Ontology, Metadata, and Semiotics

John F. Sowa

Abstract. The Internet is a giant semiotic system. It is a massive collectionof Peirce's three kinds of signs: icons, which show the formof something; indices, which point to something; andsymbols, which represent something according to some convention.But current proposals for ontologies and metadata have overlookedsome of the most important features of signs. A sign has threeaspects: it is (1) an entity that represents (2) anotherentity to (3) an agent. By looking only at the signsthemselves, some metadata proposals have lost sight of the entitiesthey represent and the agents — human, animal,or robot — which interpret them. With itsthree branches of syntax, semantics, and pragmatics, semiotics providesguidelines for organizing and using signs to represent somethingto someone for some purpose. Besides representation, semiotics alsosupports methods for translating patterns of signs intended for onepurpose to other patterns intended for different but related purposes.This article shows how the fundamental semiotic primitives arerepresented in semantically equivalent notations for logic, includingcontrolled natural languages and various computer languages.

Presented at ICCS'2000 in Darmstadt, Germany, on August 14, 2000.Published in B. Ganter & G. W. Mineau, eds., Conceptual Structures:Logical, Linguistic, and Computational Issues,Lecture Notes in AI #1867, Springer-Verlag, Berlin, 2000, pp. 55-81.

1. Problems and Issues

Ontologies contain categories, lexicons contain word senses,terminologies contain terms, directories contain addresses,catalogs contain part numbers, and databases contain numbers,character strings, and BLOBs (Binary Large OBjects).All these lists, hierarchies, and networks are tightly interconnectedcollections of signs. But the primary connections are not in the bitsand bytes that encode the signs, but in the minds of the peoplewho interpret them. The goal of various metadata proposals is to makethose mental connections explicit by tagging the data with more signs.Those metalevel signs themselves have further interconnections, whichcan be tagged with metametalevel signs. But meaningless data cannotacquire meaning by being tagged with meaningless metadata.The ultimate source of meaning is the physical world and the agentswho use signs to represent entities in the world and their intentionsconcerning them.

The study of signs, called semiotics, was independentlydeveloped by the logician and philosopher Charles Sanders Peirceand the linguist Ferdinand de Saussure.The term comes from the Greek sêma (sign); Peirceoriginally called it semeiotic, and Saussure called itsemiology, but semiotics is the most common term today.As Saussure (1916) defined it, semiology is a field that includesall of linguistics as a special case. But Peirce (CP 2.229)had an even broader view of that includes every aspectof language and logic within the three branches of semiotics:

1. Syntax."The first is called by Duns Scotus grammatica speculativa.We may term it pure grammar." Syntax is the studythat relates signs to one another.

2. Semantics."The second is logic proper," which "is the formal science ofthe conditions of the truth of representations."Semantics is the study that relates signs to things in the worldand patterns of signs to corresponding patterns that occuramong the things the signs refer to.

3. Pragmatics."The third is... pure rhetoric. Its task is to ascertain the lawsby which in every scientific intelligence one sign gives birthto another, and especially one thought brings forth another."Pragmatics is the study that relates signs to the agents who usethem to refer to things in the world and to communicate theirintentions about those things to other agents who may have similaror different intentions concerning the same or different things.

According to Peirce, semiotics is the science that studies the useof signs by "any scientific intelligence." By that term,he meant "any intelligence capable of learning by experience,"including animal intelligence and even mindlike processesin inanimate matter. By Peirce's criteria, computer techniquesfor processing knowledge bases and databases could be calledcomputational semiotics.

Unfortunately, most word processors deal only with a small subsetof syntax. They have produced what St. Laurent (1999) callsthe WYSIWYG disaster:"Plain text, dull though it may be, is much easier to manage than theoutput of the average word processor or desktop publishing program."In practice, the slogan "What you see is what you get" actually meansWYSIAYG: "What you see is all you get." The text is so overburdenedwith formatting tags that there is no room for semantics or pragmatics.The so-called Rich Text Format (RTF) is semantically the mostimpoverished representation for text ever devised. Formattingis an aspect of signs that makes them look pretty, but it failsto address the more fundamental question of what they mean.

To address meaning, the markup languages in the SGML family were designedwith a clean separation between formatting and meaning. When properlyused, SGML and its successor XML use tags in the text to representsemantics and put the formatting in more easily manageable style sheets.That separation is important, butthe semantic tags themselves must have a clearly defined semantics.Most XML manuals, however, provide no guidelines for representingsemantics. Following is an excerpt from one of the proposed standardsfor representing resources in XML:

A resource can be anything that has identity. Familiar examplesinclude an electronic document, an image, a service (e.g.,"today's weather report for Los Angeles"), and a collectionof other resources. Not all resources are network "retrievable";e.g., human beings, corporations, and bound books in a librarycan also be considered resources. (Berners-Lee, et al. 1998)

In that report, an electronic document is considered familiar, buthuman beings are unfamiliar "resources" mentioned only as an afterthought.Yet without the people, the document and its contents have no meaning.

Many of the ontologies for web objects ignore physical objects,processes, people, and their intentions.A typical example is SHOE (Simple HTML Ontology Extensions), whichhas only four basic categories: String, Number, Date, and Truth(Heflin et al. 1999).Those four categories, which are needed to describe the syntaxof web data, cannot by themselves describe the semantics.Strings contain characters that represent statements that describethe world; numbers count and measure things; dates are time units tiedto the rotation of the earth; and truth is a metalanguage term aboutthe correspondence between a statement and the world. Those categoriescan only be defined in terms of the world, the people in the world,and the languages people use to talk about the world. Without suchdefinitions, the categories are meaningless tags that confer no meaningupon the data they are attached to.

In discussing the Resource Description Framework (RDF), which is basedon the XML facilities, Bray (1998) presented a broader view of the kindsof categories that web-based metadata should represent:

It seems unlikely that one PropertyType standing by itself is aptto be very useful. It is expected that these will come in packages;for example, a set of basic bibliographic PropertyTypes like Author,Title, Date, and so on. Then a more elaborate set from OCLC, anda competing one from the Library of Congress.These packages are called Vocabularies; it's easy to imaginePropertyType vocabularies describing books, videos, pizza joints,fine wines, mutual funds, and many other species of Web wildlife.

This is a good statement of one issue, but it raises other issues:How are the packages related to one another?How is the Date property of the OCLC package related to the Vintageproperty of a wine package? Can packagesinherit type definitions from other packages?If two packages are competing, is there any way to define conversionrules for translating or redefining the types of one in terms of another?A human reader may know that a wine vintage can be comparedto an OCLC date, but without a formal definition, the computer cannot.

Ironically, the computer networks that make it easierto transmit data have made it more difficult to share data.In continuing his discussion, Bray raised further issues:

Nobody thinks that everyone will use the same vocabulary (nor shouldthey), but with RDF we can have a marketplace in vocabularies.Anyone can invent them, advertise them, and sell them.The good (or best-marketed) ones will survive and prosper.Probably, most niches of information will come to be dominated by a smallnumber of vocabularies, the way that library catalogues are today.

There are already thousands, if not millions of competing vocabularies.The tables and fields of every database and the lists of items in everyproduct catalog for every business in the world constitute incompatiblevocabularies. When product catalogs were distributed on paper,any engineer or contractor could read the catalogsfrom different vendors and compare the specifications.But minor variations in the terminology of computerized catalogscan make it impossible for a computer system to comparecomponents from different vendors.

By standardizing the notations, XML and RDF take an important first step,but that step is insufficient for data sharing without some wayof comparing, relating, and translating the vocabularies. Phipps (2000)warned that standardizing the vocabularies may create even moredifficulties "by hiding complexities behind superficial agreements":

To connect from the heart of my e-business to the heart of yours would beimpossibly expensive in shared systems without XML, but even with it thesystem analysis needed to create the translation is a significant task.We should not assume that XML is a panacea or that the standardizationof vocabularies will automatically bring interoperability. XML providesus with a medium to express our understanding of the meaning of data,but we still have to discern realities and differences of meaningswhen we exchange data.

More important than standardizing vocabularies is the developmentof methods for defining and translating vocabularies.To have a sound semantics and pragmatics, those methods must relatethe terms in the vocabularies to the things they refer to andto the people who use them to communicate information about those things.

The purpose of this paper is to analyze the differences of meaning,to explore their implications for web-based metadata, and to showhow the methods of logic and ontology can be used to define,relate, and translate signs from one vocabulary to another.Among the methods discussed in this paper are Peirce's systems of logic,ontology, and semiotics, which are presented in more detail in thebook KnowledgeRepresentation by Sowa (2000).

2. Signs of Signs

Metalanguage consists of signs that signify something about other signs,but what they signify depends on what relationships those signshave to each other, to the entities they represent, andto the agents who use those signs to communicate with other agents.Figure 1 shows the basic relationships in a meaning triangle(Ogden and Richards 1923). On the lower left is an icon that resemblesa cat named Yojo. On the right is a printed symbol that representshis name. The cloud on the top gives an impression of the neuralexcitation induced by light rays bouncing off Yojo and his surroundings.That excitation, called a concept, is the mediatorthat relates the symbol to its object.

Figure 1. The meaning triangle

The triangle in Figure 1 has a long history.Aristotle distinguished objects, the words that refer to them,and the corresponding experiences in the psychê.Frege and Peirce adopted that three-way distinction from Aristotleand used it as the semantic foundation for their systems of logic.Frege's terms for the three vertices of the triangle wereZeichen (sign) for the symbol, Sinn (sense)for the concept, and Bedeutung (reference) for the object.As an example, Frege cited the terms morning star andevening star. Both terms refer to same object, the planet Venus,but their senses are very different: one means a star seenin the morning, and the other means a star seen in the evening.Following is Peirce's definition of sign:

A sign, or representamen, is something which stands tosomebody for something in some respect or capacity. It addressessomebody, that is, creates in the mind of that person an equivalentsign, or perhaps a more developed sign. That sign which it createsI call the interpretant of the first sign. The sign standsfor something, its object. It stands for that object, notin all respects, but in reference to a sort of idea, which I havesometimes called the ground of the representamen.(CP 2.228)

The terms morning star and evening star are distinct signsthat create different concepts or interpretants in the mindof the listener. Both concepts stand for the same object, but in respectto a different ground, which depends on the time of the observation.

Aristotle observed that symbols could symbolize other symbols,as "written words are symbols of the spoken." Frege said that hislogic could be used as a language to talk about the logic itself.But Peirce went further than either of them in recognizing thatmultiple triangles could be linked together in different waysby attaching a vertex of one to a vertex of another.By stacking another triangle on top, Figure 2 representsthe concept of representing an object by a concept.The upper triangle relates the cloud that suggests the concept of Yojoto the symbol [Cat: Yojo], which is a printable symbolfor the more elusive neural excitation. At the very top isa cloud for the neural excitation that occurs when some personrecognizes that Yojo is being represented by a printed symbol.

Figure 2. Concept of representing an object by a concept

Meaning triangles can be linked side by side to represent signs of signsof signs. On the left of Figure 3 is the triangle of Figure 1,which relates Yojo to his name. The middle triangle relatesthe name Yojo to the quoted string "Yojo".The rightmost triangle relates that character string to its encodingas a bit string 0x596F6A6F. In each of the three triangles,the symbol is related to its object by a different metalevel process:naming, quoting, or representing.At the top of each triangle, the clouds that represent the unobservableneural excitations have been replaced by concept nodesthat serve as printable symbols of those excitations.The concept node [Cat: Yojo] is linked by theconceptual relation node (Name) to a node for the conceptof the name [Word: "Yojo"], which is linked by the conceptualrelation node (Repr) to a node for the concept of the characterstring itself [String: 'Yojo'].The resulting combination of concept and relation nodes is an exampleof a conceptual graph (CG).

Figure 3. Object, name of object, symbol of name, and character string

To deal with meaning, semiotics must go beyond relationships betweensigns to the relationships of signs, the world, and the agentswho observe and act upon the world.Symbols are highly evolved signs that are related to actual objectsby previously established conventions. People agree to those conventionsby relating the symbols to more primitive signs, such asicons, which signify their objects by some structural similarity,and indices, which signify their objects by pointing to them.All these signs can be related to one another by linking series oreven arrays of triangles. Additional triangles could showhow a name is related to the person who assigns the name, to the reasonfor giving an object one name rather than another, or to an index thatpoints to some location where the object may be found.

Different kinds of applications require different levels of detailin the metadata. For information retrieval (IR), a simple string searchcan often find a web page with the desired information. To findinformation about Yojo the cat, it could search for the strings "Yojo"and "cat"; to find information about Queequeg's ebony idol in the novelMoby Dick, it could search for the strings "Yojo" and "Queequeg".IR systems depend on a human reader to decide which stringsto search for and to interpret the results that are retrieved.Systems that go beyond simple search must be able to distinguishthe physical object Yojo from an icon that resembles the object,the name of the object, and the character string that represents the name.Following is an interchange between a human user who asked a questionand a computer system that did not make those distinctions:

Q: What is the largest state in the US?A: Wyoming.

To answer questions about sizes, the computer would use the greater-thanoperator to compare numbers. When it applied that operator tothe character strings, it found the last state in alphabetical order,which does not happen to be the largest state in either area orpopulation. A loosely defined system of metadata may be adequatefor finding information, but inadequate for any further processing.As Phipps observed, superficial agreements about vocabulary may hidecomplexities that make interoperability impossible.

3. Logical Primitives

The second branch of semiotics is semantics, or as Peirce called it,logic proper — the subject that studieswhat it means for a pattern of signs to represent a true propositionabout the things the signs refer to.The first complete system of first-order logic (FOL) wasthe Begriffsschrift by Gottlob Frege (1879), who developeda notation that no one else, not even his very few students, ever adopted.The second complete system was the algebraic notation for predicatecalculus, independently developed by Charles Sanders Peirce (1883, 1885).With minor modifications, it became the most commonly used versionof logic during the twentieth century. It is a much betternotation than Frege's Begriffsschrift, but for many people,it is "too mathematical." The third complete system was Peirce'sexistential graphs of 1897, which he called hischef d'oeuvre — a strong claimby a man who invented the most widely used version of logic today.

With existential graphs, Peirce set out to determine the simplest,most primitive forms for expressing the elements of logic.Although he developed a graphical notation for expressing those forms,they can be expressed equally well in a natural language,an algebraic notation, or many different linear, graphical, or evenspoken representations. The following table lists Peirce's five semanticprimitives, each illustrated with an English example. Since thesefive elements are primitive, they cannot be formally defined in termsof anything more primitive; instead, the middle column of the tablebriefly states their "informal meaning."

Table 1. Five semantic primitivesPrimitiveInformal MeaningEnglish ExampleExistenceSomething exists.There is a dog.CoreferenceSomething is the same as something.The dog is my pet.RelationSomething is related to something.The dog has fleas.ConjunctionA and B.The dog is running, and the dog is barking.NegationNot A.The dog is not sleeping.

The five primitives in Table 1 are available in every natural languageand in every version of first-order logic.They are called semantic primitives because they go beyond syntacticrelations between signs to semantic relations between signs and the world.Any notation that is capable of expressing these five primitivesin all possible combinations must include all of FOL as a subset.As an example, the WHERE clause of the SQLquery language can express each of these primitives and combine themin all possible ways; therefore, first-order logic is a subset of SQL.Different languages may use different notations for representingthe five primitives:

• Existence.In most natural languages, existence is implied by mentioningsomething. For emphasis, languages also provide an explicitexistential quantifier such as the word some.In the algebraic notation for logic, existence may be expressedby an explicit symbol, such as ?. In SQL, existence isstated implicitly by mentioning something orexplicitly by using the keyword EXISTS.

• Coreference.To say that two different signs refer to the same thing, naturallanguages use a variety of methods, both explicit and implicit:pronouns, determiners, inflections, and forms of the verb be.Most linear notations for logic use variables and the equal sign,and graphic notations use connecting lines or ligatures.Like other linear notations, SQL uses variables and the equal sign.

• Relation.Content words in natural languages express some information aboutat least one entity, known as the referent of the word,but they may also relate or imply other entities as well.The verb give, for example, refers to an act of giving,but it also implies a giver, a gift, and a recipient.In SQL, relations are called tables.

• Conjunction.In both natural and artificial languages, conjunction may beexpressed implicitly by making one statement after anotheror explicitly by a word like and ora symbol like ∧.SQL uses the keyword AND.

• Negation.All natural languages and most versions of logicprovide words, inflections, or symbols to express negation.The biggest variations from one language to another arein the methods for distinguishing the context or scopeof what is negated from what is not negated.SQL uses the keyword NOT with parentheses to show scope.

Other logical operators can be defined in terms of these five primitives.Table 2 shows three of the most common: the universal quantifier,implication, and disjunction. These operators do not qualifyas semantic primitives because they are not as directly observableas the five in Table 1. Seeing Yojo, for example, isevidence that some cat exists, but there is no way to perceive every cat.Seeing two things together is evidence of a conjunction, and not seeingsomething is evidence of a negation. But there is no direct wayof perceiving an implication or a disjunction: post hoc doesnot imply propter hoc, and seeing one alternative of a disjunctiondoes not indicate what other alternatives are possible. Althoughthe three operators of Table 2 can be defined in terms of the fiveprimitives, any assertion they make about the world can only be verifiedindirectly and usually with less certainty than the basic primitives.

Table 2. Three defined logical operatorsOperatorEnglish ExampleTranslation to PrimitivesUniversalEvery dog is barking.not(there is a dog and not(it is barking))ImplicationIf there is a dog, then it is barking.not(there is a dog and not(it is barking))DisjunctionA dog is barking, or a cat is eating.not(not(a dog is barking) and not(a cat is eating))

Instead of choosing existence and conjunction as primitives, Fregechose the universal and implication as primitives. Then he definedexistence and conjunction in terms of his primitives. The resultwas not as readable as Peirce's algebraic notation, but it wassemantically equivalent. Peirce's existential graphs (EGs) werealso semantically equivalent to both of the other notations,but they had the simplest of all mappings to the five primitives.SQL also uses existence, conjunction, and negation as its threebasic primitives, but it provides the keyword OR as well.SQL has no universal quantifier, which must be representedby a paraphrase of the form NOT EXISTS... NOT.To add logical operators to RDF, Berners-Lee (1999) proposedthe tags and ,which can be combined with the implicit conjunction of RDFto define the operators of Table 2.

To illustrate various notations for logic and their relationshipsto RDF, consider a typical sentence that might be usedin a database specification: Every human being hastwo distinct parents, who are also human beings.Since this sentence introduces numbers and plurals, which go beyondthe five primitives, start with the simpler sentenceSome human has a parent, who is also human.Figure 4 shows an existential graph that represents the sentence.

Figure 4. EG for Some human has a parent who is human.

In an existential graph, the words represent predicates or relations,and the bars represent existential quantifiers. The two bars inFigure 4 represent two individuals who are human, and the one on theleft has the one on the right as a parent. In the algebraic notation,each bar would be assigned a variable, such as x or y,and an existential quantifier, represented by the symbol ?.As a result, Figure 4 would map to the following formula:

(?x)(?y)(Human(x) ∧ HasParent(x,y) ∧ Human(y)).

The symbol ∧ in the formula representsconjunction, which is implicit in the EG and RDF notations.Figure 4 could be represented by a triple in RDF: the firsthuman could be treated as an RDF resource, the HasParentrelation as an RDF property type, and the second humanas an RDF value. The existence of the human on the left would beindicated by the proposed RDF quantifier ,and the one on the right by .

The EG in Figure 5 would require an additional relationor property type before it could be represented in RDF.It represents the sentence Some human has one parent who is human,another parent who is human, and the two are not identical.

Figure 5. EG for Some human has two distinct human parents.

The bar that represents (?x) in Figure 4is connected to both copies of the HasParent relation in Figure 5.Two bars represent each of the human parents. If they were connected,they would represent the same individual; but to represent distinctindividuals, the connection must be negated.In existential graphs, Peirce used an oval to indicate negation;in Figure 5, the oval negates part of the connecting bar.In the algebraic notation, Figure 5 would be represented by thefollowing formula:

(?x)(?y)(?z)   (Human(x) ∧ HasParent(x,y) ∧ Human(y)      ∧ HasParent(x,z) ∧ Human(z) ∧ y≠z).

The inequality y≠zcorresponds to the negated connecting bar in Figure 5.In EGs, the bar that represents a variable also represents coreference,and its negation represents inequality. Notations that have variables,such as predicate calculus, SQL, and RDF, must also have a coreferenceoperator, such as = and its negation ≠. With new property typesfor Equal and NotEqual, Figure 5 could be represented in RDFby three existential quantifiers linked together by three RDF triples.

The small oval in Figure 5 is sufficient to negate the connection betweenthe bar for one parent y and the bar for the other parent z,but an oval can be made as large as necessary to show the scope ofnegation. To show a universal quantifier, Table 2 shows thattwo negations are necessary, which are represented by a pairof large ovals in Figure 6. Literally, the resulting graph may be readIt is false that there exists a human being who does not havetwo distinct parents. It corresponds to the following formula:

~(?x)(Human(x) ∧ ~(?y)(?z)   (HasParent(x,y) ∧ Human(y)      ∧ HasParent(x,z) ∧ Human(z) ∧ y≠z)).

Two copies of the proposed RDF tag and its ending tag could be nested to provide the equivalent of the two nested ovalsin Figure 6. To make RDF equivalent to existential graphs,however, new RDF rules would be needed to restrict the scopeof the quantified variables to the contexts enclosed by the tags and .

Figure 6. EG for Every human has two distinct human parents.

As Table 2 illustrates, a pair of negations canrepresent either a universal quantifier or an implication. The EGin Figure 6 may be read in either way. If the two ovals are consideredan implication, Figure 6 could be read If there exists a human,then that human has a parent who is human and another parent who ishuman and the two parents are distinct. Another option is to readan existential quantifier nested between two ovals as the universalquantifier ?, which is expressed by the English word every.Then Figure 6 could be read Every human has a parent who is humanand another parent who is human and the two parents are distinct.By using the defined operators of Table 2, the formula could be rewrittenin a form that shows the universal quantifier ?and the implication ? explicitly:

(?x)(Human(x) ? (?y)(?z)   (HasParent(x,y) ∧ Human(y)      ∧ HasParent(x,z) ∧ Human(z) ∧ y≠z)).

In English, this formula may be read For every x, if x is human,then there exist a y and a z such that x has the human y as parent,x has the human z as parent, and y and z are not the same individual.

With their minimal number of operators, Peirce's EGs have a singlecanonical form instead of the multiple synonymous sentences in languageswith more built-in operators, such as English and predicate calculus.That property, which is sometimes an advantage, can be a disadvantagewhen the most natural or convenient translation is not obvious.Conceptual graphs (Sowa 1984, 2000) are a graphic notationfor logic based on existential graphs, but with extended featuresthat support more direct translations to natural languages.Figure 7 shows a conceptual graph that correspondsto the existential graph in Figure 6.

Figure 7. CG for If there is a human, then he or she has two distinct human parents.

Logically, the CG in Figure 7 is semantically equivalent to the EGin Figure 6. To indicate the intended reading, the CG has two boxesexplicitly labeled If and Then instead of the EG ovals.Unlike EGs, which use a bar to represent existential quantification,coreference, and connections between relations, those three functionsare distinguished in CGs: boxes, called concept nodes, representquantification; dotted lines represent coreference; and solid linesrepresent connections between the concept nodes and the relation nodes.The node T in the Thencontext, which is coreferent with the node [Human]in the If context,corresponds to a pronoun, such as he, she, or it.Altogether, Figure 7 may be read If there is a human, then he or shehas two distinct human parents. To improve the readability of logicexpressed in RDF, Berners-Lee also proposed the tags and as synonyms for .

Natural languages have a variety of quantifiers, such as the wordsevery, some, or all, the numbers two, seventeen,or half, and the phrases more than six or at leastas many. Those generalized quantifiers can be definedin logic by adding Peano's axioms to define numbers and set theoryto define collections, but it is convenient to have such quantifiersbuilt into the notation. In CGs, the default quantifier is theexistential ?, which is normally representedby a blank, but concept nodes may also contain defined quantifiers,such as the symbol ? or @every torepresent the English word every. The CG in Figure 8 isequivalent to Figure 7 by the definition ofthe quantifier ?.It maps to the following formula in typed predicate calculus:

(?x:Human)(?y,z:Human)   (HasParent(x,y) ∧ HasParent(x,z) ∧ y≠z).

In typed logic, monadic predicates such as Human(x)are replaced by type labels associated with the variables.The typed formula is more concise, but logically equivalentto the untyped formulas that represent the EG of Figure 6.

Figure 8. CG for Every human has two distinct human parents.

Figure 8 could be represented in RDF with the proposed quantifier, but CGs also support othergeneralized quantifiers that have not yet been considered for RDF.As an example, Figure 9 simplifies Figure 8 by introducingthe generic plural symbol {*} to represent a setand the number 2 to represent its count or cardinality.The resulting CG can be mapped to the following formula:

(?x:Human)(?s:Set)(?y∈s)   (Count(s,2) ∧ HasParent(x,y) ∧ Human(y)).

This formula may be read For every x of type Human, there existsan s of type Set such that for every y in s, the count of s is 2,x has y as parent, and y is human. The generalized quantifier{*}@2 in the CG maps to two quantified variables in predicatecalculus: a variable s that ranges over sets and a variabley that ranges over the elements of the set s.

Figure 9. CG for Every human has a set of two human parents.

The CG in Figure 9 is closer to English, but it still isn't quite assimple as the more natural sentence Every human has two parents.That sentence could be expressed directly by the CG in Figure 10.

Figure 10. CG for Every human has two parents.

In English, the HasParent relation is normally expressedby the verb have combined with the noun parent.That noun belongs to a large class of role words,such as spouse, pilot, lawyer, assistant, pet, weed, crop, entrance,obstacle, or facility.Syntactically, those words resemble nouns like man, woman, dog,or tree; but semantically, they imply some relationshipsto external entities. In the ontology of the bookKnowledge Representation (Sowa 2000), the primitive relation Hasis used to form dyadic relations by combining with concept types thatrepresent roles. Figure 11 shows how the HasParent relationis defined in terms of the relation Has and the role type Parent.

Figure 11. Definition of the HasParent relation.

Figure 11 defines the HasParent relation as a synonym for a conceptualgraph that has two concepts designated as formal parameters:the symbol λ₁ marks the first parameter as a humanwho has a parent that is coreferent with another human, marked asthe second parameter by the symbol λ₂.It may be read The HasParent relation is defined as a relationbetween two humans; the first has a parent who is the second.With this definition, Figure 10 can be mapped to or from Figure 9.With appropriate definitions of sets and numbers, Figure 9 can be mappedto or from Figure 8, which can be mapped to or from the existential graphor any of the algebraic formulas in typed or untyped predicate calculus.To support equivalent definitions, RDF would require a tag such as or to mark a formal parameter.

In addition to the semantic primitives of Table 1,Peirce distinguished a context-dependent primitive, which he calledan indexical.In natural languages, indexicals are represented by pronouns,by deictic words such as this and that,and by noun phrases marked by the definite article the.In conceptual graphs, indexicals are marked by the # symbol.The concept [Cat], for example, represents some unspecifiedcat that happens to exist; but the concept [Cat: #] representsthe cat that was most recently mentioned in the current context.Peirce observed that proper names are also indexicals. Within thecontext of this article, the name Yojo may refer to a cator to Queequeg's ebony idol. On the Internet, it also refersto some Japanese people, to others who have adopted that wordas a nickname, and to an organization of young journalists.The ambiguity of names and their context dependencies are major concernsaddressed by the naming conventions of the Internet. Those conventionsare semiotic features that can be represented by metalevel typesand relations in conceptual graphs and other logic-based notations.

In summary, the algebraic notation for logic, which is popularwith mathematicians, is only one of an open-ended numberof semantically equivalent notations. The five semantic primitivesof Table 1 and the mechanisms for defining the other operatorsof first-order logic can be adapted to a wide variety of notations,including natural languages and the web-oriented notations of XML and RDF.

1. Logic can be and has been represented in a wide variety of graphicand linear notations of varying degrees of readability and suitabilityfor different kinds of applications. EGs and CGs are graphic examples,and the Knowledge Interchange Format (KIF) is an equivalent linear form.Other linear versions can be written with the syntactic conventionsof SQL, RDF, or even natural languages.

2. For better readability, it is possible to represent the logicaloperators in controlled natural languages, which use a subsetof the syntax and vocabulary of natural languages. Although the taskof translating unrestricted natural languages to any formal notationis still a research problem, it is much easier to translate conceptualgraphs and other formal notations to a stylized version of naturallanguage, such as the English translations of the CGs in this article.

3. Besides notation, logic has rules of definition and inference,which allow one representation to be translated to or fromother synonymous representations. Figures 6 through 10 can betranslated automatically to or from one another or the equivalentformulas in predicate calculus — providedthat an appropriate ontology has been defined.With its formally defined semantics, logic provides the meansfor generating semantically equivalent translationsto and from other languages with radically different syntax.

For better readability, any of the logical notations mentionedin this section can be translated to controlled natural languages.One important application, for example, is the generation of comments andhelp messages automatically from the implementation. Such translationswould guarantee that the comments and help would always be up to date,consistent with the implementation, and immediately availablein every supported national language.

4. Combining Logic with Ontology

Pure logic is ontologically neutral.It makes no presuppositions about what exists or may existin any domain or any language for talking about the domain.To represent knowledge about a specific domain, it must be supplementedwith an ontology that defines the categories of things in that domainand the terms that people use to talk about them.The ontology defines the words of a natural language,the predicates of predicate calculus, the concept and relation typesof conceptual graphs, the classes of an object-oriented language,or the tables and fields of a relational database. To illustratethe issues of defining an ontology, consider the conceptual graph inFigure 12, which represents the sentence Yojo is chasing a mouse.

Figure 12. CG for Yojo is chasing a mouse.

Figure 12 uses three concepts and two conceptual relations.The concept [Cat: Yojo] represents a cat named Yojo;[Chase] represents an instance of chasing; and[Mouse] represents a mouse. The conceptual relation(Agnt) indicates that Yojo is the agent of chasing,and (Thme) indicates that the mouse is the themeor the one that is being chased. The CG is logically equivalentto the following formula in typed predicate calculus:

(?x:Cat)(?y:Chase)(?z:Mouse)   (name(x,"Yojo") ∧ agnt(y,x) ∧ thme(y,z)).

This formula and the CG in Figure 12 introduce several ontologicalassumptions: there exist entities of types Cat, Chase, and Mouse;some entities have character strings as names; and Chase can be linkedto concepts of other entities by relations of type Agent and Theme.

The representation of actions by distinct concepts follows Peirce'sontology, which represents an action such as chasing with threedistinct entities: the one that is chasing, the one that is being chased,and the act of chasing itself. The relations (Agnt) and(Thme) are examples of the case relations orthematic roles of linguistics. Instead of Peirce's ontology,which is also called event semantics, some logicians wouldrepresent the verb is chasing by a single predicate,such as chases:

(?x:Cat)(?y:Mouse)   (name(x,"Yojo") ∧ chases(x,y)).

The ontology of this formula could also be used in a conceptual graph:

[Cat: Yojo]→(Chases)→[Mouse].

This CG, which is written in the linear notation for CGs,can be translated to Figure 12 by defining the relation(Chases) in terms of the concept [Chase]:

Chases ::=   [Animate: λ1]←(Agnt)←[Chase]→(Thme)→[MobileEntity: λ2].

With this definition of (Chases), the ontology of the previousCG can be translated to or from the ontology assumed in Figure 12.

Although the Chases relation allows shorter graphs and formulasthan the concept [Chase], it introduces other complexities intothe ontology. A general representation for tenses and modality,for example, would require a proliferation of relation types, such asHasChased, WillChase, and MustHaveBeenChasing.Furthermore, the dyadic relation chases(x,y) makesno provision for attaching adverbs and other modifiers to the verb.Figure 13 takes advantage of the more general representationto define the concept type Chase in terms of a graph for an animateagent (parameter #1) that is following a mobile entity (parameter #2)in a rapid manner.

Figure 13. Definition of the concept type Chase

Figure 13 is only a partial definition because it representsa necessary, but not a sufficient condition. Runnersin a race, for example, might be following one another rapidly,but only because they are pursuing a common goal. A more completedefinition must include the purpose, which might be different fordifferent senses of the word chase. Figure 14 defines one sense,called ChaseHunt, in which the purpose of the agent is to catchthe mobile entity that is being chased.

Figure 14. Definition of the concept type ChaseHunt

In Figure 14, the purpose relation (Purp) links the action toa situation that would occur upon the successful completion of the chase.According to Peirce, purpose is a triadic relation, of which two argumentsare shown explicitly in Figure 14. The implicit third argument is theagent of Chase, whose intention is to bring about the desired situation.That situation is nested inside a context box because its intentionalstatus is different from the context of the act of chasing. If the chaseis unsuccessful, the act of catching might never occur. Figure 15 definesanother concept type ChaseAway, in which the agent's purpose is notto catch the mobile entity, but to cause it to leave its current location.

Figure 15. Definition of the concept type ChaseAway

The ontology of situations and their representation in contexts is basedon Peirce's logic combined with ideas developed in artificial intelligence,linguistics, philosophy, and logic over the past 40 years (Sowa 2000).A context box may enclose modal or intentional situations,as in Figures 14 and 15, or it may enclose temporally or spatiallyseparated parts of a larger situation. In Figure 16,the large situation with its sequence of nested situationsrepresents the following passage in English:

At 10:17 UTC, there was a situation involving a cat named Yojo anda mouse. Yojo chased the mouse. Then he caught the mouse.Then he ate the head of the mouse.

These sentences show how indexicals are used to makecontext-dependent references. When new entities are first mentioned,they are introduced with the indefinite article, as in the phrasesa situation, a cat named Yojo, and a mouse.The two middle sentences refer to the mouse with the definite articlethe and to the cat with the name Yojo orthe pronoun he. In the last sentence, the headof the mouse, which had not been mentioned explicitly, is markedwith the definite article because the introduction of the mouseimplicitly introduces all of its expected parts.In Figure 16, the indexicals are marked with the # symbol:the pronoun he is represented as #he, and the definitearticle the is represented with the # symbol by itself.

Figure 16. Nested situations with unresolved indexicals

The large context box of Figure 16 encloses the entire situation,which occurred at the point in time (PTim) of 10:17 UCT. It containsconcept nodes that represent the cat Yojo, the mouse, and three nestedsituations connected by the (Next) relation. Before that CG canbe translated to predicate calculus, the indexicals must be resolvedto links or labels that explicitly show the coreferences.To avoid multiple line crossings, Figure 17 introducesthe coreference labels *x for Yojo and *yfor the mouse.Subsequent references use the same labels, but with the prefix? in the bound occcurrences of [?x]for Yojo and [?y] for the mouse.The # symbol in the concept [Head: #] of Figure 16 is erasedin Figure 17, since the head of a normal mouse is uniquely determinedwhen the mouse itself is identified.

Figure 17. Nested situations with indexicals resolved

After the indexicals have been resolved to coreference labels,Figure 17 can be translated to the following formulain typed predicate calculus:

(?s1:Situation)(pTim(s1,"10:17 UTC")   ∧ dscr(s1,      (?s1,s2,s3:Situation)(?x:Cat)(?y:Mouse)(name(x,"Yojo")         ∧ dscr(s2, (?u:Chase)(agnt(u,x) ∧ thme(u,y)))         ∧ dscr(s3, (?v:Catch)(agnt(v,x) ∧ thme(v,y)))         ∧ dscr(s4,            (?w:Eat)(?z:Head)(agnt(w,x) ∧ ptnt(w,z) ∧ part(y,w)))         ∧ next(s2,s3) ∧ next(s3,s4)))).

The description predicate dscr(s,p), whichcorresponds to the context boxes of Figure 16, is a metalevel relationbetween a situation s and a proposition p that describess. Figure 17 or its translation to predicate calculus could alsobe paraphrased in a version of controlled English that uses variablesto show coreference explicitly: At 10:17 UTC, there was a situation sinvolving a cat x named Yojo and a mouse y. In the situation s,x chased y; then x caught y; then x ate the head of y.

The contexts of conceptual graphs are based on Peirce's logicof existential graphs and his theory of indexicals.Yet the CG contexts happen to be isomorphic to the similarly nesteddiscourse representation structures (DRS), whichHans Kamp (1981a,b) developed for representing and resolvingindexicals in natural languages. When Kamp published his first versionof DRS, he was not aware of Peirce's graphs. When Sowa (1984) publishedhis book on conceptual graphs, he was not aware of Kamp's work.Yet the independently developed theories converged on semanticallyequivalent representations; therefore, Sowa and Way (1986) wereable to apply Kamp's techniques to conceptual graphs.Such convergence is common in science;Peirce and Frege, for example, started from very different assumptionsand converged on equivalent semantics for FOL, which 120 years later isstill the most widely used version of logic. Independently developed,but convergent theories that stand the test of time are a more reliablebasis for standards than the consensus of a committee.

5. Extracting Logic from Language

Since all combinations of the five primitives of Table 1can be expressed in every natural language, it is possible to representfirst-order logic in a subset of any natural language.Such a subset, called a stylized or controlled NL,can be read by anyone who can read the unrestricted NL.As examples, the English paraphrases of the CGs and formulasin this article represent a version of controlled English.With an appropriate selection of syntax rules, that subset could beformalized as a representation of FOL that would be semanticallyequivalent to any of the common notations for logic.

For most people, no training is needed to read a controlled NL,but some training is needed to write it. For computers, it is easyto translate a controlled NL to or from logic, but fully automatedunderstanding of unrestricted NL is still an unsolved research problem.To provide semiautomated tools for analyzing unrestricted language,Doug Skuce (1995, 1998, 2000) has designed an evolving seriesof knowledge extraction (KE) systems, which he called CODE,IKARUS, and DocKMan (Document-based Knowledge Management).The KE tools use a version of controlled English called ClearTalk,which is intelligible to both people and computers.As input, the KE tools take documents written in unrestricted NL,but they require assistance from a human editor to generateClearTalk as output. Once the ClearTalk has been edited and approved,further processing by the KE tools is fully automated. The ClearTalkstatements can either be stored in a knowledge base or be written asannotations to the original documents. Because of the way they'regenerated, the comments that people read are guaranteedto be logically equivalent to the computer implementation.

The oldest logic patterns expressed in controlled natural languageare the four types of statements used in Aristotle's systemof syllogisms. Each syllogistic rule combines a major premiseand a minor premise to draw a conclusion.Following are examples of the four sentence patterns:

1. Universal affirmative. Every employee is a person.

2. Particular affirmative. Some employees are customers.

3. Universal negative. No employee is a competitor.

4. Particular negative. Some customers are not employees.

These patterns and the syllogisms based on them are usedin many controlled language systems, including ClearTalk.For inheritance in expert systems and object-oriented systems,the major premise is a universal affirmative statementwith the verb is, and the minor premise is either a universalaffirmative or a particular affirmative statementwith is, has, or other verbs.For database and knowledge base constraints, the major premise isa universal negative statement that prohibits undesirable conjunctions,such as employee and competitor.

Other important logic patterns are the if-then rules usedin expert systems. In some rule-based systems, the controlled languageis about as English-like as COBOL, but others are much more natural.Attempto Controlled English (Fuchs et al. 1998; Schwitter 1998) isan example of a rich, but unambiguous language that uses a versionof Kamp's theory for resolving indexicals. Following are two ACE rulesused to specify operating procedures for a library database:

If a copy of a book is checked out to a borrower   and a staff member returns the copythen the copy is available.

If a staff member adds a copy of a book to the library   and no catalog entry of the book existsthen the staff member creates a catalog entry        that contains the author name of the book           and the title of the book           and the subject area of the book   and the staff member enters the id of the copy   and the copy is available.

Rules like these are translated automatically to theHorn-clause subset of FOL, which is the basis for Prologand many expert system languages. The subset of FOL consistingof Horn-clause rules plus Aristotelian syllogisms can be executedefficiently, but it is powerful enough to specify a Turing machine.

For database queries and constraints, natural language statementswith the full expressive power of FOL can be translated to SQL.Although many NL query systems have been developed,none of them have yet become commercially successful.The major stumbling block is the amount of effort requiredto define the vocabulary terms and map them to appropriate fieldsof the database. But if KE tools are used to design the database,the vocabulary needed for the query system can be generatedas a by-product of the design process. As an example,the RéCIT system (Rassinoux 1994; Rassinoux et al. 1998) usesKE tools to extract knowledge from medical documents writtenin English, French, or German and translates the resultsto a language-independent representation in conceptual graphs.The knowledge extraction process defines the appropriate vocabulary,specifies the database design, and adds new information to the database.The vocabulary generated by the KE process is sufficient for end usersto ask questions and get answers in any of the three languages.

Design and specification languages have multiple metalevels.As an example, the Unified Modeling Language has four levels: themetametalanguage defines the syntax and semantics of the UML notations;the metalanguage defines the general-purpose UML types;a systems analyst defines application types as instancesof the UML types; finally, the working data of an application programconsists of instances of the application types.To provide a unified view of all these levels, Olivier Gerbéand his colleagues at the DMR Consulting Group implemented design toolsthat use conceptual graphs as the representation language at every level(Gerbé et al. 1995, 1996, 1997, 1998, 2000).For his PhD dissertation, Gerbé developed an ontology for usingCGs as the metametalanguage for defining CGs themselves.He also applied it to other notations, including UML andthe Common KADS system for designing expert systems.Using that theory, Gerbé and his colleagues developedthe Method Repository System for defining, editing,and displaying the methods used by the DMR consultants.Internally, the knowledge base is stored in conceptual graphs,but externally, the graphs can be translated to web pages in eitherEnglish or French. About 200 business processes have been modeledin a total of 80,000 CGs. Since DMR is a Canadian company,the language-independent nature of CGs is important becauseit allows the specifications to be stored in the neutral CG form.Then any manager, systems analyst, or programmer can read themin his or her native language.

Translating an informal diagram to a formal notation of any kindis as difficult as translating unrestricted NL to executable programs.But it is much easier to translate a formal representation in any versionof logic to controlled natural languages, to various kinds of graphics,and to executable specifications.Walling Cyre and his students have developed KE tools for mappingboth the text and the diagrams from patent applications and similardocuments to conceptual graphs (Cyre et al. 1994, 1997, 1999).Then they implemented a scripting language for translating the CGsto circuit diagrams, block diagrams, and other graphic depictions.Their tools can also translate CGs to VHDL, a hardware design languageused to specify very high speed integrated circuits (VHSIC).

No single system discussed in this paper incorporates all the featuresdesired in a KE system, but the critical research has been done, andthe remaining work requires more development effort than pure research.Figure 18 shows the flow of information from documents to logicand then to documents or to various computational representations.The dotted arrow from documents to controlled languages requireshuman assistance. The solid arrows represent fully automatedtranslations that have been implemented in one or more systems.

Figure 18. Flow of information from documents to computer representations

For the KE tools, the unifying representation language is logic,which may be implemented in different subsets and notationsfor different tools. All the subsets, however, use the same vocabularyof natural-language terms, which map to the same ontology of conceptsand relations. From the user's point of view, a KE system communicatesin a subset of natural language, and the differences between tools appearto be task-related differences rather than differences in language.

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Copyright 2000, John F. Sowa

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