermelo-Fraenkel approach, except that there is no axiom of infinity.</Paragraph> <Paragraph position="1"> Since so many concepts used in discourse are graindependent, a theory of granularity is also fundamental (see Hobbs 1985b). A grain is defined in terms of an indistinguishability relation, which is reflexive and symmetric, but not necessarily transitive. One grain can be a refinement of another, with the obvious definition.</Paragraph> <Paragraph position="2"> The most refined grain is the identity grain, i.e., the one in which every two distinct elements are distinguishable. One possible relationship between two grains, one of which is a refinement of the other, is what we call an &quot;Archimedean relation&quot;, after the Archimedean prop-erty of real numbers. Intuitively, if enough events occur that are imperceptible at the coarser grain g2 but perceptible at the finer grain g~, the aggregate will eventually be perceptible at the coarser grain. This is an important property in phenomena subject to the heap paradox. Wear, for instance, eventually has significant consequences.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.2 SCALES </SectionTitle> <Paragraph position="0"> A great many of the most common words in English have scales as their subject matter. This includes many prepositions, the most common adverbs, comparatives, and many abstract verbs. When spatial vocabulary is used metaphorically, it is generally the scalar aspect Of space that carries over to the target domain. A scale is defined as a set of elements, together with a partial ordering and a granularity (or an indistinguishability relation). The partial ordering and the indistinguishability relation are consistent with each other: (Vx,y,z) x < y A y ~ z D x < zV x ~ z That is, ifx is less than y and y is indistinguishable from z, then either x is less than z or x is indistinguishable from z.</Paragraph> <Paragraph position="1"> It is useful to have an adjacency relation between points on a scale, and there are a number of ways we could introduce it. We could simply take it to be primitive; in a scale having a distance function, we could define two points to be adjacent when the distance between them is less than some E; finally, we could define adjacency in terms of the grain size for the scale:</Paragraph> <Paragraph position="3"> That is, distinguishable elements x and y are adjacent on scale s if and only if there is an element z which is indistinguishable from both.</Paragraph> <Paragraph position="4"> Two important possible properties of scales are connectedness and denseness. We can say that two elements of a scale are connected by a chain of adj relations:</Paragraph> <Paragraph position="6"> A scale is connected (sconnected) if all pairs of elements are connected. A scale is dense if between any two points there is a third point, until the two points are so close together that the grain size no longer allows us to determine whether such an intermediate point exists.</Paragraph> <Paragraph position="7"> Cranking up the magnification could well resolve the continuous space into a discrete set, as objects into atoms.</Paragraph> <Paragraph position="9"> This expresses the commonsense notion of continuity.</Paragraph> <Paragraph position="10"> Computational Linguistics Volume 13, Numbers 3-4, July-December 1987 243 Jerry R. Hobbs et al. Commonsense Metaphysics and Lexical Semantics A subscale of a scale has as its elements a subset of the elements of the scale and has as its partial ordering and its grain the partial ordering and the grain of the scale.</Paragraph> <Paragraph position="11"> (V sx,s2)subscale(s2,s 0 = subset(s2,s 0 A (Vx,y)\[\[x <sl Y =- x <s2 Y\] A \[x ~sl Y -~ x ~s2 Y\]\] An interval can be defined as a connected subscale:</Paragraph> <Paragraph position="13"> The relations between time intervals that Allen and Kautz (1985) have defined can be defined in a straight-forward manner in the approach presented here, but for intervals in general.</Paragraph> <Paragraph position="14"> A concept closely related to scales is that of a &quot;cycle&quot;. This is a system that has a natural ordering locally but contains a loop globally. Examples are the color wheel, clock times, and geographical locations ordered by &quot;east of&quot;. We have axiomatized cycles in terms of a ternary between relation whose axioms parallel those for a partial ordering.</Paragraph> <Paragraph position="15"> The figure-ground relationship is of fundamental importance in language. We encode it with the primitive predicate at. It is possible that the minimal structure necessary for something to be a ground is that of a scale; hence, this is a selectional constraint on the arguments of at. 1 at(x,y) : (3s)y E s A scale(s) At this point, we are already in a position to define some fairly complex words. As an illustration, we give the example of &quot;range&quot; as in &quot;x ranges from y to z&quot;:</Paragraph> <Paragraph position="17"> A (Vu)\[u ~ x ~ (3v)v ~ s I A at(u,v)\] That is, x ranges from y to z if and only ify and z are the bottom and top ofa subscale s~ of some scale s and x is a set which has elements at y and z and all of whose elements are located at points on s~.</Paragraph> <Paragraph position="18"> A very important scale is the linearly ordered scale of numbers. We do not plan to reason axiomatically about numbers, but it is useful in natural language processing to have encoded a few facts about numbers. For example, a set has a cardinality which is an element of the number scale.</Paragraph> <Paragraph position="19"> Verticality is a concept that would most properly be analyzed in the section on space, but it is a property that many other scales have acquired metaphorically, for whatever reason. The number scale is one of these.</Paragraph> <Paragraph position="20"> Even in the absence of an analysis of verticality, it is a However, we are currently examining an approach in which a more abstract concept, &quot;system&quot;, discussed in Section 3.6.3, is taken to be the minimal structure for expressing location.</Paragraph> <Paragraph position="21"> useful property to have as a primitive in lexical semantics. null The word &quot;high&quot; is a vague term asserting that an entity is in the upper region of some scale. It requires that the scale be a vertical one, such as the number scale. The verticality requirement distinguishes &quot;high&quot; from the more general term &quot;very&quot;; we can say &quot;very hard&quot; but not &quot;highly hard&quot;. The phrase &quot;highly planar&quot; sounds all right because the high register of &quot;planar&quot; suggests a quantifiable, scientific accuracy, whereas the low register of &quot;flat&quot; makes &quot;highly flat&quot; sound much worse.</Paragraph> <Paragraph position="22"> The test of any definition is whether it allows one to draw the appropriate inferences. In our target texts, the phrase &quot;high usage&quot; occurs. Usage is a set of using events, and the verticality requirement on &quot;high&quot; forces us to coerce the phrase into &quot;a high or large number of using events&quot;. Combining this with an axiom stating that the use of a mechanical device involves the likelihood of abrasive events, as defined below, and with the definition of &quot;wear&quot; in terms of abrasive events, we should be able to conclude the likelihood of wear.</Paragraph> </Section> <Section position="3" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.3 TIME: TWO ONTOLOGIES </SectionTitle> <Paragraph position="0"> There are two possible ontologies for time. In the first, the one most acceptable to the mathematically minded, there is a time line, which is a scale having some topological structure. We can stipulate the time line to be linearly ordered (although it is not in approaches that build ignorance of relative times into the representation of time (e.g., Hobbs, 1974) nor in approaches employing branching futures (e.g., McDermott, 1985)), and we can stipulate it to be dense (although it is not in the situation calculus). We take before to be the ordering on the time line:</Paragraph> <Paragraph position="2"> We allow both instants and intervals of time. Most events occur at some instant or during some interval. In this approach, nearly every predicate takes a time argument.</Paragraph> <Paragraph position="3"> In the second ontology, the one that seems to be more deeply rooted in language, the world consists of a large number of more or less independent processes, or histories, or sequences of events. There is a primitive relation change between conditions. Thus, change(e 1,e2) A p'(e Z,x) A q'(ez,x ) says that there is a change from the condition el ofp's being true of x to the condition e 2 of q's being true of x. The time line in this ontology is then an artificial construct, a regular sequence of imagined abstract events (think of them as ticks of a clock in the National Bureau of Standards) to which other events can be related. The change ontology seems to correspond to the way we experience the world. We recognize relations of causality, change of state, and copresence Jerry R. Hobbs et al. Commonsense Metaphysics and Lexical Semantics among events and conditions. When events are not related in these ways, judgments of relative time must be mediated by copresence relations between the events and events on a clock and change of state relations on the clock.</Paragraph> <Paragraph position="4"> The predicate change possesses a limited transitivity. There has been a change from Reagan's being an actor to Reagan's being president, even though he was governor in between. But we probably do not want to say there has been a change from Reagan's being an actor to Margaret Thatcher's being prime minister, even though the second event comes after the first.</Paragraph> <Paragraph position="5"> In this ontology, we can say that any two times, viewed as events, always have a change relation between them.</Paragraph> <Paragraph position="7"> That is, if there is a change from e 1 to e 2, then there is a time t I at which e I occurred and a time t 2 at which e 2 occurred, and t~ is before t 2. This does not allow us to derive change of state from temporal succession. For this, we would need axioms of the form</Paragraph> <Paragraph position="9"> That is, if x is p at time t~ and q at a later time t 2, then there has been a change of state from one to the other.</Paragraph> <Paragraph position="10"> This axiom would not necessarily be true for all p's and q's. Time arguments in predications can be viewed as abbreviations:</Paragraph> <Paragraph position="12"> The word &quot;move&quot;, or the predicate move, (as in &quot;x moves from y to z&quot;) can then be defined equivalently in terms of change,</Paragraph> <Paragraph position="14"> (The latter definition has to be complicated a bit to accommodate cyclic motion. The former axiom is all right as it stands, provided there is also an axiom saying that for there to be a change from a state to the same state, there must be an intermediate different state.) In English and apparently all other natural languages, both ontologies are represented in the lexicon. The time line ontology is found in clock and calendar terms, tense systems of verbs, and in the deictic temporal locatives such as &quot;yesterday&quot;, &quot;today&quot;, &quot;tomorrow&quot;, &quot;last night&quot;, and so on. The change ontology is exhibited in most verbs, and in temporal clausal connectives. The universal presence in natural languages of both classes of lexical items and grammatical markers requires a theory that can accommodate both ontologies, illustrating the importance of methodological principle 4.</Paragraph> <Paragraph position="15"> Among temporal connectives, the word &quot;while&quot; presents interesting problems. In &quot;e I while e2&quot;, e2 must be an event occurring over a time interval; el must be an event and may occur either at a point or over an interval. One's first guess is that the point or interval for el must be included in the interval for % However, there are cases, such as The electricity should be off while the switch is being repaired.</Paragraph> <Paragraph position="16"> which suggest the reading &quot;e 2 is included in el&quot;. We came to the conclusion that one can infer no more than that el and e 2 overlap, and any tighter constraints result from implicatures from background knowledge.</Paragraph> <Paragraph position="17"> The word &quot;immediately&quot;, as in &quot;immediately after the alarm&quot;, also presents a number of problems. It requires its argument e to be an ordering relation between two entities x and y on some scale s.</Paragraph> <Paragraph position="19"> It is not clear what the constraints on the scale are.</Paragraph> <Paragraph position="20"> Temporal and spatial scales are acceptable, as in &quot;immediately after the alarm&quot; and &quot;immediately to the left&quot;, but the size scale is not: * John is immediately larger than Bill.</Paragraph> <Paragraph position="21"> Etymologically, it means that there are no intermediate entities between x and y on s. Thus,</Paragraph> <Paragraph position="23"> However, this will only work if we restrict z to be a relevant entity. For example, in the sentence We disengaged the compressor immediately after the alarm.</Paragraph> <Paragraph position="24"> the implication is that no event that could damage the compressor occurred between the alarm and the disengagement, since the text is about equipment failure.</Paragraph> </Section> </Section> <Section position="6" start_page="0" end_page="0" type="metho"> <SectionTitle> 3.4 SPACES AND DIMENSION: THE MINIMAL STRUCTURE </SectionTitle> <Paragraph position="0"> The notion of dimension has been made precise in linear algebra. Since the concept of a region is used metaphorically as well as in the spatial sense, however, we were concerned to determine the minimal structure a system requires for it to make sense to call it a space of more than one dimension. For a two-dimensional space, there must be a scale, or partial ordering, for each dimension.</Paragraph> <Paragraph position="1"> Moreover, the two scales must be independent, in that the order of elements on one scale can not be determined from their order on the other. Formally, A (3x)\[(3yt) \[x <s, Yi A x <s2 Yl\] A (::ly2)\[x <s, A Yz A Y2 <s2 x\]\] Note that this does not allow <,2 to be simply the reverse of <s. An unsurprising consequence of this definition is that the minimal example of a two-dimensional space consists of three points (three points determine a plane), e.g., the points A, B, and C, where A<IB, A<IC, C<2A, A<2 B.</Paragraph> <Paragraph position="2"> This is illustrated in Figure 1.</Paragraph> <Paragraph position="3"> The dimensional scales are apparently found in all natural languages in relevant domains. The familiar three-dimensional space of common sense can be defined by the three scale pairs &quot;up-down&quot;, &quot;frontback&quot;, and &quot;left-right&quot;; the two-dimensional plane of the commonsense conception of the earth's surface is represented by the two scale pairs &quot;north-south&quot; and &quot;east-west&quot;.</Paragraph> <Paragraph position="4"> The simplest, although not the only, way to define adjacency in the space is as adjacency on both scales:</Paragraph> <Paragraph position="6"> A region is a subset of a space. The surface and interior of a region can be defined in terms of adjacency, in a manner paralleling the definition of a boundary in pointset topology. In the following, s is the boundary or surface of a two- or three-dimensional region r embedded in a space sp.</Paragraph> <Paragraph position="8"> Finally, we can define the notion of &quot;contact&quot; in terms of points in different regions being adjacent:</Paragraph> <Paragraph position="10"> By picking the scales and defining adjacency right, we can talk about points of contact between communication networks, systems of knowledge, and other metaphorical domains. By picking the scales to be the real line and defining adjacency in terms of eneighborhoods, we get Euclidean space and can talk about contact between physical objects.</Paragraph> </Section> <Section position="7" start_page="0" end_page="0" type="metho"> <SectionTitle> 3.5 MATERIAL </SectionTitle> <Paragraph position="0"> Physical objects and materials must be distinguished, just as they are in apparently every natural language, by means of the count noun-mass noun distinction. A physical object is not a bit of material, but rather is composed of a bit of material at any given time. Thus, rivers and human bodies are physical objects, even though their material constitution changes over time.</Paragraph> <Paragraph position="1"> This distinction also allows us to talk about an object's losing material through wear and still remaining the same object.</Paragraph> <Paragraph position="2"> We will say that an entity b is a bit of material by means of the expression material(b). Bits of material are characterized by both extension and cohesion. The primitive predication occupies(b,r, t) encodes extension, saying that a bit of material b occupies a region r at time t. The topology of a bit of material is then parasitic on the topology of the region it occupies. A part b~ of a bit of material b is a bit of material whose occupied region is always a subregion of the region occupied by b.</Paragraph> <Paragraph position="3"> Point-like particles (particle) are defined in terms of points in the occupied region, disjoint bits (disjointbit) in terms of the disjointness of regions, and contact between bits in terms of contact between their regions.</Paragraph> <Paragraph position="4"> We can then state as follows the principle of non-jointoccupancy that two bits of material cannot occupy the same place at the same time:</Paragraph> <Paragraph position="6"> That is, if bits b 1 and b 2 are disjoint, then there is no entity z that is at interior points in both bl and b2. At some future point in our work, this may emerge as a consequence of a richer theory of cohesion and force.</Paragraph> <Paragraph position="7"> The cohesion of materials is also a primitive property, for we must distinguish between a bump on the surface of an object and a chip merely lying on the surface. Cohesion depends on a primitive relation bond between particles of material, paralleling the role of adj in regions. The relation attached is defined as the transitive closure of bond. A topology of cohesion is built up in a manner analogous to the topology of regions. In addition, we have encoded the relation that bond bears to motion, i.e., that bonded bits remain adjacent and that one moves when the other does, and the relation of bond to force, i.e, that there is a characteristic force that breaks a bond in a given material.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 246 Computational Linguistics Volume 13, Numbers 3-4, July-December 1987 </SectionTitle> <Paragraph position="0"> Jerry R. Hobbs et HI. Commonsense Metaphysics and Lexical Semantics Different materials react in different ways to forces of various strengths. Materials subjected to force exhibit or fail to exhibit several invariance properties, proposed by Hager (1985). If the material is shape-invariant with respect to a particular force, its shape remains the same.</Paragraph> <Paragraph position="1"> If it is topologically invariant, particles that are adjacent remain adjacent. Shape invariance implies topological invariance. If subjected to forces of a certain strength or degree dl, a material ceases being shape-invariant. At a force of strength d z >-- d~, it ceases being topologically invariant, and at a force of strength d 3 ~ d 2, it simply breaks. Metals exhibit the full range of possibilities, that is, 0 < d~ < d2 < d3 < oo. For forces of strength d < d~, the material is &quot;hard&quot;; for forces of strength d where d~ < d < d2, it is &quot;flexible&quot;; for forces of strength d where d2 < d < d3, it is &quot;malleable&quot;. Words such as &quot;ductile&quot; and &quot;elastic&quot; can be defined in terms of this vocabulary, together with predicates about the geometry of the bit of material. Words such as &quot;brittle&quot; (dl = d2 = d3) and &quot;fluid&quot; (d2 = 0, d3 = oo) can also be defined in these terms. While we should not expect to be able to define various material terms, like &quot;metal&quot; and &quot;ceramic&quot;, we can certainly characterize many of their properties with this vocabulary.</Paragraph> <Paragraph position="2"> Because of its invariance properties, material interacts with containment and motion. The word &quot;clog&quot; illustrates this. The predicate clog is a three-place relation: x clogs y against the flow of z. It is the obstruction by x of z's motion through y, but with the selectional restriction that z must be something that can flow, such as a liquid, gas, or powder. If a rope is passing through a hole in a board, and a knot in the rope prevents it from going through, we do not say that the hole is clogged. On the other hand, there do not seem to be any selectional constraints on x. In particular, x can be identical with z: glue, sand, or molasses can clog a passageway against its own flow. We can speak of clogging where the obstruction of flow is not complete, but it must be thought of as &quot;nearly&quot; complete.</Paragraph> </Section> </Section> <Section position="8" start_page="0" end_page="0" type="metho"> <SectionTitle> 3.6 OTHER DOMAINS 3.6.1 CAUSAL CONNECTION </SectionTitle> <Paragraph position="0"> Attachment within materials is one variety of causal connection. In general, if two entities x and y are causally connected with respect to some behavior p of x, then whenever p happens to x, there is some corresponding behavior q that happens to y. In the case of attachment, p and q are both move. A particularly common kind of causal connection between two entities is one mediated by the motion of a third entity from one to the other. (This might be called a &quot;vector boson&quot; connection.) Photons mediating the connection between the sun and our eyes, raindrops connecting a state of the clouds with the wetness of our skin and clothes, a virus being transmitted from one person to another, and utterances passing between people are all examples of such causal connections. Barriers, openings, and penetration are all defined with respect to paths of causal connection.</Paragraph> </Section> <Section position="9" start_page="0" end_page="0" type="metho"> <SectionTitle> 3.6.2 FORCE </SectionTitle> <Paragraph position="0"> The concept of &quot;force&quot; is axiomatized, in a way consistent with Talmy's treatment (1985), in terms of the predications force(a,b,dO and resist(b,a,d2) ~ a forces against b with strength dl and b resists a's action with strength d2. We can infer motion from facts about relative strength. This treatment can also be specialized to Newtonian force, where we have not merely movement, but acceleration. In addition, in spaces in which orientation is defined, forces can have an orientation, and a version of the &quot;parallelogram of forces&quot; law can be encoded. Finally, force interacts with shape in ways characterized by words like &quot;stretch&quot;, &quot;compress&quot;, &quot;bend&quot;, &quot;twist&quot;, and &quot;shear&quot;.</Paragraph> </Section> <Section position="10" start_page="0" end_page="0" type="metho"> <SectionTitle> 3.6.3 SYSTEMS AND FUNCTIONALITY </SectionTitle> <Paragraph position="0"> An important concept is the notion of a &quot;system&quot;, which is a set of entities, a set of their properties, and a set of relations among them. A common kind of system is one in which the entities are events and conditions and the relations are causal and enabling relations. A mechanical device can be described as such a system in a sense, in terms of the plan it executes in its operation. The function of various parts and of conditions of those parts is then the role they play in this system, or plan.</Paragraph> <Paragraph position="1"> The intransitive sense of &quot;operate&quot;, as in The diesel was operating.</Paragraph> <Paragraph position="2"> involves systems and functionality. If an entity x operates, there must be a larger system s of which x is a part. The entity x itself is a system with parts. These parts undergo normative state changes, thereby causing x to undergo normative state changes, thereby causing x to produce an effect with a normative function in the larger system s. The concept of &quot;normative&quot; is discussed below.</Paragraph> </Section> <Section position="11" start_page="0" end_page="0" type="metho"> <SectionTitle> 3.6.4 SHAPE </SectionTitle> <Paragraph position="0"> We have been approaching the problem of characterizing shape from a number of different angles. The classical treatment of shape is via the notion of &quot;similarity&quot; in Euclidean geometry, and in Hilbert's formal reconstruction of Euclidean geometry (Hilbert, 1902) the key primitive concept seems to be that of &quot;congruent angles&quot;. Therefore, we first sought to develop a theory of &quot;orientation&quot;. The shape of an object can then be characterized in terms of changes in orientation of a tangent as one moves about on the surface of the object, as is done in some vision research (e.g., Zahn and Roskies, 1972). In all of this, since &quot;shape&quot; can be used loosely and metaphorically, one question we are asking is whether some minimal, abstract structure can be found in which the notion of &quot;shape&quot; makes sense.</Paragraph> <Paragraph position="1"> Consider, for instance, a graph in which one scale is discrete, or even unordered. Accordingly, we have been examining a number of examples, asking when it seems right to say two structures have different shapes.</Paragraph> <Paragraph position="2"> We have also examined the interactions of shape and Computational Linguistics Volume 13, Numbers 3-4, July-December 1987 247 Jerry R. Hobbs et aideg Commonsense Metaphysics and Lexical Semantics functionality (see Davis, 1984). What seems to be crucial is how the shape of an obstacle constrains the motion of a substance or of an object of a particular shape (see Shoham, 1985). Thus, a funnel concentrates the flow of a liquid, and similarly, a wedge concentrates force. A box pushed against a ridge in the floor will topple, and a rotating wheel is a limiting case of continuous toppling.</Paragraph> <Paragraph position="3"> 3.7 HITTING, ABRASION, WEAR, AND RELATED CONCEPTS For x to hit y is for x to move into contact with y with some force.</Paragraph> <Paragraph position="4"> The basic scenario for an abrasive event is that there is an impinging bit of material m that hits an object 0 and by doing so removes a pointlike bit of material bo from the surface of o:</Paragraph> <Paragraph position="6"> A hit'(e3,m,bo) That is, e is an abrasive event of a material m impinging on a topologically invariant object 0 and detaching bo if and only if b 0 is a particle of the surface s of the bit of material material b of which 0 consists at the time t at which e occurs, and e is a change from the condition el of bo's being attached to b to the negation e 2 of that condition, where the change is caused by the hitting e 3 of m against b o.</Paragraph> <Paragraph position="7"> After the abrasive event, the pointlike bit bo is no longer a part of the object o: (Ve, m, o, b 0 , e t, e2, tz)abr- e vent'(e, m, o, b o) e2 holds at time tz, then b 0 is not part of the bit of material b 2 of which 0 consists at t 2. It is necessary to state this explicitly since objects and bits of material can be discontinuous.</Paragraph> <Paragraph position="8"> An abrasion is a large set of abrasive events widely distributed through some nonpointlike region on the surface of an object: That is, e is an abrasion by m of o if and only if there is a large set bs of bits of material and e is a set of abrasive events in which m impinges on o and removes a bit bo, an element in bs, from o, and if e occurs at time t and o consists of material b at time t, then there is a subregion r of the surface s of b over which bs is widely distributed. null Wear can result from a large collection of abrasive events distributed over time as well as space (so that there may be no instant at which enough abrasive events occur to count as an abrasion). Thus, the link between wear and abrasion is via the common notion of abrasive events, not via a definition of wear in terms of abrasion.</Paragraph> <Paragraph position="9"> That is, e is a wearing by x of o if and only if there is a large set bs of bits of material and e is a set of abrasive events in which m impinges on o and removes a bit bo, an element in bs, from o, and e is widely distributed over some time interval i.</Paragraph> <Paragraph position="10"> We have not yet characterized the concept &quot;large&quot;, but we anticipate that it would be similar to &quot;high&quot;. The concept &quot;widely distributed&quot; concerns systems. If x is distributed in y, then y is a system and x is a set of entities which are located at components of y. For the distribution to be wide, most of the elements of a partition of y, determined independently of the distribution, must contain components which have elements of x at them.</Paragraph> <Paragraph position="11"> The word &quot;wear&quot; is one of a large class of other events involving cumulative, gradual loss of material w events described by words like &quot;chip&quot;, &quot;corrode&quot;, &quot;file&quot;, &quot;erode&quot;, &quot;sand&quot;, &quot;grind&quot;, &quot;weather&quot;, &quot;rust&quot;, &quot;tarnish&quot;, &quot;eat away&quot;, &quot;rot&quot;, and &quot;decay&quot;. All of these lexical items can now be defined as variations on the definition of &quot;wear&quot;, since we have built up the axiomatizations underlying &quot;wear&quot;. We are now in a position to characterize the entire class. We will illustrate this by defining two different types of variants of &quot;wear&quot;--&quot;chip&quot; and &quot;corrode&quot;. &quot;Chip&quot; differs from &quot;wear&quot; in three ways: the bit of material removed in one abrasive event is larger (it need not be point-like), it need not happen because of a material hitting against the object, and &quot;chip&quot; does not require (though it does permit) a large collection of such events: one can say that some object is chipped even if there is one chip in it. Thus, we slightly alter the definition of abr-event to accommodate these changes:</Paragraph> <Paragraph position="13"> That is, e is a chipping event by a material m of a bit of material bo from an object o if and only if bo is a part of the surface s of the bit of material material b of which o consists at the time t at which e occurs, and e is a change from the condition el of bo's being attached to b to the negation e2 of that condition.</Paragraph> <Paragraph position="14"> &quot;Corrode&quot; differs from &quot;wear&quot; in that the bit of material is chemically transformed as well as being detached by the contact event; in fact, in some way the chemical transformation causes the detachment. This can be captured by adding a condition to the abrasive event that renders it a (single) corrode event:</Paragraph> <Paragraph position="16"> That is, e is a corrosive event by a fluid m of a bit of material bo with which it is in contact if and only if b 0 is a particle of the surface s of the bit of material b of which 0 consists at the time t at which e occurs, and e is a change from the condition e~ of bo's being attached to b to the negation e 2 of that condition, where the change is caused by a chemical reaction e 3 of m with bo.</Paragraph> <Paragraph position="17"> &quot;Corrode&quot; itself may be defined in a parallel fashion to &quot;wear&quot;, by substituting corrode-event for abr-event. All of this suggests the generalization that abrasive events, chipping and corrode events all detach the bit in question, and that we may describe all of these as detaching events. We can then generalize the above axiom about abrasive events that result in loss of material to the following axiom about detaching:</Paragraph> <Paragraph position="19"> That is, if e is a detaching event by m of b 0 from o, and e is a change from e I to e 2, and e 2 holds at time t2, then bo is not part of the bit of material b 2 of which o consists at t2.</Paragraph> </Section> <Section position="12" start_page="0" end_page="0" type="metho"> <SectionTitle> 4 RELEVANCE AND THE NORMATIVE </SectionTitle> <Paragraph position="0"> Many of the concepts we are investigating have driven us inexorably to the problems of what is meant by &quot;relevant&quot; and by &quot;normative&quot;. We do not pretend to have solved these problems. But for each of these concepts we do have the beginnings of an account that can play a role in analysis, if not yet in implementation.</Paragraph> <Paragraph position="1"> Our view of relevance, briefly stated, is that something is relevant to some goal if it is a part of a plan to achieve that goal. (A formal treatment of a similar view is given in Davies, forthcoming.) We can illustrate this with an example involving the word &quot;sample&quot;. If a bit of material x is a sample of another bit of material y, then x is a part of y, and moreover, there are relevant properties p and q such that it is believed that ifp is true ofx then q is true ofy. That is, looking at the properties of the sample tells us something important about the properties of the whole. Frequently, p and q are the same property. In our target texts, the following sentence occurs: We retained an oil sample for future inspection.</Paragraph> <Paragraph position="2"> The oil in the sample is a part of the total lube oil in the lube oil system, and it is believed that a property of the sample, such as &quot;contaminated with metal particles&quot;, will be true of all the lube oil as well, and that this will provide information about possible wear on the bearings. It is therefore relevant to the goal of maintaining the machinery in good working order.</Paragraph> <Paragraph position="3"> We have arrived at the following provisional account of what it means to be &quot;normative&quot;. For an entity to exhibit a normative condition or behavior, it must first of all be a component of a larger system. This system has structure in the form of relations among its components. A pattern is a property of the system, namely, the property of a subset of these stuctural relations holding.</Paragraph> <Paragraph position="4"> A norm is a pattern established either by conventional stipulation or by statistical regularity. An entity behaves in a normative fashion if it is a component of a system and instantiates a norm within that system. The word &quot;operate&quot;, discussed in Section 3.6.3, illustrates this. When we say that an engine is operating, we have in mind a larger system -- i.e., the device the engine drives -- to which the engine may bear various possible relations. A subset of these relations is stipulated to be the norm -- the way it is supposed to work. We say it is operating when it is instantiating this norm.</Paragraph> </Section> class="xml-element"></Paper>