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Home , Context (language use), Syntax

From: AAAI Technical Report FS-95-02. Compilation copyright © 1995, AAAI (www.aaai.org). All rights reserved.

Syntax, , and of Contexts

John F. Sowa andComputers and State University of NewYork at Binghamton Abstract. The notion of context is indispensable in discussions of meaning, but the context has often been used in conflicting senses. In , the first representation of context as a formal was by the philosopher C. S. Peirce; but for nearly eighty years, his treatment was unknownoutside a small group of Peirce aficionados. In the early 1980s, three newtheories included related notions of context: Kamp’sdiscourse representation theory; Barwiseand Perry’s ; and Sowa’sconceptual graphs, which explicitly introduced Peirce’s approach to the AI community. More recently, John McCarthyand his students have begunto use a closely related notion of context as a basis for organizing and partitioning knowledgebases. Each of the theories has distinctive, but complementaryideas that can enrich the others, but the relationships betweenthem are far from clear. This paper analyzes the semantic foundations of these theories and showshow McCarthy’s ist(c,p) can be interpreted in terms of the semantic notions underlying the others.

1. Theoriesof Contexts In the AI literature, the term context has been Someof the confusionabout contexts results applied to a profusion of ideas that have not been from an in the English word. Die- clearly distingafished. Someof them concern the tionaries list two major senses of the word syntactic representation of contexts; others refer context:. to the semanticrelationship of a linguistic context ¯ The basic meaningis a section of the linguis- to a physical situation; and still others introduce tic text or that surrounds some pragmatic notions concerning the purpose or use wordor of interest. of a context in various applications. Each of these major areas can be subdivided further. ¯ The derived meaz~gis a nonlingnistic situ- Syntactically, there are three distinct aspects of ation, environment, domain, setting, back- context: ground, or milieu that includes someentity, , or topic of interest. 1. A mechanismfor grouping, associating, or packaging that can be named Thesetwo informal senses suggest intuitive crite- and referencedas a single unit. ria for distinguishing the various functions of contexts: 2. The contents of that package, which have ¯ Syntax. The syntacticfunction of context is been called anything from quoted formula to to group, delimit, or package "a section of microtheory. linguistic text." Formally, a context behaves 3. The permissible operations on the informa- like the QUOTEoperator in Lisp together tion in the package and the constraints on with the parentheses that delimit the portion importing and exporting information into of text that is quoted. and out of a package. ¯ Semantics. The quoted text of a context re- All three of these notions represent syntactic fers to something, which may be a physical mechani.mls for representing and manipulating entity or situation, a mathematical con- logical formulas without any consideration of struction, or someother expression in a na- their relationship to the real world, a possible tural or artificial . world, or somemodel of the world. Muchof the ¯ Pragmatics. The word interest, which occurs controversy about contexts results from the lack in both senses of the English definition, sug- of a formalsemantics that relates these operations gests somereason or purpose for distinguish- to a Tarski- model. Even an informal se- ing "a section of linguistic text" or "a mantics that displays the intuitive meaning of nonlinguistie situation." That purpose con- contexts in terms of real-world objects and situ- stitutes the pragmatiesor the reason whythe ations would be helpful as a guide to further text is being quoted. In Lisp, the QUOTE and formaliTation. operator blocks the execution of the standard Lisp interpreter to allow nonstandard oper-

85 ations to be performed for someother pur- ated with c. The predicate ~s-in represents the pose. In logic, a quote blocks the standard syntactic relationship of c to p; and the predicates rules of inference and allows the definition of refers-to and describes represent the semanticre- newrules for somespecial purpose. lationships of c and p to the external entity x. As this analysis indicates, the notion of context McCarthy, Guha, and Buva~ have primarily is intimately connectedwith a complexof related considered the syntactic operations associated ideas. Muchof the confusion results from which with the ~s-in componentof the ist predicate. To of them happens to be called a context:, some justify those operations, the semantics of the people apply the word to the package; and others refers-to and describes componentsmust also be to the information contained in the package, to addressed. the thing that the information is about, or to the Muchof the controversy about contexts re- possible uses of either the information or the suits from the abundanceof notation and termi- thing. The ideas themselves may be compatible, nologyin different theories, their application to but they must be carefully distinguished and diverse phenomena, and the lack of commlmi- sorted out. cation betweenthe different schools of thought. These intuitive criteria provide a basis for The purpose of this paper is to emphasize the analyzing John McCarthy’s (1993) "Notes underlying similarities and to promote cross- Formalizing Context" and relating the ideas to fertilization of ideas. Thefollowing five theories the other theories. McCarthy’sbasic notation is will be considered: the predicate ist(c,p), which may be read "the 1. Charles Sanders Peirce (1885) invented the propositionp is true in context c." In his disser- modernalgebraic notation for predicate cal- tation written under McCarthy’sdirection, R. V. culus; but a dozen years later, he developed Guha (1991) applied McCarthy’sapproach to the an alternate notation, which he called problem of partitioning a large, monolithic existential graphs (Roberts 1973). Although knowledgebase into a collection of smaller, more Peirce’s algebra and graphs had equivalent modular microtheories. Guha implemented the expressive power,the graphic structure served microtheories in the Cyc system (I.gnat & Guha as a heuristic aid that led himto explore op- 1990), in which they have becomea fundamental erations and applications that were over- mechanism for organizing and stnmturing a looked by logicians used only the knowledge base. McCarthy and Buva~ (1994) algebraic notation. In particular, Peirce’s have also applied contexts and the /st predicate graphic notation for contexts was isomorphic to the analysis and representation of natural lan- to the discourse representation structures guagediscourse. (DRSs) invented by Hans Kampeighty years Although McCarthy, Guha, and Buva~ have later. His rules of inference were based on shownthat the /st predicate can be a powerful operations of iterating and deiterating infor- tool for building knowledgebases and analyzing mation to and from contexts in a way that discourse, they have not clearly distinguished the resembles John McCarthy’ rules. syntax of contexts and from their 2. Hans Kamp(1981) developed discourse rep- semantic relationship to some domain of dis- resentation theory (DRT)to express the log- course. In fact, the ist predicate itself mixesthe ical constraints on anaphofic in syntactic notion of containment(~s-in) with the . Because of the difficulty semanticnotion of truth (/s-true-of). To clarify of expressing those constraints in the alge- these relationships, it maybe helpful to analyze braic notation for logic, Kampintroduced the the ist predicate as a of three more graphic DRSnotation, which allowed a tim- primitive predicates, ~s-in, refers-to, and pler formulation of his rules. Si~iBcantly, describes: the nested contexts in Kamp’s DRSs are ist(c,p)-- (~3x:Entity)(is-in(c,p) isomorphicto the nestof contexts in Peirce’s refers-to(c,x)^ describes(x,p)). EGs, even though Kamp had no previous knowledgeof them. Kampdeserves credit for Accordingto this analysis, the p is discovering the constraints on in true in context e ff and only if there exists some DRT, but DRSsand EGs are equally suit- entity x such that p is in c, c refers to x, and p able for expressingthose constraints. describes x. The formula distinguishes the ab- stract context c from somenonlingulstic entity x, 3. Jon Barwise and John Perry (1983) devel- which represents the "situation, environment, oped situation semantics as a theory of domain, setting, background, or milieu" assoei- meaning in natural langq_~age. Unlike

86 Montague’s approach (1975), which related 5. John McCarthyis one of the founding fathers the semanticsof languageto potentially infi- of AI, whose collected work (McCarthy nite models of the real world or possible 1990) has frequently inspired and sometimes worlds, Barwiseand Perry adopted finite sit- revolutionized the application of logic to uations as their basis. Each situation is a knowledge representation. His work on bounded region of space-time containing context, although published later than the physical objects and processes, as well as previous four approaches, has grownout of other situations. A great deal of research has ideas based on his earlier work. McCarthy’s been done within the paradigm of situation /st predicate is the key to relating that work semantics(Barwise et al. 1991), including ef- to the ongoing research in the other forts to mergeit or at least reconcile it with paradisnns. If the/st predicate can be defined DRT (Cooper & Kamp 1991). An impor- in termsof the other theories, then any results tant is how and whether it can be obtained in one approach can be translated merged or reconciled with McCarthy’s con- to any of the others. Besides defining con- texts as well. texts, McCarthy has been emphasizing his John Sowa (1984) developed conceptual lifting rules for importing and exporting in- . formation into and out of the quoted text or graphs as a system of logic and reaching based on the semantic networks of AI and package. Such rules, which resemble Peirce’s the existential graphs of C. S. Peirce. The rules of iteration and deiteration, are essential nodes called concepts correspond to typed, for allowing quoted information to be un- quantified variables in a sorted predicate cal- quoted and used. caius. A context is a defined as a concept of Besidesthese five theories, there is a long history type Proposition, whosereferent field con- of related ideas in logic, philosophy,, tains one or more conceptual graphs that and AI. The most important ones for a theory state the proposition. Later papers (Sowa of context include indexicals, possible worlds, Way1986; Sowa1991) used the CGcontexts metalanguage, revision, and ontology re- to represent Kamp’s DRSsand Barwise and vision. Yet the various ideas and theories were Perry’s situations. In a paper on "Crystalliz- developed by people with different intuitions, ing Theories out of KnowledgeSoup," Sowa which they applied to different problems of (1990) proposed the use of contexts for par- knowledgerepresentation. Trying to unify and titioning a knowledgebase into a collection clarify those intuitions by defining the terms of of smaller "chunks" that could be assembled one theory in those of another runs the risk of into theories appropriate to any particular distorting the insights of both. This paper will application. In his dissertation, Guha(1991) explore the implications of these definitions to cited the knowledge soup paper, which he determine whether the benefits of unification and said was "in the same spirit as the workde- clarification outweighany possible distortions of scribed in this document." the original in~ghts.

2. Peirce’s Contexts First-order predicate calculus was independently commonalgebraic notation for logic later aban- invented by Gottlob Frege (1879) and Charles doned it for a graph representation, which he SandersPeirce (1885). Frege used a tree notation, called his "chef d’oeuvre" and "the luckiest find which no one else ever adopted. But Peirce de- of mycareer." With the existentialgraphs that he veloped an algebraic notation, which through the invented in 1897, Peiree developed an aspect of textbook by Ernst Schr6der (1890) and with logic that waslargely ignored by the mathematical change of symbols by C-iuseppe Peano became logicians of the twentieth century. In relating the modemsystem of predicate calculus. Long Peirce’s later logic and philosophy to situation before Bertrand Russell learned logic from Frege semantics, Burke (1991) said "Peiree anticipated and Peano, it had becomea flourishing subject in his own waysome of the concerns of situation based on the Peirce-Schr6der foundations. theory (or rather, he happenedto be workingbe- The early history of modemlogic is a fasci- fore it went out of fashion to wrestle with such nating tale that has been recounted by Roberts concerns)." A century later, those concerns are (1973) and Houser et al. (1995). The main point back in fashion, and Peirce is once again in the for this paper is that the man whoinvented the avant garde of modemlogic.

87 The three primitives of existential graphs qualifiers is of the graphs nested inside. (EGs) include the ova/enclosure, which delimits Existence, conjunction, and negation provide a a context, the line of identity, whichcorresponds completerepresentation for all of first-order logic. to an existentially quantified variable, and As an example, the middle diagram in Figure 1 juxtaposition, which represents conjunction. The is an existential graphfor the If a farmer default interpretation of an oval with no other ownsa donkey, then he beats it.

DbcounmRepresentc~n Structure Pmflffoned Semcm~Network

Figure1. Threerepresentations for "If a farmerowns a donkey,then he beatsit."

The EG in Figure 1 has two ovals, which The variables x and y in the antecedent box have represent . It also has two lines of implicit existential quantifiers; Kampdefined the identity, represented as linked bars: one line, scoping rules for the DRSto include consequent whichconnects farmer to the left side of ownsand box within the of the antecedent. As in beats, represents an existentially quantified vari- existential graphs, conjunctionis implicitly shown able (3x); the other line, which connects donkey by juxtaposition. Altogether, the DRSmay be to the right side of ownsand beats represents an- read If there exists a farmer x and a donkeyy and other variable (3y). WhenFigure 1 is translated x ownsy, then x beats y. to the algebraic notation, farmer and donkey map Although the DRSand EG notations look to monadic predicates; owns and beats map to quite different, they are exactly isomorphic:they dyadic predicates. The implicit conjunctions can have the same three primitives and exactly the be represented with Peano’s symbol ^: samescoping rules for variables or lines of iden- ~(~) (qy) (farmer(x) donkey(y) ^ tity. Whatmakes this coincidence remarkable is owns(x,y) ^ ~beats(x,y)). that in the dozens of notations for semantic net- Peirce called a nest of two ovals, as in Figure 1, works in the 1960s and 1970s, no one else redis- a scroll, which he used to represent material im- covered Peirce’s conventions. The notation that plication, since "(p^~q) is equivalent to p=q. comesclosest is the partitioned semantic network Using the ~ symbol, the above formula may be by Gary Hendrix (1975), which is illustrated rewritten as the rightmost diagram of Figure 1. l.ike Peirce and Kamp, Hendrix took the existential (Vx)(Vy)((farmer(x) donkey(y) ^ as the default, represented conjunction owns(x,y)) ~ beats(x,y)). by juxtaposition, and used a graphic enclosure for The algebraic formula with the = symbolil- partitioning contexts. But unlike Peirce and lustrates a peculiar of logic in Kamp, Hendrix allowed overlapping contexts: with natural : in order to preserve the two overlapping boxes in Figure 1 represent scope, the implicit existential quantifiers in the the antecedent and the consequent of the impli- a farmer and a donkey must be movedto cation. the front of the formula and be tsanslated to uni- With Overlapping contexts, Hendrix had no versal quant~ers. This puzzling feature of logic need for scoping rules. Althoughthe farmer and has posed a problem for linguists. In his dis- donkey nodes each occurred only once, the over- course representation structures, Hans Kamp lap allowed them to occur ~multaneouslyin both (1981) resolved it by introducing a newsymbol contexts. Yet the overlapping contexts proved to for implication with different scoping rules. The be unwieldy: with more than three contexts, it diagram on the left of Figure 1 shows a DRSfor becameimpossible to draw partitioned nets on a the donkeysentence. The twoboxes connected by an arrowrepresent the Englishpair if-then~ plane. Furthermore, the nestingof clausesin

88 natural languages has a more direct mappingto cussion of these points, see Roberts (1973), Sow^ the Peirce-Kamp nested contexts than to (1984, 1993), and Houseret al. (1995). Hendrix’s overlapping contexts. Kamp’srules for In discussing modality, Peiree imagined the resolving anaphom in DRSs could be stated graphs drawn on "a book of separate sheets, equally well in terms of EGs, but not in terms of tacked together at points." The upper sheet re- overlapping contexts. presents "a universe of existing individuals," while Besides notation, Peirce defined rules of in- the other sheets "represent altogether different ference for EGs, which in manyrespects are the universes with which our discourse has to do." simplest and most elegant inference rules ever Graphs on those sheets mayrepresent "conceived devised for any version of logic. A typical theo- propositions whichare not realized." Peirce said rem that requires 43 steps to prove with Russell that a necessarily true proposition could be con- and Whitehead’srules of 1910 takes only 8 steps sidered as replicated on aU the sheets in the book, with Peirce’s rules of 1897. Peirce’s roles are a while a possible proposition might occur on only generalization of natural deduction, which one. In the algebraic notation, Peirce used the GerhardGentzen discovered 40 years later, l.ike symbolco as an index for "a state of things," to Gentzen, Peirce took the empty set as his only whichhe applied a universal quantifier for neces- axiom, but Peirce’s proofs are simpler than sity and an existential quantifier for possibility. Gentzen’s because the nesting of contexts elimi- With his interpretation of necessity as truth "un- nates the bookkeeping needed for making and der all circumstances," Peirce was following discharging assumptions -- the most error-prone Leibniz and anticipating Kripke. aspect of Gentzen’s system. For further dis-

Figure2. EGfor "Youcan lead a horseto water~but youcan’t makehim drink."

In 1906, Peirce introducedcolors or tinctures ~(3x)(3y)(3z)(person(x) horse~) ^ to represent modalities. Figure 2 shows one of water(z) ~(Oleads-to(x,y,z) ^ Peirce’s examples,but with shadinginstead of the ~Omakes-drink(x,y,z) ) original red for possibility. The graph contains With the symbol = for implication, this formula four ovals: the outer two are associated to form becomes a scroll for if-then; the inner two represent possi- bility (shading) and impossibility (shading inside (Vx)(¥y)(Vz) ((person(x) ^ a negation). The outer oval maybe read If there water(z))(Oleads-to(x,y,z) ^ exists a person, a horse, and water;, the next oval ~ ~makes-drink(x,y,z) ). maybe read then it is possible for the person to This version maybe read For all x, y, and z, if x lead the horse to the waterand not possible for the is a person,y is a horse, andz is water, then it is person to makethe horse drink the water. possible for x to lead y to z, and not possible for The notation leads to represents the triadic x to makey drink z. predicate leads-to(’x~v,z), and _makes_drink_re- As a systematic wayof representing the kinds presents makes-drink(x~v,z).In the algebraic no- of contexts, Peirce adopted the traditional tation with 0 for possibility, Figure 2 mapsto heraldic tinctures, whichwere classified as metal, the following formula: color, or fur. He applied that three-way dis- tinct.ion to actual, modal, and intentional con- texts:

89 1. Metal: argent, or, fer, and plomb. Peirce usual cases in whichsuch precision is re- used argent (white background)for "the ac- quired, is denoted either by using tual or true in a general or ordinary sense," modifications of the heraldic tinctures, and the other metallic tinctures for "the ac- markedin something like the usual man- tual or true in somespecial sense." A state- ner in pale ink uponthe surface, or by ment about the physical world, for example, scribing the graphs in colored inks. wouldbe actual in an ordinary sense. Peirce By 1906, based on also considered mathematical idealizations, Peirce’s algebraic notation had becomea flour- such as Cantor’s hierarchy of infinite sets, to ishing field of research, and his graphs were ig- be "actual," but not in the samesense as or- nored. There were several reasons for the neglect: dinaryphysical entities. the notation and terminology were unfamiliur; 2. Color: azure, gules, vert, and purpure. Peirce most logicians, whohad a strong background in distinguished four basic modalities: azure for mathematics, had already foundthe algebraic logical possibility (dark blue) and subjective notation congenial to their tastes; and si~if- possibility (light blue); gules for objective icantly, Peirce’s novel applications of his graph possibility; vert for "what is in the logic to modality, intentionality, and mood";and purpure for "free- metalangn,~ were outside the main interests of domor ability." Each of these modalities the logicians of his time. Today, however, could be combinedwith negation: he defined Peirce’s contributions are central to research on necessaryin the usual wayas not-possibly-not contexts: and obligatory as not-freedom-not (an antic- 1. Representation of contexts by nests of enclo- ipation of deontic logic). sures, which separate or partition groups of 3. Fur: sable, ermine, vair, and potent. The four propositions of different modalstatus. furs correspond to propositional attitudes: 2. First-order logic based on three operators: sab/e for "the metaphysically, or rationally, existence (line of identity), conjunction or secondarily necessitated"; erm/ne for pur- (juxtaposition), and negation (oval enclosure pose or intention; vair for "the on a white background). commarlded"; and potent for "the compelled." 3. Sound and complete rules of inference for first-order logic basedon operations of draw- Peirce’s three-wayclassification is highly sugges- ing or erasing graphs and importing or ex- tive, but incomplete. He wrote that the complete porting graphs into and out of contexts. classification of "all the conceptionsof logic" was "a labor for generations of analysts, not for 4. Tinctures for distingmishing the purpose or one." But throughout his analyses, he clearly "nature" of a context from its logical opera- distinguished the logical operators represented by tors (for which he used only the basic three the graphs, from the tinctures, which, he said, do -- existence, conjunction, and negation). not represent 5. A three-way classification of the use of con- differences of the predicates, or texts for representing actuality (metal), significations of the graphs, but of the modality(color), or intentionality (fur). predetermined objects to which the 6. The use of graphs as a metalanguage for graphs are intended to refer. Conse- talking about graphs. quently, the Iconic idea of the System 7. Completestatement of the rules of inference requires that they should be represented, for existential graphs in existential graphs not by differentiatiom of the Graphs themselves. themselves but by appropriate visible characters of the surfaces upon whichthe Peirce’s later , although fragmentary,in- Graphs are marked. complete, and mostly unpublished, are no more fragmentary and incomplete than many modem It seemsthat Pcirce did not consider the tinctures publications about contexts. Although (or per- to be part of logic itself, but of the metalangua~ haps because) he did not use the word context, for describing howthe logic applies to the uni- Peiree was moreconsistent in distinguishing the verse of discourse: syntax (oval enclosures), the semantics("the uni- The nature of the universe or universes verse or universes of discourse"), and the prag- of discourse (for several maybe referred matics (the tinctures that "denote" the "nature" to in a single assertion) in the rather un- of those universes).

9O 3. Contexts in ConceptualGraphs

Conceptualgraphs are extensionsof existential ¯ The arrows or numbers on the arcs of con- graphs with new features based on the semantic ceptual relations distinguish the arguments networks of AI and the linguistic research on more clearly than Peirce’s unlabeled lines. thematic roles and generalized quantifiers. The For dyadic relations, the arrow pointing to- primary difference is in the treatment of lines of wards the circle is the first , and the identity. In existential graphs, the lines serve two arrow pointing away is the second argument. different purposes: they represent existential For relations with more than two arguments, quantifiers, and they show how the arguments are the arcs are numbered l, 2 ..... n; the arrow connected to the relations. In conceptual graphs, on the n-th arc points away from the circle, those two functions are split: boxes called con- and the other arcs point towards the circle. cepts contain the quantifiers, and arcs marked ¯ In an existential graph, any point on a line with arrows show the connections of arguments of identity could be considered as a separate to circles called conceptual relations. This sepa- quantified variable. Conceptual graphs con- ration of functions has several important conse- centrate the point of quantification in the quences: concepts rather than the lines. A blank ¯ Conceptshave a placeto representa type la- field represents the quantifier 3, but bel for each quantifier.Conceptual graphs the universal quantifier V or other generalized thereforecorrespond to a typedor sorted quantifiers mayalso occur in the referent field logic,unlike the untyped existential graphs. of a concept. ¯ Conceptsmay also containa or other ¯ Peirce’s lines of identity could cross context specification of the referent of the concept, boundaries, but a concept may only occur in as in [Cat: Yojo], where the type is Cat and a single context. Whentwo concepts in dif- the particular individual is named Yojo. The ferent contexts refer to the same individual, area on the left of the colon is called the type they must be associated by labels field, and the area on the right is called the or by a dotted line called a coreference link. referent field. To illu~t~ette these features, Figure 3 shows two ¯ When an existential graph or an tmtyped equivalent conceptual graphs for the donkey sen- formula in predicate calculus is mapped to a tence. The CGon the left uses the basic notation conceptual graph, the type label r may be with the -’ symbol to mark negation and with used to mark the universal type, which is a dotted lines for coreference links. The CG on the supertype of all others. The type T imposes right uses an extended notation with the types If no restrictions on the quantifier or referent. and Then defined as negated propositions and The concept [T: Yojo], for example, would with coreference shown by the labels x and y. represent an entity named Yojo whose type The concepts marked *x and *y are the defining was unknown or unspecified. nodes with implicit existential quanfifiers, and the concepts marked ?x and ?y are bound nodes within the scope of *x and *y.

Figure 3. Twoconceptual graphs for "If a farmerowns a donkey, then he beats it."

The scoping rules for the CG-s in Figure 3 are Figure 1. The CGs may be read If a farmer x the same as the rules for the DRS and EG in owns a donkey y, then x beats y. The circles rep-

91 resent dyadic conceptual relations, where the ar- the thematic roles or case relations used in lin- row pointing towards the circle marks the first guistics: experiencer (EXPR); theme (THME); argument and the arrow pointing awaymarks the agent (AGNT); and patient (PTNT). For second argument. The two CGs in Figure 3 venience, there is also a linear notation that wouldcorrespond to the foUowingformula in al- makesCGs easier to type: gebraic notation: 7[ [Farmer:*x]÷(EXPR)÷[Own]÷(THME)÷[Donkey: ~(3X:Farmer) (~y:Oonkey) (3z:Own) (expr(z,x) 7[ [?x]÷(AGNT)÷[Beat]÷(PTNT)÷[?y] ^ thme(z,y)A ~(3w:Beat) (agnt(w,x) [If: [Farmer:*x]÷(EXPR)÷[Own]÷(THME)÷[Donkey: ^ ptnt(w,y)) [Then:[?x]÷(AGNT)÷[Beat]÷(PTNT)-~[?y] This representation follows C. S. Peirce and In the graphic form of Figure 3, coreference may DonaldDavidson in reifying with the event be shownby dotted lines, but coreference labels variables z and w. The dyadic relations represent must be used in the linear notation.

+ ]Tln~. 19:29:32 GMr]--~~ InteNal: @13 .~ ~11me: 19.’29:45 GMT]

Figure 4. CG for "A cat chased a mouse for an interval of 13 seconds from 19:29".32 GMTto 19:29:45 GMT."

In CGs, a context is defined as a concept In mostapplications, the abbreviated form shown whosereferent field contains nested conceptual in Figure 4 is used.When necessary, the type la- graphs. Since every context is also a concept, it bel of the context determines how the nested can have a type label, coreference links, and at- graph is interpreted. This use of the type label tached conceptual relations. In Figure 4, the corresponds to type in programming graph for a cat chasing a mouseis nested inside lang~_m~s. a concept of type Situation. The conceptual To relate Figure 4 to McCarthy’sist predi- graph in the inner context describes the situation. cate, first expandthe abbreviations;then translate Attached to that context is the relation DURfor the expandedgraph to predicate calculus: duration, which is linked to a concept for an in- terval of 13 seconds. The relations FROMand (Bs:S i tuation) (Bi: Interval TOshow that the interval lasted from 19:29:32 (duration(s,/]a measure(i,13sec) GMTto 19:29:45 GMT. from(i,lg:2g:32GMT) ^ to(i, lg:2g:45GMT) ^ dscr(s, Whena conceptual graph occurs in a context (Rz:Cat)(By:Mouse) (Bz:Chase) of type Graph,it is used as a literal; in a context (agnt(z,x)A thme(z,y)) of type Proposition, it states a proposition; in a context of type Situation,it describes a situation. Nowit is possible to relate parts of this formula The following concept, by itself, may be read A to the earlier definition of the/st predicate: situation of a cat chasinga mouse. ist(e,p)_--- (~Ix:Entity)(is-in(c,p) refers-to(c,x)^ describes(x,p)). [Situation:[Cat] ÷(AGNT) ÷[Chase]÷ (THME) ÷ [Mouse] J. Theentity x canbe identified with the situation Thisgraph maybe considered an abbreviation for the following graph, whichsays that there exists s, and the proposition p with the proposition that a situationwhose description (DSCR) is the a cat is chasing a mouse.But there is no entity proposition that a cat is chasing a mouse: in this that can be identified with McCarthy’scontext c. In McCarthy’ssense, c is [Situati on]÷ (DSCR) ÷ [Propo siti on: supposedto be larger than the single proposition. [Cat]÷(AGNT) ÷[Chase] ÷ (THME)÷ [Mouse] In fact, the context maybe so rich that no finite set of formulas could exhaust its full content.

92 This discussion raises a question about the dis- been central to situation semantics, which may tinction betweena total description of everything help to clarify the semantics of both CGcontexts that is knowableabout a situation and a partial and McCarthy’scontexts. description in a single formula. That issue has

4. Situations and Propositions Barwiseand Perry (1983) developedsituation se- As an illustration of the waysituations are manticsas a reaction again~ the potentially infi- related to partial descriptions, Figure 5 showsa nite models of Kripke’s and Montague’s modal concept of type Situation, which is linked to two and intensional . Each situation is a finite images and a description. The image relation configuration of some aspect of the world in a (IMAG)links the situation to two different kinds limited region of space and time. It may be a of imagesof that situation: a picture and the as- static configuration that remains unchangedfor a soeiated sound. The description relation (DSCR) period of time, or it may include processes and links it to a proposition that desc~bes someas- events that are causing changes. It mayinclude pect of the situation, which is linked by the people and things with their actions and thoughts; statement relation (STMT)to three different it maybe real or imaginary; and its time maybe statements of the proposition in three different present, past, or future. A situation maybe as languages: an E~gllsh sentence, a conceptual large as the solar system or as small as an atom, graph, and a formula in the KnowledgeRepre- and it maycontain nested situations for smaller sentation Lang~ (KIF) by Genesereth and or moredetailed aspects of the world. Fikes (1992).

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"Aplumber Is c~nflnga pipe." (exists((’?x Idmber) (?y=my) (?z " (==1(=a=t ~ ?d(tl~ ~ .’M)

Graph:

Figure5. A situation of a plumbercarrying a pipe

A proposition may be defined as an equiv- proposition. To avoid complexities, the equiv- alence class of statements in one or more lan- alence mappingwill be defined first for formal guages. No concrete statement in any specific languages like KIF and CGs. Letfbe a mapping language /s the proposition. Instead, a concept from language M1to language ~2 that defines the of type Proposition, such as the one in Figure 5, equivalence;let s be any statement in "~1; and let represents a class of intertranslatable statements t =fl-l(](s)) be the result of mappings 2 and in one or more concrete languages. Each state- back again. Thenfmustobey the following three ment in that class is said to express the same constraints:

93 1. Truth preserving: The statements s and t closest to what McCarthycalls a context is the must be provably equivalent according to the set of all infons entailed by a situation: {o [ rules of inference of language~1- s N o}. If the DSCRrelation is interpreted as 2. preserving: Exactly the same entailment of a partial description, McCarthy’s nonlogical symbols must appear in both s context c may be considered a complete de- and t. scription of a situation: 3. Negation preserving: When s and t axe completeDscr(s,c) = c= {a [ s N ~r}. mapped to Peirce Normal Form (with ne- For Figure 5, the context c would include - gation, conjunction, and the existential positions or infons saying that the plumber is quantifier as the only logical operators), they carrying a toolbox, he works for AcmePlumbing contain exactly the same numberof negation Co., he is draggingthe pipe with a clankety noise, operators. etc. Thepredicate is-in(c,p) could then be inter- These three conditions ensure that the statements preted as provability. s and t axe highly similar, if not identical. Al- With this analysis, the following formula re- though Fermat’s Last Theoremand 2 + 2 = 4 are presents an interpretation of the /st predicate in both true, the proof is far from trivial, and the terms of situations and infons (or propositions): two statements require different . Condition #2 allows variables to be renamedand i st(c,p)- (3s:Situation)(provable(c,p) permits pvq to be replaced by ~(~p^"q); but completeDscr(s,¢)^ dscr(s,p)). ensures that a statement like Thesyntactic predicate is-in(c,p) is interpreted as (Vx)(dog(x)=dog(x)) does not get mapped to provable(c,p). The two semantic predicates are (Vx)(cat(x)~cat(x)). Condition #3 allows pap to definedin termsof entailment:refers-to(c,s) be mappedto p, but it prevents ~~p from being completeDscr(s,c); and dser(s,p) is s ~p. In mappedto p. Generalizing this definition to na- pure first-order framework, s ~p would be re- tural languages should be possible, but that dundant, since it would be implied by c [--p and would raise further issues beyond the scope of the definition of c. Including it, however,could this paper. accommodate nonstandard logics in which As Figure 5 illustrates, the proposition stated provability is not equivalent to entailment. by any of the three statements represents a tiny The previous definition was presented at the fraction of the total information available. Both IJCAI’95 workshop on context. In the dis- the sound image and the picture image capture eussions, McCarthymade the point that identi- information that is not in the sentence, but even fying a context c with the set of all entailments they are only partial representations. A picture of a situation s does not completely capture his may be worth a thousand , but a situation intuitions about context. As aa example, sup- can be worth a thousand pictures. To relate a pose that John everything that Marybe- situation to the information it contains, Keith lieves. Theneither of their belief systems would Devlin (1991) introduced the term infon for an entail exactly the same propositions. But abstract informationobject that is closely related McCarthywould like to say that the context of to the previous definition of proposition. John’s beliefs is not the same as the context of Devlin’s basic formula is s ~ o, wheres is a Mary’sbeliefs. That distinction implies that no situation and o is an infon that is semantically purely extensional definition can capture the full entailed by s. Devlin’s constructionthat comes import of McCarthy’snotion of context.

5. Extensions,, and Intentions The difficulty of giving a precise definition of shownthat a purely extensional definition of sit- McCarthy’scontexts is closely related to the dif- uation is inadequate, for manyof the same rea- ficulty of giving a precise definition of situations. sons that McCarthyobjected to an extensional In their 1983 book, Barwiseand Perry identified definition of context. The following examplesil- a situation with a boundedregion of space-time, lustrate the problems: and they tried to use that definition to support a ¯ A college lecture might be considered a situ- theory of verbs, such as ation boundedby a 50-minute time periodin believe, think, hope, fear, wish, want, and plan. a spatial region enclosed by the walls of a Yet further research on situation semantics has classroom. But if the time were movedfor-

94 ward by 30 minutes, the region wouldinclude Peirce’s notion of sign was broad enoughto in- the last half of one lecture and the first half clude situations, contexts, propositions or infons, of another. That time shift would create a and their expression in any language, including very unnatural "situation." English and logic. His notion ofgroundis crucial: ¯ An even more unnatural transformation it acknowledgesthat someagent’s purpose, in- wouldbe to shift the spatial region to the left tention, or "conception" is essential for deter- by half the width of a classroom. Then it miningthe scope of a situation or context. wouldinclude part of one class listening to Unlike the two-waydistinction of extensions one teacher, part of another class listening to and intensions, Peirce drew a three-way dis- a different teacher speaking on a different tinction in his basic categories of Firstness, topic, and a wall that separated the two dis- Secondness, and Thirdness: cussions. First is the conceptionof being or exist- ¯ Anotherpossible tranfformation is to fix the hag independent of anything else. Second coordinate system relative to the sun instead is the conceptionof being relative to, the of the earth. Thenthe space-time region that conception of reaction with, something included the class at the beonningof the lec- else. Third is the conception of medi- ture would quickly shift into deep space as ation, wherebya first and a second are the earth moved; by the end of the hour, it broughtinto relation. would contain nothing but an occasional hy- drogen atom. His three kinds of tinctured contexts, whichwere discussed in Section 2, are applications of these Theseexamples illustrate a fundamentaldifficulty categories: for any purely extensional definition. An arbi- trarily chosenspace-time region is useless for de- 1. Metal represents the "Actual" or what can fining a "situation" or "context." Instead, a be described in classical first-order logic useful or "meaningful" choice of region depends without modalities or intentionality. critically on someagent’s purpose or intention. 2. Colorrepresents the varieties of possibilities, But if purposeor intention is critical to the deft- objective, subjective, and deontic. This topic nition of situation or context, then the explication has been thoroughly explored in the modem of propositional attitudes in terms of situations modal and intensional logics, whoseseman- becomescircu,!~: the intentionality that is ex- tics have been defined in terms of infinite plained by the theory depends on the families of possible worlds. intentionality that went into the choice of situ- ation. 3. Fur represents the intentionality (with a instead of an S) whereby an agent selects Peirce considered intentionality as funda- somecontext for the representation of some mental to his theory of signs or . Al- object for somepurpose. Intentionality, in though he did not use the word context, he would Peirce’s terms, is the mediating Thirdness have defined a context as a kind of sign, to which that determines the context for directing at- he wouldapply his basic definition: tention to someobject for somepurpose. A sign, or representamen, is something Peirce’s third category has never been ad~uately which stands to somebodyfor something studied or represented in any of the modern in somerespect or capacity. It addresses modaland intensional logics. It is central to the somebody,that is, creates in the mindof issues that McCarthy, Baxwise and Perry, and that person an equivalent sign, or per- others have been trying to capture in their theo- haps a more developed sign. That sign des. But those theories cannot be completed whichit creates I call the interpretant of without representing intentionality with a T. the first sign. The sign stands for some- Peirce never completed his theory of thing, its object. It stands for that object, intentionality, but at least he madea good begin- not in all respects, but in to a ning. As he said, the complete classification of sort of idea, which I have sometimes all the conceptionsis "a labor for generations of called the ground of the representamen. analysts, not for one." (CP 2.228)

95 References Barwise, Jon, Jean Mark Gawron, Gordon STAN-CS-TN-94-13, Stanford University. Plotkin, & Syun Tutiya, eds. (1991) Situation Availablefrom http://sail.stanford.edu. Theoryand its Applications, CSLI, Stanford, CA. Peirce, Charles Sanders (1885) "On the algebra Barwise, Jon, & John Perry (1983) Situations and of logic," AmericanJournal of Mathematics,vol. Attitudes, MITPress, Cambridge, MA. 7, pp. 180-202. Reprinted in Peirce (W)vol. Burke, Tom (1991) "Peirce on truth and Peirce, Charles Sanders (CP) Collected Papers of partiality," in Barwiseet al. (1991)pp. 115-146. C. S. Peirce, ed. by C. Hartahome,P. Weiss, & Cooper, Robin, & Hans Kamp(1991) "Negation A. Burks, 8 vols., Harvard University Press, in situation semantics and discourse represen- Cambridge, MA,1931-1958. tation theory," in Barwise et al. (1991) pp. 311-333. Peirce, Charles Sanders (W) Writings of Charles S. Peirce, vols. 1-5, Indiana University Press, Devlin, Keith (1991) "Situations as mathematical Bloomington, 1982-1993. ," in Barwiseet al. (1991) pp. 25-39. Roberts, Don D. (1973) The Existential Graphs Frege, Gottlob (1879) Begriffsschrift, translated of Charles S. Peirce, Mouton,The Hague. in Jean van Heijenoort, ed. (1967) FromFrege to GOdel, Harvard University Press, Cambridge, Schr6der, Ernst (1890-1895)Vorleswagen i~er die MA,pp. 1-82. Algebra der Logik, 3 vols., Teubner, Leipzig. Reprinted by Chelsea Publishing Co., Bronx, Genesereth, Michael R., & Richard E. Fikes NY, 1966. (1992) KnowledgeInterchange Format, Reference Manual, Version 3.0, Report Logic-92-1, Com- Sowa, John F. (1984) Conceptual Structures: puter Science Department, Stanford University. Information Processing in and Machine, Addison-Wesley, , MA. Guha, R. V. (1991) Contexts: A Formalization and Some Applications, technical report Sowa,John F. (1990) "Crystallizing theories out ACT-CYC-423-91, MCC,Austin, TX. of knowledge soup," in Z. W. Ras & M. Zemankova,eds., Intelligent Systems: State of Houser, N., D. D. Roberts, & J. Van Evra, eds. (1995) Studies in the Logic of Charles Sanders the Art and Future Directions, Ellis Horwood, Peirce, Indiana University Press, Bloomington. NewYork, p. 456-487. Kamp,Hans (1981) "Events, discourse represen- Sowa, John F. (1991) "Towards the expressive tations, and temporal references," Langages64, power of natural language," in J. F. Sowa, ed. 39-64. (1991) Principles of Semantic Networks: Explo- rations in the Representation of Knowledge, Lenat, Dougla~B., & R. V. Guha(1990) Building Morgan Kanfmana Publishers, San Mateo, CA, Large Knowledge-Based Systems, Addison- pp. 157-189. Wesley, Reading, iVIA. Sowa, John F. (1993) "Logical foundations for McCarthy, John (1990) Formalizing Common representing object-oriented systems," Journal of Sense, Ablex, Norwood,NJ. Experimental and Theoretical AI, vol. 5, nos. McCarthy, John (1993) "Notes on formalizing 2&3, pp. 237-261. context," Proc. IJCAI-93, Chamb6ry,France, pp. 555-560. Sowa, John F., & Eileen C. Way(1986) "Imple- menting a semanticinterpreter for conceptual McCarthy, John, & Sa~ Buva~(1994) Formaliz- graphs," IBM Journal of Research and Develop. ing Context, Technical Note merit 30:.1, pp. 57-69.

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