Syntax, Semantics, and Pragmatics of ContextsSyntax, Semantics, and Pragmatics of Contexts John F. Sowa philosophy and Computers and Cognitive Science State University of New York

Syntax, Semantics, and Pragmatics of Contexts

John F. Sowaphilosophy and Computers and Cognitive Science

State University of New York at Binghamton

Abstract. The notion of context is indispensable in discussions of meaning, but the word context hasoften been used in conflicting senses. In logic, the first representation of context as a formal object wasby the philosopher C. S. Peirce; but for nearly eighty years, his treatment was unknown outside a smallgroup of Peirce aficionados. In the early 1980s, three new theories included related notions of context:Kamp’s discourse representation theory; Barwise and Perry’s situation semantics; and Sowa’s conceptualgraphs, which explicitly introduced Peirce’s approach to the AI community. More recently, JohnMcCarthy and his students have begun to use a closely related notion of context as a basis for organizingand partitioning knowledge bases. Each of the theories has distinctive, but complementary ideas thatcan enrich the others, but the relationships between them are far from clear. This paper analyzes thesemantic foundations of these theories and shows how McCarthy’s ist(c,p) predicate can be interpretedin terms of the semantic notions underlying the others.

1. Theories of Contexts

In the AI literature, the term context has beenapplied to a profusion of ideas that have not beenclearly distingafished. Some of them concern thesyntactic representation of contexts; others referto the semantic relationship of a linguistic contextto a physical situation; and still others introducepragmatic notions concerning the purpose or useof a context in various applications. Each ofthese major areas can be subdivided further.Syntactically, there are three distinct aspects ofcontext:

1. A mechanism for grouping, associating, orpackaging information that can be namedand referenced as a single unit.

2. The contents of that package, which havebeen called anything from quoted formula tomicrotheory.

3. The permissible operations on the informa-tion in the package and the constraints onimporting and exporting information intoand out of a package.

All three of these notions represent syntacticmechani.mls for representing and manipulatinglogical formulas without any consideration oftheir relationship to the real world, a possibleworld, or some model of the world. Much of thecontroversy about contexts results from the lackof a formal semantics that relates these operationsto a Tarski-style model. Even an informal se-mantics that displays the intuitive meaning ofcontexts in terms of real-world objects and situ-ations would be helpful as a guide to furtheranalysis and formaliTation.

Some of the confusion about contexts resultsfrom an ambiguity in the English word. Die-tionaries list two major senses of the wordcontext:.¯ The basic meaning is a section of the linguis-

tic text or discourse that surrounds someword or phrase of interest.

¯ The derived meaz~g is a nonlingnistic situ-ation, environment, domain, setting, back-ground, or milieu that includes some entity,subject, or topic of interest.

These two informal senses suggest intuitive crite-ria for distinguishing the various functions ofcontexts:¯ Syntax. The syntactic function of context is

to group, delimit, or package "a section oflinguistic text." Formally, a context behaveslike the QUOTE operator in Lisp togetherwith the parentheses that delimit the portionof text that is quoted.

¯ Semantics. The quoted text of a context re-fers to something, which may be a physicalentity or situation, a mathematical con-struction, or some other expression in a na-tural or artificial language.

¯ Pragmatics. The word interest, which occursin both senses of the English definition, sug-gests some reason or purpose for distinguish-ing "a section of linguistic text" or "anonlinguistie situation." That purpose con-stitutes the pragmaties or the reason why thetext is being quoted. In Lisp, the QUOTEoperator blocks the execution of the standardLisp interpreter to allow nonstandard oper-

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

ations to be performed for some other pur-pose. In logic, a quote blocks the standardrules of inference and allows the definition ofnew rules for some special purpose.

As this analysis indicates, the notion of contextis intimately connected with a complex of relatedideas. Much of the confusion results from whichof them happens to be called a context:, somepeople apply the word to the package; and othersto the information contained in the package, tothe thing that the information is about, or to thepossible uses of either the information or thething. The ideas themselves may be compatible,but they must be carefully distinguished andsorted out.

These intuitive criteria provide a basis foranalyzing John McCarthy’s (1993) "Notes Formalizing Context" and relating the ideas tothe other theories. McCarthy’s basic notation isthe predicate ist(c,p), which may be read "theproposition p is true in context c." In his disser-tation written under McCarthy’s direction, R. V.Guha (1991) applied McCarthy’s approach to theproblem of partitioning a large, monolithicknowledge base into a collection of smaller, moremodular microtheories. Guha implemented themicrotheories in the Cyc system (I.gnat & Guha1990), in which they have become a fundamentalmechanism for organizing and stnmturing aknowledge base. McCarthy and Buva~ (1994)have also applied contexts and the /st predicateto the analysis and representation of natural lan-guage discourse.

Although McCarthy, Guha, and Buva~ haveshown that the /st predicate can be a powerfultool for building knowledge bases and analyzingdiscourse, they have not clearly distinguished thesyntax of contexts and propositions from theirsemantic relationship to some domain of dis-course. In fact, the ist predicate itself mixes thesyntactic notion of containment (~s-in) with thesemantic notion of truth (/s-true-of). To clarifythese relationships, it may be helpful to analyzethe ist predicate as a conjunction of three moreprimitive predicates, ~s-in, refers-to, anddescribes:

ist(c,p) -- (~3x:Entity)(is-in(c,p) refers-to(c,x) ^ describes(x,p)).

According to this analysis, the proposition p istrue in context e ff and only if there exists someentity x such that p is in c, c refers to x, and pdescribes x. The formula distinguishes the ab-stract context c from some nonlingulstic entity x,which represents the "situation, environment,domain, setting, background, or milieu" assoei-

ated with c. The predicate ~s-in represents thesyntactic relationship of c to p; and the predicatesrefers-to and describes represent the semantic re-lationships of c and p to the external entity x.McCarthy, Guha, and Buva~ have primarilyconsidered the syntactic operations associatedwith the ~s-in component of the ist predicate. Tojustify those operations, the semantics of therefers-to and describes components must also beaddressed.

Much of the controversy about contexts re-suits from the abundance of notation and termi-nology in different theories, their application todiverse phenomena, and the lack of commlmi-cation between the different schools of thought.The purpose of this paper is to emphasize theunderlying similarities and to promote cross-fertilization of ideas. The following five theorieswill be considered:

1. Charles Sanders Peirce (1885) invented themodern algebraic notation for predicate cal-culus; but a dozen years later, he developedan alternate notation, which he calledexistential graphs (Roberts 1973). AlthoughPeirce’s algebra and graphs had equivalentexpressive power, the graphic structure servedas a heuristic aid that led him to explore op-erations and applications that were over-looked by logicians who used only thealgebraic notation. In particular, Peirce’sgraphic notation for contexts was isomorphicto the discourse representation structures(DRSs) invented by Hans Kamp eighty yearslater. His rules of inference were based onoperations of iterating and deiterating infor-mation to and from contexts in a way thatresembles John McCarthy’s lifting rules.

2. Hans Kamp (1981) developed discourse rep-resentation theory (DRT) to express the log-ical constraints on anaphofic references innatural language. Because of the difficultyof expressing those constraints in the alge-braic notation for logic, Kamp introduced thegraphic DRS notation, which allowed a tim-pler formulation of his rules. Si~iBcantly,the nested contexts in Kamp’s DRSs areisomorphic to the nest of contexts in Peirce’sEGs, even though Kamp had no previousknowledge of them. Kamp deserves credit fordiscovering the constraints on anaphora inDRT, but DRSs and EGs are equally suit-able for expressing those constraints.

3. Jon Barwise and John Perry (1983) devel-oped situation semantics as a theory ofmeaning in natural langq_~age. Unlike

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Montague’s approach (1975), which relatedthe semantics of language to potentially infi-nite models of the real world or possibleworlds, Barwise and Perry adopted finite sit-uations as their basis. Each situation is abounded region of space-time containingphysical objects and processes, as well asother situations. A great deal of research hasbeen done within the paradigm of situationsemantics (Barwise et al. 1991), including ef-forts to merge it or at least reconcile it withDRT (Cooper & Kamp 1991). An impor-tant question is how and whether it can bemerged or reconciled with McCarthy’s con-texts as well.

John Sowa (1984) developed conceptualgraphs as a system of logic and reachingbased on the semantic networks of AI andthe existential graphs of C. S. Peirce. Thenodes called concepts correspond to typed,quantified variables in a sorted predicate cal-caius. A context is a defined as a concept oftype Proposition, whose referent field con-tains one or more conceptual graphs thatstate the proposition. Later papers (Sowa Way 1986; Sowa 1991) used the CG contextsto represent Kamp’s DRSs and Barwise andPerry’s situations. In a paper on "Crystalliz-ing Theories out of Knowledge Soup," Sowa(1990) proposed the use of contexts for par-titioning a knowledge base into a collectionof smaller "chunks" that could be assembledinto theories appropriate to any particularapplication. In his dissertation, Guha (1991)cited the knowledge soup paper, which hesaid was "in the same spirit as the work de-scribed in this document."

5. John McCarthy is one of the founding fathersof AI, whose collected work (McCarthy1990) has frequently inspired and sometimesrevolutionized the application of logic toknowledge representation. His work oncontext, although published later than theprevious four approaches, has grown out ofideas based on his earlier work. McCarthy’s/st predicate is the key to relating that workto the ongoing research in the otherparadisnns. If the/st predicate can be definedin terms of the other theories, then any resultsobtained in one approach can be translatedto any of the others. Besides defining con-texts, McCarthy has been emphasizing hislifting rules for importing and exporting in-formation into and out of the quoted text orpackage. Such rules, which resemble Peirce’srules of iteration and deiteration, are essentialfor allowing quoted information to be un-quoted and used.

Besides these five theories, there is a long historyof related ideas in logic, philosophy, linguistics,and AI. The most important ones for a theoryof context include indexicals, possible worlds,metalanguage, belief revision, and ontology re-vision. Yet the various ideas and theories weredeveloped by people with different intuitions,which they applied to different problems ofknowledge representation. Trying to unify andclarify those intuitions by defining the terms ofone theory in those of another runs the risk ofdistorting the insights of both. This paper willexplore the implications of these definitions todetermine whether the benefits of unification andclarification outweigh any possible distortions ofthe original in~ghts.

2. Peirce’s Contexts

First-order predicate calculus was independentlyinvented by Gottlob Frege (1879) and CharlesSanders Peirce (1885). Frege used a tree notation,which no one else ever adopted. But Peirce de-veloped an algebraic notation, which through thetextbook by Ernst Schr6der (1890) and with change of symbols by C-iuseppe Peano becamethe modem system of predicate calculus. Longbefore Bertrand Russell learned logic from Fregeand Peano, it had become a flourishing subjectbased on the Peirce-Schr6der foundations.

The early history of modem logic is a fasci-nating tale that has been recounted by Roberts(1973) and Houser et al. (1995). The main pointfor this paper is that the man who invented the

common algebraic notation for logic later aban-doned it for a graph representation, which hecalled his "chef d’oeuvre" and "the luckiest findof my career." With the existentialgraphs that heinvented in 1897, Peiree developed an aspect oflogic that was largely ignored by the mathematicallogicians of the twentieth century. In relatingPeirce’s later logic and philosophy to situationsemantics, Burke (1991) said "Peiree anticipatedin his own way some of the concerns of situationtheory (or rather, he happened to be working be-fore it went out of fashion to wrestle with suchconcerns)." A century later, those concerns areback in fashion, and Peirce is once again in theavant garde of modem logic.

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The three primitives of existential graphs(EGs) include the ova/enclosure, which delimitsa context, the line of identity, which correspondsto an existentially quantified variable, andjuxtaposition, which represents conjunction. Thedefault interpretation of an oval with no other

qualifiers is negation of the graphs nested inside.Existence, conjunction, and negation provide acomplete representation for all of first-order logic.As an example, the middle diagram in Figure 1is an existential graph for the sentence If a farmerowns a donkey, then he beats it.

Dbcounm Representc~n Structure Pmflffoned Semcm~ Network

Figure 1. Three representations for "If a farmer owns a donkey, then he beats it."

The EG in Figure 1 has two ovals, whichrepresent negations. It also has two lines ofidentity, represented as linked bars: one line,which connects farmer to the left side of owns andbeats, represents an existentially quantified vari-able (3x); the other line, which connects donkeyto the right side of owns and beats represents an-other variable (3y). When Figure 1 is translatedto the algebraic notation, farmer and donkey mapto monadic predicates; owns and beats map todyadic predicates. The implicit conjunctions canbe represented with Peano’s symbol ̂:

~(~) (qy) (farmer(x) donkey(y) ^

owns(x,y) ̂ ~beats(x,y)).Peirce called a nest of two ovals, as in Figure 1,a scroll, which he used to represent material im-plication, since "(p^~q) is equivalent to p=q.Using the ~ symbol, the above formula may berewritten as

(Vx)(Vy)((farmer(x) donkey(y) ^owns(x,y)) ~ beats(x,y)).

The algebraic formula with the = symbol il-lustrates a peculiar feature of logic in comparisonwith natural languages: in order to preservescope, the implicit existential quantifiers in thephrases a farmer and a donkey must be moved tothe front of the formula and be tsanslated to uni-versal quant~ers. This puzzling feature of logichas posed a problem for linguists. In his dis-course representation structures, Hans Kamp(1981) resolved it by introducing a new symbolfor implication with different scoping rules. Thediagram on the left of Figure 1 shows a DRS for

The variables x and y in the antecedent box haveimplicit existential quantifiers; Kamp defined thescoping rules for the DRS to include consequentbox within the scope of the antecedent. As inexistential graphs, conjunction is implicitly shownby juxtaposition. Altogether, the DRS may beread If there exists a farmer x and a donkey y andx owns y, then x beats y.

Although the DRS and EG notations lookquite different, they are exactly isomorphic: theyhave the same three primitives and exactly thesame scoping rules for variables or lines of iden-tity. What makes this coincidence remarkable isthat in the dozens of notations for semantic net-works in the 1960s and 1970s, no one else redis-covered Peirce’s conventions. The notation thatcomes closest is the partitioned semantic networkby Gary Hendrix (1975), which is illustrated the rightmost diagram of Figure 1. l.ike Peirceand Kamp, Hendrix took the existentialquantifier as the default, represented conjunctionby juxtaposition, and used a graphic enclosure forpartitioning contexts. But unlike Peirce andKamp, Hendrix allowed overlapping contexts:the two overlapping boxes in Figure 1 representthe antecedent and the consequent of the impli-cation.

With Overlapping contexts, Hendrix had noneed for scoping rules. Although the farmer anddonkey nodes each occurred only once, the over-lap allowed them to occur ~multaneously in bothcontexts. Yet the overlapping contexts proved tobe unwieldy: with more than three contexts, itbecame impossible to draw partitioned nets on a

the donkey sentence. The two boxes connectedby an arrow represent the English pair if-then~ plane. Furthermore, the nesting of clauses in

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natural languages has a more direct mapping tothe Peirce-Kamp nested contexts than toHendrix’s overlapping contexts. Kamp’s rules forresolving anaphom in DRSs could be statedequally well in terms of EGs, but not in terms ofoverlapping contexts.

Besides notation, Peirce defined rules of in-ference for EGs, which in many respects are thesimplest and most elegant inference rules everdevised for any version of logic. A typical theo-rem that requires 43 steps to prove with Russelland Whitehead’s rules of 1910 takes only 8 stepswith Peirce’s rules of 1897. Peirce’s roles are ageneralization of natural deduction, whichGerhard Gentzen discovered 40 years later, l.ikeGentzen, Peirce took the empty set as his onlyaxiom, but Peirce’s proofs are simpler thanGentzen’s because the nesting of contexts elimi-nates the bookkeeping needed for making anddischarging assumptions -- the most error-proneaspect of Gentzen’s system. For further dis-

cussion of these points, see Roberts (1973), Sow^(1984, 1993), and Houser et al. (1995).

In discussing modality, Peiree imagined thegraphs drawn on "a book of separate sheets,tacked together at points." The upper sheet re-presents "a universe of existing individuals," whilethe other sheets "represent altogether differentuniverses with which our discourse has to do."Graphs on those sheets may represent "conceivedpropositions which are not realized." Peirce saidthat a necessarily true proposition could be con-sidered as replicated on aU the sheets in the book,while a possible proposition might occur on onlyone. In the algebraic notation, Peirce used thesymbol co as an index for "a state of things," towhich he applied a universal quantifier for neces-sity and an existential quantifier for possibility.With his interpretation of necessity as truth "un-der all circumstances," Peirce was followingLeibniz and anticipating Kripke.

Figure 2. EG for "You can lead a horse to water~ but you can’t make him drink."

In 1906, Peirce introduced colors or tincturesto represent modalities. Figure 2 shows one ofPeirce’s examples, but with shading instead of theoriginal red for possibility. The graph containsfour ovals: the outer two are associated to forma scroll for if-then; the inner two represent possi-bility (shading) and impossibility (shading insidea negation). The outer oval may be read If thereexists a person, a horse, and water;, the next ovalmay be read then it is possible for the person tolead the horse to the water and not possible for theperson to make the horse drink the water.

The notation leads to represents the triadicpredicate leads-to(’x~v,z), and _makes_drink_ re-presents makes-drink(x~v,z). In the algebraic no-tation with 0 for possibility, Figure 2 maps tothe following formula:

~(3x)(3y)(3z)(person(x) horse~) ^water(z) ~(Oleads-to(x,y,z) ^~Omakes-drink(x,y,z) )

With the symbol = for implication, this formulabecomes

(Vx)(¥y)(Vz) ((person(x) ^ water(z)) (Oleads-to(x,y,z) ^~ ~makes-dri nk(x,y,z) ).

This version may be read For all x, y, and z, if xis a person, y is a horse, and z is water, then it ispossible for x to lead y to z, and not possible forx to make y drink z.

As a systematic way of representing the kindsof contexts, Peirce adopted the traditionalheraldic tinctures, which were classified as metal,color, or fur. He applied that three-way dis-tinct.ion to actual, modal, and intentional con-texts:

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1. Metal: argent, or, fer, and plomb. Peirceused argent (white background) for "the ac-tual or true in a general or ordinary sense,"and the other metallic tinctures for "the ac-tual or true in some special sense." A state-ment about the physical world, for example,would be actual in an ordinary sense. Peircealso considered mathematical idealizations,such as Cantor’s hierarchy of infinite sets, tobe "actual," but not in the same sense as or-dinary physical entities.

2. Color: azure, gules, vert, and purpure. Peircedistinguished four basic modalities: azure forlogical possibility (dark blue) and subjectivepossibility (light blue); gules for objectivepossibility; vert for "what is in theinterrogative mood"; and purpure for "free-dom or ability." Each of these modalitiescould be combined with negation: he definednecessary in the usual way as not-possibly-notand obligatory as not-freedom-not (an antic-ipation of deontic logic).

3. Fur: sable, ermine, vair, and potent. The fourfurs correspond to propositional attitudes:sab/e for "the metaphysically, or rationally,or secondarily necessitated"; erm/ne for pur-pose or intention; vair for "thecommarlded"; and potent for "thecompelled."

Peirce’s three-way classification is highly sugges-tive, but incomplete. He wrote that the completeclassification of "all the conceptions of logic" was"a labor for generations of analysts, not forone." But throughout his analyses, he clearlydistinguished the logical operators represented bythe graphs, from the tinctures, which, he said, donot represent

differences of the predicates, orsignifications of the graphs, but of thepredetermined objects to which thegraphs are intended to refer. Conse-quently, the Iconic idea of the Systemrequires that they should be represented,not by differentiatiom of the Graphsthemselves but by appropriate visiblecharacters of the surfaces upon which theGraphs are marked.

It seems that Pcirce did not consider the tincturesto be part of logic itself, but of the metalangua~for describing how the logic applies to the uni-verse of discourse:

The nature of the universe or universesof discourse (for several may be referredto in a single assertion) in the rather un-

usual cases in which such precision is re-quired, is denoted either by usingmodifications of the heraldic tinctures,marked in something like the usual man-ner in pale ink upon the surface, or byscribing the graphs in colored inks.

By 1906, mathematical logic based onPeirce’s algebraic notation had become a flour-ishing field of research, and his graphs were ig-nored. There were several reasons for the neglect:the notation and terminology were unfamiliur;most logicians, who had a strong background inmathematics, had already found the algebraicnotation congenial to their tastes; and si~if-icantly, Peirce’s novel applications of his graphlogic to modality, intentionality, andmetalangn,~ were outside the main interests ofthe logicians of his time. Today, however,Peirce’s contributions are central to research oncontexts:

1. Representation of contexts by nests of enclo-sures, which separate or partition groups ofpropositions of different modal status.

2. First-order logic based on three operators:existence (line of identity), conjunction(juxtaposition), and negation (oval enclosureon a white background).

3. Sound and complete rules of inference forfirst-order logic based on operations of draw-ing or erasing graphs and importing or ex-porting graphs into and out of contexts.

4. Tinctures for distingmishing the purpose or"nature" of a context from its logical opera-tors (for which he used only the basic three-- existence, conjunction, and negation).

5. A three-way classification of the use of con-texts for representing actuality (metal),modality (color), or intentionality (fur).

6. The use of graphs as a metalanguage fortalking about graphs.

7. Complete statement of the rules of inferencefor existential graphs in existential graphsthemselves.

Peirce’s later writings, although fragmentary, in-complete, and mostly unpublished, are no morefragmentary and incomplete than many modempublications about contexts. Although (or per-haps because) he did not use the word context,Peiree was more consistent in distinguishing thesyntax (oval enclosures), the semantics ("the uni-verse or universes of discourse"), and the prag-matics (the tinctures that "denote" the "nature"of those universes).

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3. Contexts in Conceptual Graphs

Conceptual graphs are extensions of existentialgraphs with new features based on the semanticnetworks of AI and the linguistic research onthematic roles and generalized quantifiers. Theprimary difference is in the treatment of lines ofidentity. In existential graphs, the lines serve twodifferent purposes: they represent existentialquantifiers, and they show how the arguments areconnected to the relations. In conceptual graphs,those two functions are split: boxes called con-cepts contain the quantifiers, and arcs markedwith arrows show the connections of argumentsto circles called conceptual relations. This sepa-ration of functions has several important conse-quences:

¯ Concepts have a place to represent a type la-bel for each quantifier. Conceptual graphstherefore correspond to a typed or sortedlogic, unlike the untyped existential graphs.

¯ Concepts may also contain a name or otherspecification of the referent of the concept,as in [Cat: Yojo], where the type is Cat andthe particular individual is named Yojo. Thearea on the left of the colon is called the typefield, and the area on the right is called thereferent field.

¯ When an existential graph or an tmtypedformula in predicate calculus is mapped to aconceptual graph, the type label r may beused to mark the universal type, which is asupertype of all others. The type T imposesno restrictions on the quantifier or referent.The concept [T: Yojo], for example, wouldrepresent an entity named Yojo whose typewas unknown or unspecified.

¯ The arrows or numbers on the arcs of con-ceptual relations distinguish the argumentsmore clearly than Peirce’s unlabeled lines.For dyadic relations, the arrow pointing to-wards the circle is the first argument, and thearrow pointing away is the second argument.For relations with more than two arguments,the arcs are numbered l, 2 ..... n; the arrowon the n-th arc points away from the circle,and the other arcs point towards the circle.

¯ In an existential graph, any point on a lineof identity could be considered as a separatequantified variable. Conceptual graphs con-centrate the point of quantification in theconcepts rather than the lines. A blankreferent field represents the quantifier 3, butthe universal quantifier V or other generalizedquantifiers may also occur in the referent fieldof a concept.

¯ Peirce’s lines of identity could cross contextboundaries, but a concept may only occur ina single context. When two concepts in dif-ferent contexts refer to the same individual,they must be associated by coreference labelsor by a dotted line called a coreference link.

To illu~t~ette these features, Figure 3 shows twoequivalent conceptual graphs for the donkey sen-tence. The CG on the left uses the basic notationwith the -’ symbol to mark negation and withdotted lines for coreference links. The CG on theright uses an extended notation with the types Ifand Then defined as negated propositions andwith coreference shown by the labels x and y.The concepts marked *x and *y are the definingnodes with implicit existential quanfifiers, and theconcepts marked ?x and ?y are bound nodeswithin the scope of *x and *y.

Figure 3. Two conceptual graphs for "If a farmer owns 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 xthe same as the rules for the DRS and EG in owns a donkey y, then x beats y. The circles rep-

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resent dyadic conceptual relations, where the ar-row pointing towards the circle marks the firstargument and the arrow pointing away marks thesecond argument. The two CGs in Figure 3would correspond to the foUowing formula in al-gebraic notation:

~(3X: Farmer) (~y:Oonkey) (3z:Own) (expr(z,x)^ thme(z,y) A ~(3w:Beat) (agnt(w,x)^ ptnt(w,y))

This representation follows C. S. Peirce andDonald Davidson in reifying verbs with the eventvariables z and w. The dyadic relations represent

the thematic roles or case relations used in lin-guistics: experiencer (EXPR); theme (THME);agent (AGNT); and patient (PTNT). For venience, there is also a linear notation thatmakes CGs easier to type:7[ [Farmer: *x]÷(EXPR)÷[Own]÷(THME)÷[Donkey:

7[ [?x]÷(AGNT)÷[Beat]÷(PTNT)÷[?y]

[If: [Farmer: *x]÷(EXPR)÷[Own]÷(THME)÷[Donkey: [Then: [?x]÷(AGNT)÷[Beat]÷(PTNT)-~[?y]

In the graphic form of Figure 3, coreference maybe shown by dotted lines, but coreference labelsmust 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 GMT to 19:29:45GMT."

In CGs, a context is defined as a conceptwhose referent field contains nested conceptualgraphs. Since every context is also a concept, itcan have a type label, coreference links, and at-tached conceptual relations. In Figure 4, thegraph for a cat chasing a mouse is nested insidea concept of type Situation. The conceptualgraph in the inner context describes the situation.Attached to that context is the relation DUR forduration, which is linked to a concept for an in-terval of 13 seconds. The relations FROM andTO show that the interval lasted from 19:29:32GMT to 19:29:45 GMT.

When a conceptual graph occurs in a contextof type Graph, it is used as a literal; in a contextof type Proposition, it states a proposition; in acontext of type Situation, it describes a situation.The following concept, by itself, may be read Asituation of a cat chasing a mouse.

[Situation: [Cat] ÷(AGNT) ÷[Chase]÷ (THME) ÷ [Mouse]

This graph may be considered an abbreviation forthe following graph, which says that there existsa situation whose description (DSCR) is theproposition that a cat is chasing a mouse:

[Si tuati on]÷ (DSCR) ÷ [Propo si ti on:[Cat] ÷(AGNT) ÷[Chase] ÷ (THME)÷ [Mouse]

J.

In most applications, the abbreviated form shownin Figure 4 is used. When necessary, the type la-bel of the context determines how the nestedgraph is interpreted. This use of the type labelcorresponds to type coercion in programminglang~_m~s.

To relate Figure 4 to McCarthy’s ist predi-cate, first expand the abbreviations; then translatethe expanded graph to predicate calculus:

(Bs: S i tuati on) (Bi: Interval (duration(s,/] a measure(i,13sec) from(i, lg:2g:32GMT) ^ to(i, lg:2g:45GMT)^ dscr(s,

(Rz:Cat) (By:Mouse) (Bz:Chase)(agnt(z,x) A thme(z,y))

Now it is possible to relate parts of this formulato the earlier definition of the/st predicate:

ist(e,p) _--- (~Ix:Entity)(is-in(c,p) refers-to(c,x) ^ describes(x,p)).

The entity x can be identified with the situations, and the proposition p with the proposition thata cat is chasing a mouse. But there is no entityin this translation that can be identified withMcCarthy’s context c. In McCarthy’s sense, c issupposed to be larger than the single proposition.In fact, the context may be so rich that no finiteset of formulas could exhaust its full content.

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This discussion raises a question about the dis-tinction between a total description of everythingthat is knowable about a situation and a partialdescription in a single formula. That issue has

been central to situation semantics, which mayhelp to clarify the semantics of both CG contextsand McCarthy’s contexts.

4. Situations and Propositions

Barwise and Perry (1983) developed situation se-mantics as a reaction again~ the potentially infi-nite models of Kripke’s and Montague’s modaland intensional logics. Each situation is a finiteconfiguration of some aspect of the world in alimited region of space and time. It may be astatic configuration that remains unchanged for aperiod of time, or it may include processes andevents that are causing changes. It may includepeople and things with their actions and thoughts;it may be real or imaginary; and its time may bepresent, past, or future. A situation may be aslarge as the solar system or as small as an atom,and it may contain nested situations for smalleror more detailed aspects of the world.

As an illustration of the way situations arerelated to partial descriptions, Figure 5 shows aconcept of type Situation, which is linked to twoimages and a description. The image relation(IMAG) links the situation to two different kindsof images of that situation: a picture and the as-soeiated sound. The description relation (DSCR)links it to a proposition that desc~bes some as-pect of the situation, which is linked by thestatement relation (STMT) to three differentstatements of the proposition in three differentlanguages: an E~gllsh sentence, a conceptualgraph, and a formula in the Knowledge Repre-sentation Lang~ (KIF) by Genesereth andFikes (1992).

Sour~l:

$1tuat~n~ Picture:

CLANKET¥scrape ---7

"e,-,p

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

Graph:

Figure 5. A situation of a plumber carrying a pipe

A proposition may be defined as an equiv-alence class of statements in one or more lan-guages. No concrete statement in any specificlanguage /s the proposition. Instead, a conceptof type Proposition, such as the one in Figure 5,represents a class of intertranslatable statementsin one or more concrete languages. Each state-ment in that class is said to express the same

proposition. To avoid complexities, the equiv-alence mapping will be defined first for formallanguages like KIF and CGs. Letfbe a mappingfrom language M1 to language ~2 that defines theequivalence; let s be any statement in "~1; and lett =fl-l(](s)) be the result of mapping s 2 andback again. Thenfmust obey the following threeconstraints:

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1. Truth preserving: The statements s and tmust be provably equivalent according to therules of inference of language ~1-

2. Vocabulary preserving: Exactly the samenonlogical symbols must appear in both sand t.

3. Negation preserving: When s and t axemapped to Peirce Normal Form (with ne-gation, conjunction, and the existentialquantifier as the only logical operators), theycontain exactly the same number of negationoperators.

These three conditions ensure that the statementss and t axe highly similar, if not identical. Al-though Fermat’s Last Theorem and 2 + 2 = 4 areboth true, the proof is far from trivial, and thetwo statements require different vocabularies.Condition #2 allows variables to be renamed andpermits pvq to be replaced by ~(~p^"q); but ensures that a statement like(Vx)(dog(x)=dog(x)) does not get mapped to(Vx)(cat(x)~cat(x)). Condition #3 allows pap tobe mapped to p, but it prevents ~~p from beingmapped to p. Generalizing this definition to na-tural languages should be possible, but thatwould raise further issues beyond the scope ofthis paper.

As Figure 5 illustrates, the proposition statedby any of the three statements represents a tinyfraction of the total information available. Boththe sound image and the picture image captureinformation that is not in the sentence, but eventhey are only partial representations. A picturemay be worth a thousand words, but a situationcan be worth a thousand pictures. To relate asituation to the information it contains, KeithDevlin (1991) introduced the term infon for anabstract information object that is closely relatedto the previous definition of proposition.

Devlin’s basic formula is s ~ o, where s is asituation and o is an infon that is semanticallyentailed by s. Devlin’s construction that comes

closest to what McCarthy calls a context is theset of all infons entailed by a situation: {o [s N o}. If the DSCR relation is interpreted asentailment of a partial description, McCarthy’scontext c may be considered a complete de-scription of a situation:

completeDscr(s,c) = c= {a [ s N ~r}.

For Figure 5, the context c would include pro-positions or infons saying that the plumber iscarrying a toolbox, he works for Acme PlumbingCo., he is dragging the pipe with a clankety noise,etc. The predicate is-in(c,p) could then be inter-preted as provability.

With this analysis, the following formula re-presents an interpretation of the /st predicate interms of situations and infons (or propositions):

i st(c,p) - (3s:Situation)(provable(c,p) completeDscr(s,¢) ^ dscr(s,p)).

The syntactic predicate is-in(c,p) is interpreted asprovable(c,p). The two semantic predicates aredefined in terms of entailment: refers-to(c,s) completeDscr(s,c); and dser(s,p) is s ~p. In pure first-order framework, s ~p would be re-dundant, since it would be implied by c [--p andthe definition of c. Including it, however, couldaccommodate nonstandard logics in whichprovability is not equivalent to entailment.

The previous definition was presented at theIJCAI’95 workshop on context. In the dis-eussions, McCarthy made the point that identi-fying a context c with the set of all entailmentsof a situation s does not completely capture hisintuitions about context. As aa example, sup-pose that John believes everything that Mary be-lieves. Then either of their belief systems wouldentail exactly the same propositions. ButMcCarthy would like to say that the context ofJohn’s beliefs is not the same as the context ofMary’s beliefs. That distinction implies that nopurely extensional definition can capture the fullimport of McCarthy’s notion of context.

5. Extensions, Intensions, and Intentions

The difficulty of giving a precise definition ofMcCarthy’s contexts is closely related to the dif-ficulty of giving a precise definition of situations.In their 1983 book, Barwise and Perry identifieda situation with a bounded region of space-time,and they tried to use that definition to support atheory of propositional attitude verbs, such asbelieve, think, hope, fear, wish, want, and plan.Yet further research on situation semantics has

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shown that a purely extensional definition of sit-uation is inadequate, for many of the same rea-sons that McCarthy objected to an extensionaldefinition of context. The following examples il-lustrate the problems:

¯ A college lecture might be considered a situ-ation bounded by a 50-minute time period ina spatial region enclosed by the walls of aclassroom. But if the time were moved for-

ward by 30 minutes, the region would includethe last half of one lecture and the first halfof another. That time shift would create avery unnatural "situation."

¯ An even more unnatural transformationwould be to shift the spatial region to the leftby half the width of a classroom. Then itwould include part of one class listening toone teacher, part of another class listening toa different teacher speaking on a differenttopic, and a wall that separated the two dis-cussions.

¯ Another possible tranfformation is to fix thecoordinate system relative to the sun insteadof the earth. Then the space-time region thatincluded the class at the beonning of the lec-ture would quickly shift into deep space asthe earth moved; by the end of the hour, itwould contain nothing but an occasional hy-drogen atom.

These examples illustrate a fundamental difficultyfor any purely extensional definition. An arbi-trarily chosen space-time region is useless for de-fining a "situation" or "context." Instead, auseful or "meaningful" choice of region dependscritically on some agent’s purpose or intention.But if purpose or intention is critical to the deft-nition of situation or context, then the explicationof propositional attitudes in terms of situationsbecomes circu,!~: the intentionality that is ex-plained by the theory depends on theintentionality that went into the choice of situ-ation.

Peirce considered intentionality as funda-mental to his theory of signs or semiotics. Al-though he did not use the word context, he wouldhave defined a context as a kind of sign, to whichhe would apply his basic definition:

A sign, or representamen, is somethingwhich stands to somebody for somethingin some respect or capacity. It addressessomebody, that is, creates in the mind ofthat person an equivalent sign, or per-haps a more developed sign. That signwhich it creates I call the interpretant ofthe first sign. The sign stands for some-thing, its object. It stands for that object,not in all respects, but in reference to asort of idea, which I have sometimescalled the ground of the representamen.(CP 2.228)

Peirce’s notion of sign was broad enough to in-clude situations, contexts, propositions or infons,and their expression in any language, includingEnglish and logic. His notion ofgroundis crucial:it acknowledges that some agent’s purpose, in-tention, or "conception" is essential for deter-mining the scope of a situation or context.

Unlike the two-way distinction of extensionsand intensions, Peirce drew a three-way dis-tinction in his basic categories of Firstness,Secondness, and Thirdness:

First is the conception of being or exist-hag independent of anything else. Secondis the conception of being relative to, theconception of reaction with, somethingelse. Third is the conception of medi-ation, whereby a first and a second arebrought into relation.

His three kinds of tinctured contexts, which werediscussed in Section 2, are applications of thesecategories:

1. Metal represents the "Actual" or what canbe described in classical first-order logicwithout modalities or intentionality.

2. Color represents the varieties of possibilities,objective, subjective, and deontic. This topichas been thoroughly explored in the modemmodal and intensional logics, whose seman-tics have been defined in terms of infinitefamilies of possible worlds.

3. Fur represents the intentionality (with a instead of an S) whereby an agent selectssome context for the representation of someobject for some purpose. Intentionality, inPeirce’s terms, is the mediating Thirdnessthat determines the context for directing at-tention to some object for some purpose.

Peirce’s third category has never been ad~uatelystudied or represented in any of the modernmodal and intensional logics. It is central to theissues that McCarthy, Baxwise and Perry, andothers have been trying to capture in their theo-des. But those theories cannot be completedwithout representing intentionality with a T.Peirce never completed his theory ofintentionality, but at least he made a good begin-ning. As he said, the complete classification ofall the conceptions is "a labor for generations ofanalysts, not for one."

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