The Information Flow Framework: A Descriptive Category Metatheory

THE INF ORMA TION FLO W FRAMEW ORK: A DESCRIPTIVE CA TEGOR Y MET A TH EOR Y R OBER T E. KENT A BSTRACT . The Information Flow F ra mework (IFF) 1 is a descriptive categ ory meta theory currently under developmen t that provides an imp ortant practical application of catego r y theory . It is an exp eriment in foundatio ns, which follows a bo ttom-up a pproach to logica l de- scription. The IFF for ms the structural asp ect o f the IEEE P1 600.1 Standa rd Upp er Ontology (SUO) pro ject. The categ orical appro ach of the IFF provides a pr incipled framework for the mo dular design of ob ject-lev el on tologies . The IFF represents meta- logic, and a s suc h op erates at the structural lev el of ontologies. In the IFF, there is a pr ecise b oundary betw een the metalevel and the ob ject lev el. The mo dular archi- tecture of the IFF c o nsists of metalevels, namespaces and meta-ont olo g ies. There are four metalevels, Lower , Upp er , T o p a nd Ur , corres po nding to the set-theoretic distinc- tions be tw een the “ small”, the “la rge”, the “very large” and the “ generic”, r esp ectively . Each metalevel serv ices the lev els b elow by providing a metala nguage used to decla r e and axio matize those levels. Corresp o nding to the four metalevels are the four nested metalanguage s, meta - lower ⊒ met a - upper ⊒ meta - top ⊒ meta - ur , where each metalan- guage axiomatization includes sp ecia liz ation of the one immediately ab ove. Within each metalevel, the terminology is partitioned into names paces, and v arious namespaces ar e collected tog ether into meaning ful comp osites called meta- ontologies. The IFF contains meta-ontologies repr esenting category theory , informatio n ﬂow, formal concept ana lysis, institutions, multitudes, equatio na l logic, ﬁrst order logic, the common log ic standard, etc. All of the v arious meta-o nt olo gies in the IFF are anchored to the IFF metastack Set ⊆ Cls ⊆ C ol ⊆ Ur . The IFF development is la rgely driven by the principles of c onc ep tual warr ant , c ate gori c al design and institu tional lo gic . The main application of the IFF is institutional: the notion of ins titutions and their mo rphisms are b eing ax- iomatized in the upp er metalevels of the IFF, and the low er metalevel of the IFF has axiomatized v arious institutions in which sema nt ic integration has a natural expr e s sion. “ Philosop hy c annot b e c ome scientiﬁc al ly he althy without an immense te chnic al vo c abulary. We c an har d ly imagine our gr e at-gr andsons turning o ver the le aves of this dictionary with- out amusement over t he p aucity of wor ds with which their gr andsir es attempte d t o hand le metaphysics and lo gic. L ong b efor e that day, it wil l have b e c ome indisp ensably r e quisite, to o, that e ach o f these t erms should b e c onﬁne d to a single me a ning which, however br o ad, must b e fr e e fr om al l vagueness. Th is wil l involve a r evolution in terminolo gy; for in its pr esent c onditio n a philosophi c al thought of any pr e cisio n c an seldom b e expr esse d without lengthy explanations. ” — Charles Sanders Peirce, Collected P ap ers 8:16 9 2000 Mathema tics Sub ject Classiﬁcation: 18A15 . Key words and phrases: descriptive categ o ry metatheory , ontologies, metalangua ges, ins titutions. c  Rob er t E. Ken t, 200 4 . Permission to cop y for priv ate use granted. 1 Several years ago there was a discussion thr ead on the SUO email list, ent itled “comp osing ont olog ies using morphisms and colimits” and in volving Michael Uschold, John So wa, and others. This discussion was concerned with the question of the use of category theory (v ersus set theory) in the IFF pro ject as a foundation for knowledge representation and log ic. Michael Healy for warded this q uestion to the category ema il list. This pap e r can b e regarded as an ex tended answer to th is question. 1 2 1. Intro ducti on Recall reading the classic Cate gories for the Working Mathematician ( CWM ) [12] b y Saunders Mac Lane. This standar d reference oﬀers an intro duction to the basic concepts of category theory , suc h as the catego r ies 2 “category”, “functor”, “natural transformatio n” , “adjunction”, “monad” and “(co)limit”, and the relatio nships (some functional) “ob j ect”, “morphism” = “arro w” (terms a r e equal when they are synon yms), “source”, “target”, “equiv alent t o ”, “isomorphic to”, etc. C WM discusse s a collection of w ell-deﬁned concepts suc h as thes e, eac h b eing represen ted b y a word o r phrase in natural langua g e augmen ted b y mathematical no tation. In addition, CWM contains a collection of logical expressions used to repre sen t its meaning. The log ical expressions of CWM are built up fro m its categories and relationships using logical connectiv es and q uantiﬁe rs, or are expressed using comm utativ e diagrams. F or example, consider a page c hosen at random from CWM , suc h as page 78. Some of the categorical concepts encoun tered on t his page include : “free category”, “small g raph”, “functor”, “forgetf ul func tor”, “ U : Cat → Grph ” = “the forgetful functor from the category of small categories to the category of small graphs”, “morphism”, “unique map”, “category”, “natural isomorphism”, “ Set ” = “the category of small sets”, “function”, “ bijection”, “ natural”, “isomorphism”, “adjunction”, “bifunctor”, “ X op × A → Set ” = “a Set -v alued bifunctor”, “pair of ob jects”, “ hom-set”, etc. Some of the lo gical expressions encoun tered on this page include: “fo r all mo dules A , B , and C o v er a comm utativ e ring K , there is an isomorphism Hom( A ⊗ K B , C ) ∼ = Hom( A, B ⊗ K C )” and “for a ll A -morphisms k : a → a ′ and all X -morphisms h : x ′ → x t he dia grams A ( F x, a ) A ( F x, a ′ ) X ( x, G a ) X ( x, G a ′ ) A ( F x, k ) X ( x, G k ) φ x,a φ x,a ′ ✲ ✲ ❄ ❄ A ( F x, a ) A ( F x ′ , a ) X ( x, G a ) X ( x ′ , Ga ) A ( F h, a ) X ( h, Ga ) φ x,a φ x ′ ,a ✲ ✲ ❄ ❄ comm ute”. Some questions immediately arise. Can C WM b e formalized, either approx - imately or fully? Should it b e formalized? What meaning is captured by forma lizat io n? What meaning is missed? Should w e w ait for a more mature foundations of category theory b efore formalizing C WM ? W ould a formalization of CWM b e itself a candidate foundations of category theory? This pap er discusses an eﬀort called the Infor ma t io n Flo w F ramew ork (IFF) [6], whic h can b e view ed a s an appro ximate formalization of CWM , since a substan tial p ortion of CWM has already b een axiomatized in the IFF, but whic h was originally formulated to b e a descriptiv e category metatheory for ob ject-lev el on tologies. 2. Ontologi es Since the IFF is b eing oﬀered as a metatheory fo r ontologies, it is appropria te to ask the question “What is an on tology?”. The term “ontology” w as introduced in 1 7th cen tury 2 The philosophica l notion of category . After all, the ter m “catego ry” in “catego ry theory” w as pur- loined from Aristotle and Ka nt . 3 Virtual Pro duct Delivery Physical Pro duct Pro duct Customer Physical Lo cation Lo cation Order Price isa of to isa has isa has acco rding to ❅ ❅ ❅ ❘    ✠ ❅ ❅ ❅ ❘ ❄ ❅ ❅ ❅ ■    ✒ ❄ ❄ ✲ ✲ ✲ Figure 1 : An e-commerce o n tology frag ment philosoph y to represen t t he study of b eing, but had origins in Aristotle’s metaphy sics. In the 20th cen tury , the term “on tology” w as b orrow ed b y the art iﬁcial in telligence comm u- nit y to represen t a collection of things that exist. The meaning ha s since ev olv ed, and no w repres en ts conceptual mo deling and kno wledge engineering. The v ario us kinds of on- tologies form a sp ectrum according to forma lity , with catalogs and glossaries at one p ole, taxonomies in the middle, and formal axiomatic theories at the other p ole. One p opular deﬁnition is that “an o ntology is a formal, explicit speciﬁcation of a shared conceptual- ization. It is an abstract mo del of some phenomena in the w orld, explicitly represen ted as concepts, relationships and constraints, whic h is mac hine-readable and incorp orates the consensual know ledge of some comm unity .” Since it is an abstract mo del of some phenomena in the world, it is a seman tic conception. Because it uses concepts, relation- ships and constraints , it is logic-orien ted. Because it is ma chine-readable, it is formal and explicit. And since it incorp orates the consensual kno wledge of some comm unit y , it is shared and relative. F ig ure 1 illustrates a fragment of an E-commerce sc hema (o n tology) tak en from the pap er 3 [2], where concepts are represen ted as no des, relatio nships are r ep- resen ted as edges, and constraints are represen ted as parallel pairs o f edge paths, limits and copro ducts. W e distinguish b etw een ob ject-lev el and metalev el on tologies. An ob ject-lev el o n tol- ogy represen ts some a sp ect of the “real w orld”. Three lev els of ob ject-leve l ontology ha v e b een men tioned. Low er ob ject-lev el ( do main) ontologies represen t the conceptual structures of some communit y; e.g., phy sics, chemis try , biology , go v ernmen t, business , etc. O ne example of a domain on tology is a ph ysics ontology , consisting of the o rganiza- tion of the communit y of ph ysicists, a collection of ph ysical subtheories, suc h as gravit y , heat, motio n, electricit y , magnetism, . . . , and t echnic al terminology . Another example of 3 This pa per by C.N.G Da mpney and Michael Johnso n sug gests rega r ding ca teg ory theory as a meta - ontology , whic h ﬁts exactly the spirit of the Information Flow F r amework (IFF) pro ject. 4 a domain ontology is The Gene Ontolo gy (GO) , a n actual bio-ontology for the genetics comm unit y , consisting of the organization of the comm unit y of geneticists, subtheories, suc h as molecular-function, biological-pro cess, cellular-component, . . . , and tec hnical ter- minology . Middle ob ject-leve l on tologies help org anized do ma in on tologies b y axiomatiz- ing some abstract facet, suc h as the conceptual structures of organizatio ns, t he mo del of an en terprise, the nature of gov ernme nts , a template for educational institutions, the mathematical structures of ev en ts and pro cesse s, . . . . Upp er ob ject-level on tologies are limited to concepts that are meta, generic, abstract and philosophical. Upp er- lev el on- tologies contain the most general classiﬁcations of entities. Hence, they tend to b e domain indep enden t a nd p o ssess high r eusability . Examples of upp er-level ontologies include t he W o r dNet online lexical database, the Standard Upp er On tology ( SUO), the Cyc upp er on tology , the High P erformance Kno wledge Base pro ject (HPKB), the Suggested Upp er Merged Ontology (SUMO), . . . . Of course, t here may b e o v erlap b et w een domain a nd middle-lev el on tologies, and b et w een middle-lev el a nd upp er-leve l on tologies. And o f ten, upp er-lev el ontologies contain meta-infor ma t ion. A meta-lev el on tology is an on tolog y ab o ut ontologies — it represen ts some asp ect of the organization of ob ject-lev el ontologies. In this paper, w e iden tify meta-on tologies and metatheories. Ho w ev er, ob ject-lev el ontologies and theories are not iden tical, since ob ject- lev el ontologies can b e either p opulated or not. W e iden tify unp opulated ob ject-lev el on tologies with theories [a k a sc hemas], and w e iden tify p opula t ed ontologies with (lo cal) logics [ak a databases]. Unp opulated o n tologies hav e only type information. P opulated on tologies hav e b oth t yp e and instance information, plus the classiﬁc ation relationships b et w een these tw o kinds of things. This pap er is ab out a metatheory called the Infor- mation Flow F ramew ork (IFF ), whic h is descriptiv e and category- t heory- based. The fact that the IFF is a des criptiv e metatheory inv o lv es use of the constrain t called “ conceptual w arran t”. The fact that the IFF is a category-theory-based metatheory in v olv es use of the motiv ator called “the categorical design principle”. 3. Descri ptive Metatheor ies Distinctions are imp ortan t in on tology dev elopmen t. Tw o distinc tions in par t icular are imp ortant for the IFF: the monolithic v ersus mo dular distinction and the prescriptiv e v ersus descriptiv e distinction 4 . How ever, these distinctions are sometimes confused — although prescriptiv e approac hes ar e o f ten monolithic and descriptiv e approac hes ar e often modular , these t w o distinctions are conceptually diﬀeren t. A monolithic ontology is one-size-ﬁts-all. The monolithic approac h is not compatible with the need for contin ual revision and consistency che c king. The mo dular approa c h, whic h adv o cates the lattice and con text of theories, is very compatible with t hese needs. The monolithic-mo dular distinction is imp ortant for the maintenance of ob ject-leve l ontologies (see the discussion b elo w). Ho w ev er, in this section we ar e mainly interes ted in the presc riptive -descriptiv e distinction, whic h is imp ortan t for meta-lev el on tologies; that is, metatheories. 4 Compare the prescriptive-semantic distinctio n discussed in the pap er [2] b y Dampney and Johnson. 5 All natural languages are inheren tly dynamic living en tities. Ho w ev er, the dynamic gro wth of some languages suc h as English is greatly aided b y the w ay that it is cataloged. In 1755 the great English lexicographer a nd literary critic Samuel Johnson published his Dic tion a ry of the English L anguage [5]. This w ork set t he standard for all mo dern English dictionaries. Ho w ev er, this standard of lexicograph y is distinctly diﬀeren t from that used in the relat ed F renc h and Ita lia n lang uages. The linguistic purity of F renc h and Italian is to some exten t regulated b y the Acad ´ emie F ran¸ caise 5 and the Accadem ia della Crusca 6 , resp ectiv ely . These tw o b o dies were established “to prescrib e the use of the language” [19]. As the full title 7 of Jo hnson’s w ork implies, mo dern English dictionaries do not pr escrib e how the English language should b e used, but instead describ e how the language actually is used. The f o llo wing quota tion fr o m the preface to [5 ] expands on t his descriptiv e attitude. “ Every incr e ase of know le dge , whether r e al or fancie d, wil l pr o duc e new wor ds, o r c ombina- tion of wo r ds. When the mind is unchaine d fr om ne c essity, it wil l r ange after c o nvenienc e; when it is left at lar ge in the ﬁelds of sp e culation, it wil l shift opinions; as any cust om is disuse d, the wor ds that expr esse d it mu st p erish with it; as any opinion gr ows p op ular, it wil l innovate sp e e ch in the same pr op ortion as it alters pr actic e. ” An on tolog y is similar to a dictionary , but has greater detail and structure. Bo t h dic- tionaries and on tologies come in t w o basic philosophies: prescriptiv e or descriptiv e. A descriptiv e dictionar y or ontology describes actual usage. Most mo dern dictionaries are descriptiv e with the Oxford English Dictionary ( O ED) as a leading example. As discussed b elo w, the Information Flo w F ramew ork (IFF) follow s a similar descriptiv e philosoph y . In this pap er w e are mainly interes ted in the ana logy b et w een dictionaries and meta - on tologies. First, there is a correspondence b etw ee n the builders of dictionaries and those of meta-ontologies. Corresp onding to the lexicographers 8 , suc h as Samue l Johnson or James Murray of the OED, who create dictionaries, are the ontologicians ( ma t hemati- cians, particularly category-theorists) who create meta-on tologies. Second, there is a corresp ondence b et w een the so ur c e material used by dictionaries and t ha t used b y meta- on tologies. As discussed in [19], the en tries placed and describ ed in dictionaries hav e three sources: (i) terms b o rro w ed from other dictionaries, (ii) new terms used to express concepts in works of literature, and (iii) new terms used to express concepts in ev eryda y sp eec h. By analogy , the concepts axiomatized in meta-on tolog ies or ig inate fro m three sources: (i) terms b orro w ed from other meta-ontologies, (ii) new metalev el terms used to express concepts in ob ject-lev el on tologies, and (iii) new metalev el terms used to express the conceptual structure of a communit y . F or b oth dictionaries a nd on tologies, the sec- ond source is most imp or t an t. Lexicographers use w orks of lit era t ure as the main source for dictionaries, whereas on tologicians use t he meta, generic and abstract terminology of 5 1635– present [ www.acade mie-franc aise.fr ] 6 1582– present [ www.accad emiadella crusca.it ] 7 A Dictionary of the English L anguage: In Which t he Wor d s ar e De duc e d f r om Their Originals, and Il lust r ate d in Their Diﬀer ent Signiﬁc a tions by Examples fr om the Best Writers [5] 8 Compare Samuel Johnson’s deﬁnition [5] of a lexico grapher a s a “a writer of dictionaries, a harmless drudge, that busies himself in tracing the original, and detailing the signiﬁcatio n o f words”. 6 ob ject-lev el o n tologies as the main source for meta-ontologies. F or meta-ontologies, this constraining pro cess is kno wn as “conceptual warran t” and is discussed in mo r e detail b elo w. Third, there is a corresp ondence b et w een the sour c e cr e ators used b y dictionaries and those used b y meta-on tologies. Corresp onding to the literar y ﬁgures who originate new terms in w orks of literature are the kno wledge engineers who o riginate and use new metalev el terms in ob ject-level ontologies. 4. T h e Sta nd a rd Upp er On tol ogy (SUO) The IEEE P1600.1 Standard Upp er On tology (SUO) pro ject [16], whic h op erates under the auspices of the IEEE Standards Asso ciation (IEEE-SA), aims to sp ecify an upp er on tology that will pro vide a structure and a set of general concepts up on which ob ject- lev el ontologies can b e constructed. According to the IEEE-SA, “ a standard is a published do cumen t that sets out sp eciﬁcations and pro cedures designed to ensure that a material, pro duct, metho d, or service meets its purp ose and consisten tly p erforms to its in tended use” 9 . It is also understo o d that suc h a standard ma y inv o lve some agreemen t ab o ut the conformance to implemen tations of the standard. As understo o d by the SUO w orking group, an upp er o n tology is limited to concepts tha t are meta, generic, abstract and philosophical. It is an ticipated that the SUO will b e useful in data interoperability , information searc h and retriev a l, automa t ed infere ncing, and natura l lang ua ge pro cessing. The SUO follow s Rob ert’s Rules of Order — t here are v arious discussions on the email list, after w hic h the c hair can p o ssibly call fo r a v ote. The SUO has b een activ e for ab out ﬁv e y ears. During this time, it has approv ed sev eral resolutions. In chronological order, these are describ ed as follows . 2001 August: Informatio n Flow F ramew ork (IFF ) . The SUO IFF pro ject 10 w as the ﬁrst prop osal passed by t he SUO working g roup. It is a descriptiv e category metatheory that r epresen ts the struc tural asp ect of the SUO. The IFF is in v olv ed with concepts that a re meta 11 , generic a nd abstract. The philosoph y , approac h, arc hitecture and metho ds of the IFF are described in this pap er ( see Sections 1 , 2 and 3 fo r the philosoph y , Sections 5 and 6 for the approach, Sections 7, 8 , 9 and 10 for the arc hitecture and Section 11 f or the metho ds). As described in this pap er, the IF F 9 According to the dictionar y [1 3], a standar d has the sense of “ s omething establis hed by authority , custom, or g eneral consent as a mo del or example”, “something set up a nd established b y author it y as a r ule for the meas ure of quan tity , w eight, e x ten t, v alue, or q ua lity”, or “ a structure built for or serving as a ba se or suppor t” . A standard “applies to a n y deﬁnite rule, pr inc iple, or measure established b y authority”. 10 [ suo.ieee .org/IFF/ ] 11 According to the dictionar y [13], the term ‘meta-’ is a preﬁx mea ning a “more highly orga nized or more comprehensive form o f ” something, and is “used with the name of a discipline to de s ignate a new but related discipline desig ned to deal cr itically with the or ig inal one”. Examples include: a metathe ory is a theory whose s ub ject matter is another theo r y; a metalanguage is a la nguage used to talk a bo ut other languages ; and metamathematics is the ﬁeld of study concer ned with the formal str uc tur e and prop erties of mathema tical systems. 7 migh t b e called the Standard Meta Ontology ( SMO). In its a pplicational asp ect, the IFF pro ject uses tec hniques and concepts from the ﬁelds of information ﬂo w ([1]), formal concept analysis ([3]), category theory ( [12]) and institutions ([4]). 2003 June: Suggested Upp er Merged On tology (SUMO). The SUO SUMO pro ject 12 is sp onsored and dev eloped by the T eknow ledge Corp oratio n 13 . The SUMO prop osal initially failed its v ote, principally due to a dv o cacy of a monolithic philosophy for on tology dev elopme nt. Ab out this time, there w as a vigorous debate o n the SUO email lis t ab o ut the mono lithic–mo dular distinction (See Sec tion 11 for a discussion ab out mo dularit y .). The mo dular philosoph y for on tology dev elopmen t is adv o cated b y the SUO Lattice of Theories pro j ect. With passage o f this prop osal, there is an expresse d inten t tha t SUO SUMO working group will collab orate with t he SUO Lattice of Theories w orking group. 2003 June: Lat t ice of Theories (LOT). The SUO Lattice of Theories pro ject in tends t o dev elop a standard for on tology sp eciﬁcation and registration. The standard will b e based on the con tributions of other SUO candidate pro jects. The standard will sp ecify an on tology registry , such as the metada t a registries sp eciﬁed by the In- ternational Organization for Standardization (ISO) standard ISO/IEC IS 11 179-3, but with extensions that are needed in order to deﬁne a nd r elat e on tolog ies. The on tology registry shall b e orga nized as a collection o f mo dules, related in a gener- alization/sp ecialization hierarc h y . Eac h mo dule shall consist of a theory together with do cumen tation and other metadata. The theory shall consist of a xioms and deﬁnitions stated in a logic-based lang ua ge, suc h as those in t he Common Logic (CL) framework. The standard shall include the sp eciﬁcation of a metho dology for testing the theory part of a ny mo dule for consistency , relating theories to one another in the generalization/sp ecialization hierarc h y , and com bining t w o o r more theories to deriv e a new theory t ha t is larger and mor e specialized than the theo- ries from whic h it w as deriv ed (See Section 11 for a discussion and mathematical form ulation of the “ la ttice of theories” construction.). 2003 Octob er: OpenCyc O ntology . The Op enCyc on tology 14 , whic h con tains a b out 5,000 concepts and 50,000 axioms (ak a rules), is t he o p en source ve rsion of the Cyc on tology . The Cyc ontology , whic h con tains ab out 300,0 00 concepts a nd 3 million axioms, is a large a nd general knowle dge base whose inten ded use is for commonsense reasoning. It has sp en t ov er 600 p erson-y ears in a dev elopmen t eﬀort o v er t he past 17 y ears. The SUO w orking g roup w ould lik e to disman tle the O p enCyc on tology in to meaningful comp o nen ts and reassem ble them within the ﬂexible and dynamically c hangeable structure of the “lattice of theories” fra mework (Again, see 12 [ suo.ieee .org/SUO/ SUMO/ ] 13 [ www.tekn owledge.c om ] 14 [ www.open cyc.org ] 8 Section 11 for a discussion and mathematical f orm ulation of the “latt ice of theories” construction.). 2003 Octob er: SUO 4D On tology . The SUO 4D On tology pro j ect 15 in tends to dev elop an on tolog y based on the 4-dimensional paradigm. A starting p oint fo r the de- v elopmen t o f this on tology is con tained in the ISO standard ISO/FD IS 15926-2, a lifecycle integration of pro cess plant data including oil and gas production facilities. The approach will b e to dev elop the on tology as reusable comp o nen ts within the institutional asp ect of the IFF and to dev elop mappings to other ontologies within this framew ork. 2004 Ma y: Multi-Sour ce O ntology (MSO). The SUO MSO is based at the W ebKB- 2 knowle dge serv er 16 , a know ledge serv er that p ermits W eb users to browse and up date priv ate know ledge bases on their mac hines and a la rge shared kno wledge base on the serv er machine. The on tology of the shared kno wledge base is currently an in tegration of v arious top-lev el on tologies (e.g. So w a, D olce, the Lifecycle Inte gratio n Sc hema, the Natural Semantic Metalangua g e, O WL, DAML+OIL, KIF and the Dublin Core) and a lexical on tology deriv ed from a n extension and correction of the noun-related part of W ordNet 1.7. 5. T h e IFF Desig n Guidelines The Informat io n Flow F ramew ork (IF F) (F igure 2) is a descriptiv e category metatheory that is in tended t o represen t the structural asp ect of the SUO. In the dev elopmen t of the IFF, certain guidelines hav e prov en to b e very imp orta n t. The se are all predicated on the g oal of building a category-theoretic metatheory for ob ject-lev el o ntologies. Ini- tially , this metatheory w as designed to represen t ﬁrst order lo gic, its languages, theories, mo del-theoretic structures a nd (lo cal) log ics, including satisfaction and ﬁbra tions based at languages. Later, this metatheory incorp orated the theory of institutions o f G oguen and Burstall. The most imp o rtan t guideline in the dev elopmen t o f the IFF — what w e migh t call the meta-g uideline — is to follow the intuitions of the w orking category-theorist. Suc h in tuitions represen t naiv e category theory 17 . In practice, w e initially form ulate any IFF mo dule as a set-theoretic axiomatizatio n using a ﬁrst order expression. Then, b y elimi- nating quantiﬁers and lo gical connectiv es, w e attempt to mo v e, morph or tr a nsform this set-theoretic a xiomatization tow ard a catego r y-theoretic axiomatization. A conv enie nt rule-of-thum b is to “kee p it simple”. F r o m the foundatio nal standp o in t, this means that w e start with no assumptions at all. How eve r, in view of the Cantor dia gonal argument, as a ﬁrst step w e assume a slender hierar ch y called the IFF metastac k. 15 [ suo.ieee .org/SUO/ SUO-4D/ ] 16 [ www.webk b.org ] 17 The mea ning o f naive here is not pejorative. It means [1 3] primitiv e, natural, in tuitiv e, ﬁrs t- formed, primary , not evolved or elemen tal. 9 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ C O R E generic very lar ge lar ge smal l metal ev el ob ject lev el Figure 2 : The IF F Architec ture (iconic v ersion) The IFF metastack , whic h is the core hierarch y o f the IFF, is represen ted a s the cen- tral a xis or spindle in Figure 2. The con ten t o f the metastack is the axiomatization for sets and functions (and fo r conv enie nce, binary relations) in four core mo dules at four diﬀeren t set-theoretic lev els, the “small” in the low er metalev el, the “large” in the upp er metalev el, the “v ery large” in the to p metalev el and the “generic” in the ur metalev el. Curren tly , the axiomatization for t he metastack has a “conical” shape 18 , with an axiom- atization for categories in tro duced in the ur metalev el, an a xiomat izat io n for ﬁnite limits in tro duced in the top metalev el, an axiomatization for exp onents and ﬁnite colimits in- tro duced in the upp er metalev el, and an axiomatization for sub ob ject classiﬁers 19 and general limits/colimits in tro duced in the low er metalev el 20 . This eﬀectiv ely distributes top os axioms ov er the four metalev els. The categories Set ⊆ Cls ⊆ Col ⊆ Ur in the IFF metastac k, which corresp ond to the four metalev els men tioned ab ov e, are axiomatized in the low er core ( IFF-LC O ), the upp er core ( IFF -UCO ), the top core ( IF F-TCO ) and the ur ( IFF -UR ) meta-ontologies, respectiv ely . Here, all inclusions preserv e composition and iden tities (are functorial), the ﬁrst t w o preserv e ﬁnite limits, and the ﬁrst preserv e s the 18 The conica l shape of this ax io matization seems natural so far, even with the axiomatization for 2-catego ries. How ever, with the prop er place men t of the axioma tization for institutions in the upp er metalevel, any strong a ppea l for applicatio n of the categ orical design pr inciple to this institutional ax- iomatization would call for ex po nent s in the top metalevel. This w ould give the metasta ck axiomatizatio n more of a “cylindrica l” shap e. 19 This axiomatiza tio n has not b een r ealized so far, sinc e it is not needed, and hence, at this time would violate the principle of c o nceptual warrant (see b elow). How ever, we anticipate its need. 20 Assuming a co nic a l shap e, an abstr act metastack M = h M 4 ⊇ M 3 ⊇ M 2 ⊇ M 1 i is deﬁne to be a category M 4 at the 4 th metalevel (Ur) r estricted to a ﬁnit ely complete category M 3 at the 3 rd metalevel (T op) restr icted to a car tesian closed catego ry M 2 at the 2 nd metalevel (Uppe r) r estricted to a topos M 1 at the 1 st metalevel (Lower). 10 C k +1 C k ✻ ✁ ✂ subset A k +1 A k B k +1 B k f k +1 f k ✻ ✁ ✂ ✻ ✁ ✂ ✲ ✲ restriction c ommutative diagr am A k +1 r k +1 R k +1 B k A k R k r k B k +1 π A k +1 π B k +1 π A k π B k ✻ ✁ ✂ ✻ ✁ ✂ ✻ ✁ ✂ ✛ ✛ ✲ ✲ abridgmen t multipul lb ack diagr am Figure 3 : The three f undamen tal generic relations Cartesian closed structure 21 . These preserv a tion pro p erties represen t the fa ct that the axiomatization at eac h met- alev el includes the sp ecialization of the axiomatization at the next higher metalev el. Sp ecialization is a tec hnical term here, which means the exact adaptatio n of the core terminology and axiomatization at one metalev el to the next lo w er metalev el b y use of the three fundamen tal generic r elat io ns of subset, (function) restriction and (binary rela- tion) abridgmen t t ha t are diagra mmatically describ ed in Figure 3, where k is a particular metalev el, k + 1 is the next higher metalev el, the set C k is a subset of the set C k +1 , the function f k is a restriction of the function f k +1 , and the relat io n r k with extent R k is an abridgmen t o f the relation r k +1 with exten t R k +1 . T able 1 gives a brief g lance at the IFF co de that represen ts sp ecialization of t he cor e set notions. Another guideline is that quantiﬁcation should b e o v er a sp eciﬁc collection. This ca- pabilit y is realized in the IFF metas hell (see Section 10 on metalangua g es), and hence all metalang uages, b y the abbreviation called restricted quan tiﬁcation. A third g uideline in the IFF dev elopmen t, is to use functions wherev er p ossible. This encourages the elim- ination of quantiﬁers. A ﬁnal guideline is tha t as a general rule, univers es are k ept a w a y from p ow er op erators (and hence also exp onen t op erators). This is part o f the injunction to av oid con tradictions (paradox es) due to Can tor (a nd Russell etc.). In pra ctice, w e ha v e little or no need for univ erses — we use the term “thing” in the Ur and T o p cor e on tologies, but this has minimal use. 6. T h e IFF Developmen t P hases The IFF has gone through tw o phases of dev elopmen t. The IFF Category Theory (meta) On tology ( IFF-CA T ) is distinguished by b eing the ﬁrst meta-ontology axiomatized during 21 Assuming a conical shap e, a morphism of abstr act met astacks H : M → M ′ consists of a functor (morphism of categ ories) H 4 : M 4 → M 4 at the 4 th metalevel, restricted to a mor phism of ﬁnitely complete categorie s H 3 : M 3 → M 3 at the 3 rd metalevel, restricted to a morphism of cartesian closed catego ries H 2 : M 2 → M 2 at the 2 nd metalevel, restricted to a (geometr ic) morphism of top o s es H 1 : M 1 → M 1 at the 1 st metalevel. 11 IFF-UR : “ An Ur-obje ct r epr esents the notion of a gene ri c set. ” (forall (?x (object ?x)) (thing ?x)) IFF-TC O : “ A c ol le ction r epr esents the notion of a very lar ge set. Ther e is an Ur-obje ct of al l c ol le ctions. Any c ol lection is a UR- obje ct. The Ur-obje ct of al l c ol le ctions is not it self a c ol le ction. ” (ur:obje ct collect ion) (forall (?c (collec tion ?c)) (ur:obje ct ?c)) (not (collec tion collecti on)) IFF-UC O : “ A class r epr esents the notion of a lar ge set. Ther e is a co l le ction of al l classes. Classes ar e ma inly use d in IF F to sp e cify the obje ct and mo r- phism c ol le ctions of lar ge c ate gories such as the c ate gory of al l smal l set s and functions Set and the c ate gory of al l smal l classiﬁc ations and infomorphisms Clsn . Every class is a c ol le ction. The co l le ction of al l classes is not itself a class. ” (vlrg.se t:collection class) (ur:subo bject class vlrg.set: collection) (not (class class)) T able 1 : Co de f or t he IF F Metastack the ﬁrst phase of dev elopmen t. It indicated necessary ing redien ts in the core a xiomat iza- tion now kno wn as the metastack . The I FF-CAT ha s a xiomatizations for larg e categories, large functors, large natura l transformations, large a djunctions, large mona ds, and lim- its/colimits. Perhaps IFF-CA T is in some sense an en try-lev el axiomatization for the cate- gory of a ll categories. A non-starter for the remaining part of the ﬁrst phase was a top os axiomatization. This receiv ed ob jections f rom the SUO working group, in part due to its lac k of suppo r t b y motiv ating ex amples. Rejection of the top os axiomatization prompted the idea of conceptual warran t. The second phase of the IFF was designed b otto m-up and follow ed conceptual w arran t as its guideline. The most impor t a n t metalog ic termi- nology incorp orated in the second pha se references the v arious concepts related to ﬁnite limits. F or example, the desire to axiomatize comp osition of class functions in t he upp er metalev el requires the pullbac k concept at the top metalev el. The IFF is no w w ell within its third phase of dev elopmen t, whic h in v olv es conv ersion of t he truth construction to the theory of institutions and reorganization of the metastac k using the “adj unctive axioma- tization” tec hniq ue illustrated in La wv ere and Rosebrugh [10]. A ma j or goal for the third phase is to complete the categorical design principle at the lo w er metaleve l and to initiate it at the upp er metalev el. In order to accomplish this, w e ha v e extended comp osition to a ll lev els of the metastac k and w e hav e introduced the impo rtan t exp onen t metalogic concept in the upp er core. A fo urt h phase is en visioned in the f uture, when the concepts of ﬁbrations and indexed categories will b e axiomatized. These are imp ortant for the 12 axiomatizations of institutions and ﬁbring logics. During the IFF dev elopmen t, f our concepts hav e ev en tually emerged as imp ortant. One of these concepts, the IFF metastac k, which originated in phase three of IFF deve lop- men t, was dis cussed ab ov e. The other three concepts are principles for IFF dev elopmen t. In ch ronolo g ical order these are ( 1 ) the principle of conceptual w arrant, (2) the principle of categorical design and (3) the principle of institutional logic. Conceptual w arran t re- stricts the introduction o f upp er metalev el terminology , whereas categorical design forces the in tro duction o f this terminology , principally in the core. Principle: conceptual w arran t. All IFF terminology should require con- ceptual warran t f o r their existence : any term that app ears in (and is a x- iomatized b y) a metalanguage should reference a concept needed in a lo w er metalev el or ob ject leve l axiomatization. The principle of conceptual w arran t originat ed in phase one of IFF deve lopmen t. The ter- minology app earing in an y standardization meta-on tology will exert authority . Because of t his, in selecting whic h terminology to sp ecify in the IFF, w e utilize the notion of “con- ceptual w arrant”. W arrant means evidence for, or tok en of, author izatio n. Conceptual w arran t is an adaptation of the libra r ianship notion of literary w arran t. According to the Library of Congress, its collections serv e as the literary w arrant (i.e., the lit era t ure on whic h the con trolled v o cabular y is based) for the Library of Congress sub ject headings system. In the same fashion, the ob ject-lev el and low er metalev el terminology of the IFF serv es as the conceptual w arrant for the IFF upp er metalev el axiomat izat io n. Principle: categorical design. The design of a mo dule at an y particular metalev el should a dhere to the prop ert y that its axiomatic represen tation is strictly category-theoretic: All axioms use terms from the metalanguage at that metalev el. No axioms use explicit logical notat ion: No v ariables, quan- tiﬁcation (‘ ∀ ’, ‘ ∃ ’) or logical connectiv e s (‘ ∧ ’, ‘ ∨ ’, ‘ ¬ ’, ‘ ⇒ ’,‘ ⇔ ’) are used. The principle of categorical design originated in phas e tw o of IFF dev elopmen t. The goal of this principle has b een to simplify the IFF axiomatization — ﬁrst order expression w ould b e reduced to t erm- r ewriting. The p eripheral (non-core) mo dules in the low er IFF metalev el hav e the tripartit e form: outer c ategory name space, inner ob ject and morphism namespaces (see Section 9 for more discussion on this and Figure 7 fo r a visualization). The outer namespace fully conforms to the categorical design principle. The inner names- paces conform to it to a great exten t (80– 90%). The categorical design principle was originally expres sed for the lo w er metalev el (“the small”), but with comp osition extended up w ard this principle also seems a ppropriate fo r the upp er metalev el (“t he large”); in particular, it see ms appropriate for axiomatizing the IFF Category T heory (meta) On tol- ogy ( IF F-CAT ). F or example, T able 2 contains co de f or the concept of a (large) category , whic h fully conforms to t he categorical design principle. 13 IFF-CA T : “ A (lar ge) c ate gory c an be thought of as a sp e cial kind of gr aph — a gra ph with monoidal pr op ert ies. It co nsists of an underlying gr aph, a c omp osition g r aph morp hism and an identity gr aph morphism, b oth with an identity obje ct function. ” (vlrg.se t:collection category ) (vlrg.ft n:function graph) (vlrg.ft n:function underl ying) (= underlyi ng graph) (= (vlrg.ftn :source graph) category) (= (vlrg.ftn :target graph) lrg.gph.o bj:graph) (vlrg.ft n:function graph- pair) (= (vlrg.ftn :source graph-pai r) catego ry) (= (vlrg.ftn :target graph-pai r) lrg.gp h.obj:multipl iable-pair) (= graph-pai r (vlrg.l im.pbk.obj:pa iring [graph graph])) (vlrg.ft n:function mu) (= (vlrg.ftn :source mu) category) (= (vlrg.ftn :target mu) lrg.gph.m or:2-cell) (= (vlrg.ftn :composition [mu lrg.gph .mor:source]) (vlrg.ft n:composition [graph- pair lrg.gph. obj:multipli cation])) (= (vlrg.ftn :composition [mu lrg.gph .mor:target]) graph) (vlrg.ft n:function eta) (= (vlrg.ftn :source eta) category ) (= (vlrg.ftn :target eta) lrg.gph. mor:2-cell) (= (vlrg.ftn :composition [eta lrg.gph.mo r:source]) (vlrg.ft n:composition (vlrg.f tn:compositi on [graph lrg.gph.o bj:object]) lrg.gph. obj:unit])) (= (vlrg.ftn :composition [eta lrg.gph.mo r:target]) graph) T able 2 : Co de f or ( la rge) categories 14 Principle: in stitutional logic 22 . All logics used in the IFF application (see Section 11) should b e fo rm ulated as institutio ns. The principle of institutional logic originated in phase three of IFF dev elopmen t. The theories of information ﬂow a nd formal concept analysis (and hence, eﬀec tive ly the theory of institutions) ha v e b een use d througho ut the IFF dev elopme nt. This use has cente red on the “tr ut h construction”. The truth construction is institutional [9], consisting o f a classiﬁcation functor clsn : Sign → CLSN o r a concept lat t ice functor clg : Sign → CLG , where Sign is the category of signatures, and CLSN and CLG are the equiv alent categories [7] of (large) clas siﬁcations and (large) concept lattices, respectiv ely . The truth classiﬁcation w as discussed in Barwise and Seligman [1] 23 . In t he IFF dev elopmen t, the truth concept lattice, which was deﬁned to b e the concept lattice of the truth classiﬁcation, its equiv alen t intens ional asp ect of closed t heories, or its equiv alen t construction of theories under en tailmen t order, w as promoted b y the IFF as the prop er r epresen tation of t he “lattice of theories” no tion adv o cated b y the SUO working g r o up. As noted in [14], truth is not dy adic b et w een models and sen tences, but triadic b et w een mo dels, sen tences and signatures. This corresp onds t o the contex tual dep endency in the semiotics of Charles Sanders P eirce, whic h uses signs, o b jects and in terpretan ts. In formal concept analysis triadic conce pt lattices [11] hav e b een used to formalize this. Ho w ev er, the represe ntation of con texts as the logics of institutions has greater a dv a ntages. 7. T h e IFF Ar c hitect ure The IFF arc hitecture (Figure 4) consists of metalev els, namespaces and meta-o n tologies. Within eac h lev el, the terminology is part it io ned in to names paces. The nu mber of names- paces and the conten t ma y v ary ov er time: new namespaces ma y b e cre ated or o ld names- paces ma y b e deprecated, and new terminology and axiomatization within any particular namespace may c hange ( new ve rsions). In additio n, within eac h lev el, v arious names- paces are collected together in to meaningful comp osites called meta-on tolog ies. A t an y particular metaleve l, these meta-ontologies cov er all the namespaces at that leve l, but they ma y ov erlap. The n um b er of meta-o ntologies and the conten t o f an y meta-ontology ma y v ary ov er time: new meta-on tologies may b e created or old meta-on tologies may b e deprecated, and new namespaces within an y particular meta-on tology ma y c hange (new v ersions). T able 3 presen ts a list of IFF meta-ontologies organized by metalev el. All URLs in this table should b e preﬁxed with the string ‘ ht tp://s uo.iee e.org/IFF/ ’. The IFF terminology is managed in terms of namespace preﬁxes — eac h names pace is giv en a unique preﬁx (with p erhaps a few synon yms) in order to a v oid clash of terminol- ogy 24 . The arc hitecture of t he IFF namespace mec hanis m is ﬂat — namespace preﬁxe s 22 Suggested to the author b y Jose ph Goguen (per sonal communication). 23 See ex a mple 4.6 on pag e 71. 24 F or example, the ter m ‘ m orphis m ’ is introduced in the co ntexts of mo dels and theo ries for ﬁrst order logic to represent tw o distinct, but analogous , concepts. When these concepts need to be us e d in other 15 Ur: IFF-UR [ 30 terms, 7 pages] “generic” (category axioms) metastac k/UR.pdf T op: IFF-TC O [ 300+ terms, 60 pages] “v ery large” (collections; ﬁnite li mit axioms) metastac k/TCO.pdf IFF-2CAT (basic 2-category theory) metaleve l/top/ontolog y/2-category/version20041004.pdf Upp er: IFF-UCO [ 500+ terms , 100+ pages] “large” (classes; exp onen ts, ﬁnite l imits and colim its) metastac k/UCO.pdf metaleve l/upper/ontol ogy/core/version20020102.pdf IFF-CAT [ 220+ terms, 53 pages] (basic category theory) metaleve l/upper/ontol ogy/category-theory/version20020102.pdf IFF-CLSN [280+ terms, 78 pages] (basic i nformation ﬂow and form al concept analysis) metaleve l/upper/ontol ogy/classification/version20020102.pdf IFF-INS (rudimen tary institution theory) work-in- progress/INS/ version20031002.pdf work-in- progress/INS/ version20041014.pdf Lo w er: IFF-LCO [many , man y terms, 150+ pages] “the small ” (sets; quartets; arbitrary limits and colimits) metaleve l/lower/ontol ogy/core/version20020515.pdf metaleve l/lower/names pace/set/version20030402.pdf IFF-CL [ 100 terms, 36 pages] (common logic) metaleve l/lower/names pace/scl/version20040505-obj.pdf IFF-ONT [ 600+ terms, 194 pages, Language, Theory , Mo del, Logic namespaces] (logic and ontology) (non traditional FOL) (old v ersion) metaleve l/lower/ontol ogy/ontology/version20021205.htm IFF-OO (logic and ontology) (non traditional FOL) (new v ersion) work-in- progress/#IFF -OO IFF-FOL [ man y , many terms, 150+ pages] (logic and ontology) (traditional FOL) metaleve l/lower/ontol ogy/fol/version20040101.html T able 3 : IFF (meta) O n tologies 16 metalevel object level set cat gph dbl-ca t 2-cat ins fol ur = 4 core vlrg = 3 core category graph double category 2-category lrg = 2 core category graph double category 2-category insti tution sml = 1 core category graph double category 2-category institution . . . ﬁrst order logic obj = 0 SUO Cyc SUMO Wo rdNet SENSUS Hol es Gene Botany Onto l ingua Enterprise e-commerce Government Education H PK B Semantic W eb . . . Figure 4 : The IF F Architec ture (detailed v ersion) are like tags: b y using namespace preﬁxes the complete IFF terminolog y is the disjoin t union o f the terminology in the individual IFF namespaces. The IFF arc hitecture can b e thought of a s a t w o dimensional structure (F ig ure 4) consisting of metalev els, whic h are partitioned in to top-leve l 25 namespaces represen ting basic concepts suc h as category (‘ cat ’), graph (‘ gph ’), or institution (‘ ins ’). The v arious lev els are inde xed by the natural n um b ers ‘ 0 ’, ‘ 1 ’, ‘ 2 ’, ‘ 3 ’, ‘ 4 ’, ‘ 5 ’, . . . , or for the ﬁrst ﬁv e lev els b y their natural language correlates ‘ obj ’, ‘ sml ’, ‘ lrg ’, ‘ vlrg ’ and ‘ u r ’, starting with the o b j ect leve l indexed by ‘ 0 ’. Ov erall, v ario us namespaces ma y hav e the same na me, since they represen t the same basic concept a t diﬀeren t metalev els. F or eac h basic concept the namespace axiomati- zation at a pa rticular metalev el is in t w o parts, one generic and the other speciﬁc: the generic par t is the sp ecialization of the axiomatization just ab ov e it in the hierarch y; these axiomatizatio ns rely heavily up on the sub collection, restriction and abridgmen t re- lations for sets, functions and binary relations, respectiv ely (Figure 3); the sp eciﬁc pa r t is an extension of the basic concept; it is a strictly new axiomatization. T o lo cate any namespace one can use its lev el-concept pair. F or example, the namespace axiomatizing v ery large categories w ould b e denoted by ‘ vlrg.cat ’ or ‘ 3.cat ’. Sub-namespaces (fo ot- note 25) (not a t t o p-lev el) will need further qualiﬁcation. F or example, the sub-namespace of small graph morphisms w ould b e denoted by ‘ sml.gph. mor ’. It is assum ed that eac h basic concept has a particular metalev el that is in common use. These are underlined for the particular concepts in Figure 4. F or suc h common use namespaces, the leve l not a tion need not b e used. F or example, the namespace that axiomatizes large categories w ould b e denoted b y ‘ l rg.cat ’ o r ‘ 2. cat ’, but more simply by ‘ ca t ’. The namespace mec hanism has b een made ba c kw ard compatible, by allo wing sp ecial namespace preﬁx notat io n that is equiv alen t to the general fo r ma t just described. F or example, the namespace axiomatizing large categories w ould b e denoted by the general contexts, the namespa ce pr eﬁxes ‘ fol.mod. mor ’ a nd ‘ fol.th.m or ’ co uld b e used to distinguish them, th us resulting in the distinct repre s ent ations ‘ fol. mod.m or:mor phism ’ and ‘ f ol.th. mor:mo rphism ’. 25 This do es no t refer to the vertical dimension of the metalevel structure in Figur e 4, but ins tea d to an implicit third dimension o f the archit ecture. 17 Kernel (objects, morph isms, rela- tions and sub ordinanc y) Kernel (collections, functions, re- lations and subordinancy) Finite Limits Kernel (classes and functions) Finite Limits Finite Colimits Exp on ents IFF-UR IFF-TOP IFF-UCO Figure 5 : Core Arc hitecture (iconic v ersion) IFF-UR Mono Epi Iso Sub Obj Mo r Obj Mo r × Obj Mo r Rel Obj = ≤ ⊥ ∼ = ⌊ ⊳ λ δ γ ❄ ✻ ❄ ✲ ι = ✛ ✲ ✛ π 0 ρ π 1 ∂ 0 1 ∂ 1 ❄ ✻ ❄ µ 0 ◦ µ 1 ❄ ❄ ❄ o 0 ǫ o 1 ❄ ❄ ❄ IFF-TOP Spn ⊆ Ftn 2 ⊇ Spn op (-,-) [-,-] pbk ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❲ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✎ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❲ Inj Surj Bij Sub Col Ftn Col Ftn × Col Ftn Rel Col Col 2 = ≤ ⊥ ∼ = ⌊ ⊳ ✛ ∪ , ∩ , × , + λ δ γ ❄ ✻ ❄ ✲ ι = ✛ ✲ ✛ π 0 ρ π 1 ∂ 0 1 ∂ 1 ❄ ✻ ❄ µ 0 ◦ µ 1 ❄ ❄ ❄ ι 0 ǫ ι 1 ❄ ❄ ❄ Figure 6 : Core Arc hitecture (detailed v ersion) preﬁxes ‘ cat ’, ‘ lrg.cat ’ or ‘ 2.cat ’ o r b y the sp ecial preﬁx ‘ CAT ’ that w as used in earlier v ersions of the IFF-CAT axiomatization. Hence , fo r an y basic concept, b oth the general preﬁx and the common lev el need to b e declared, and for any namespace a xiomat izat io n the sp ecial preﬁxes also need to b e declared. The preﬁxes denoting sub-namespaces should be compatible with the top-leve l preﬁxes; for example, in t he upp er metalev el, the general preﬁx for the namespace of classes w ould b e ‘ lrg.se t ’ or ‘ 2.set ’ with the sp ecial preﬁx ‘ SET ’, whereas the general preﬁx fo r the sub-namespace of class pullbac ks would b e ‘ lrg.se t.lim.pb k ’ with the special preﬁx ‘ S ET.LIM.P BK ’. As these examples illustrate, only the lo w er case is needed fo r the g eneral namespace preﬁx notation. 18 Set F unction Relation ‘ thing’ Obj . = ‘ object’ ≤ . = ‘ subobject’ ⊥ . = ‘ disjoint’ ∼ = . = ‘ isomorphic ’ ⌊ . = ‘ restrictio n’ ⊳ . = ‘ abridgment ’ Mor . = ‘ morphism’ ∂ 0 , ∂ 1 . = ‘ source’, ‘target’ ‘ restrictio n’ ρ . = ‘ mor2rel’ Mono . = ‘ monomorphi sm’ Mor × Obj Mor . = ‘ morphism µ 0 , µ 1 . = ‘ morphism0’, ‘morphism1’ Epi . = ‘ epimorphis m’ -morphis m’ ◦ , 1 . = ‘ composition’, ‘identity’ Iso . = ‘ isomorphis m’ Rel . = ‘ relation’ o 0 , o 1 . = ‘ object0’, ‘object1’ ‘ abridgment ’ ǫ . = ‘ extent’ π 0 , π 1 . = ‘ projection0’, ‘projection 1’ Sub . = ‘ subordinat e’ λ , γ . = ‘ lesser’, ‘greater’ ι , δ . = ‘ inclusion’, ‘reflex’ T able 4: IF F-UR corresp ondences 8. T h e IFF Met a stack The left sides of Fig ures 5 a nd 6 illustrate the ar c hitecture f or the IFF Ur (meta) On tology IFF-UR , a tin y ontology at the v ery top of t he metastac k, the IFF core metalev el hierar- c h y . Its purp ose is to pro vide an in terface b etw ee n the IFF logical metashell and o t her metalev el axiomatizations. Principally , it do es this by servicing the top metalev el. A design constraint is that “things are opaque”; that is, the detailed state of aﬀa irs “inside” an ything is unknow n. Suc h detail is to b e pro vided b y a using on tology , suc h as the IFF T op Core (meta) On tology ( IFF-TCO ), the IF F 2-Category (meta) On tology ( IFF-2CAT ), and the future IFF Double-Category (meta) On tology ( IFF- DBLCAT ). The IF F-UR con tri- bution to the meta -ur me talangua ge is a simple framew ork — the IFF-T CO , IFF-2CAT and IFF-DCAT ﬁll it out. T a ble 4 show s v a rious corresp ondences b etw ee n the elemen ts on the left side of Figure 6 a nd the terminolog y of IFF-UR (the IFF-UR -term ‘ thing ’ is not repre- sen ted in Figure 6). The IFF-UR terminology consis ts of only 30 terms: 6 generic sets; 16 generic functions; and 8 g eneric relations, consisting of 3 binary endorelation on ob jects, 1 binary a nd 3 unary relations on mo r phisms, and 1 binary relation o n relations. The sub ob ject, restriction and abridgment binary relations a re imp o rtan t for the preserv ation prop erties of the metastack . The middle of F igure 5 and the right side of Figure 6 illustrate the arc hitecture for the IFF T op Core (meta) On tology IFF-TCO , w hic h is situated in the top metalev el — the highest lev el of the IFF metas tack other than Ur. It contains rudime ntary (fundamen tal) namespaces fo r collections, functions, relations and ﬁnite limits. The IFF-TCO pro vides an adequate foundation f or r epresen ting ontologies in general and for deﬁning other metalev el on tologies in particular. The fact that the IFF-T CO includes the sp ecialization of the IFF-UR is eviden t fro m the isomorphic em b edding o f the left side in to the rig ht side of Figure 6. The IFF-U R follow s Mac Lane’s b eginning a xiomat izat io n for category theory [12] in tha t it intro duces terminology and prov ides a n axiomatizatio n fo r this terminology , but it do es not giv e a formal interpretation using set theory — it only gives an inf ormal, 19 in tuitiv e in terpretation. The IFF-TC O initiates such a formal in terpretation. The IFF Upp er and Lo w er Core ( meta) On tologies, IFF -UCO and I FF-LCO , complete suc h a formal in terpretation. The IF F-TCO axiomatization con tains 317 terms (276 concepts or non-iden tical terms), partitioned according to whether the term is a basic term (collections, relations or func- tions), a diagra m term or a limit term. Ignor ing the 10 terms used to designate the four indexing collections, there are a tot a l of 30 7 terms part it io ned into 58 basic terms designating 55 basic concepts dealing with collections, (partial) functions and relations, 105 terms designating 97 concepts of ﬁnite diagrams, and 144 terms designating 120 con- cepts of ﬁnite limits pro p er. Although the IF F-TCO axiomatizates ﬁnite limits, the curren t v ersion do es not represen t ﬁniteness, but instead explicitly represen ts the sev eral ﬁnite shap es ne eded. A future v ersion of the IFF- TCO ma y exp erimen t with Dedek ind’s abstract deﬁnition of ﬁniteness [10]. T erminology ha s b een placed in the IFF T op Core On tology only wh en it is needed in t he IFF upper metalev el. All upp er metalev el on tologies impo rt and use, either directly or indirectly , the IFF T op Core On tology . This includes the upp er core, upp er graph and upp er catego ry theory meta-on tolog ies. In the third phase, a revised v ersion of the IFF Upp er Core (meta) On tology ( IFF-UC O ) will b e designed . The right side of Figure 5 illustrates the architecture for the I FF-TCO . This axiomatization will b e in a djunctiv e form. The IFF-UCO pla ys a central ro le in the IFF a xiomatization — it is the most referenced on tolog y at the metalev el. The IFF-UCO k ernel consists of the axiomatization for classes and functions. At the upp er metalev el the axiomatizat io n for relations, whic h includes order- theoretic concepts, has branc hed oﬀ into an ontology in its o wn right. The k ernel and ﬁnite limits namespace axiomatizations are the sp ecializations of those in the I FF-TCO , and as suc h rely hea vily up on the sub collection, restriction and abridg men t relations for collections, functions and binary relations, resp ectiv ely (Figure 3). The ﬁnite colimits na mespace is a categorical dual to the ﬁnite limits namespace. The exponents namespace axiomatizatio n is new. 9. T h e IFF Low er Met alevel Eac h lo w er-lev el meta-on tology in the IFF is concen trated around a single c ategory (with its asso ciated functors, natural tra nsfor ma t io ns and monads), and hence the format of eac h low er- lev el meta-on tology has a sp eciﬁc category-theoretic format consisting of an “outer” category namespace and t w o “inner” namespaces for ob jects and morphisms. In Figure 7 w e illustrate this format with the category T rm-Lang of term languages a nd term language morphisms 26 , which is axiomatized in the term language namespace trm.lang 26 A term language is simple — it cons ists of a set of (indexing ) v ariables, a set o f function types or symbols, a nd an arity function fro m function sym b ols to indicia (subsets of v ariables). A ter m lang uage morphism is a lso simple — it consis ts of a v ariable bijection and a function b e t ween function symbol sets, whic h form a set quartet b etw een so urce/tar get arity functions (preserves a r ity). W e could hav e ignored v ariable s , indicia and arity for languages, and just used natural num b ers and v alence instead, with la ng uage morphis ms require d to pr eserve v a lence. In fact, the classic Lawv ere categor y do es this — it uses natural n um b ers as ob jects, in pla ce of indicia, and uses term sequences in place of term tup les. 20 trm.lang catego ry namespace trm.la ng.ob j object namespace trm.la ng.mo r mo rphism na mespace Figure 7 : Low er Core F ormat of the IFF T erm (meta) On tology IF F-TRM . The category namespace o f tr m.lang is com- pletely compliant with t he categorical-design principle. This is illustrated in T able 5, whic h lists the logical co de for the cat ego ry T rm-Lang , the f unctor ftn : T rm - Lang → Set , the arity na tural transformat io n # : ftn ⇒ va r ◦ ℘ : T rm - Lang → Set , and the Lawv ere construction l a w : T rm - La ng → Cat 27 . The “outer” category namespace of trm.lang uses terminology from the meta-upp er metalanguage, plus the low er metalev el t erminolo g y from t he ob ject and mor phism namespaces of trm .lang . The latter are only partially (80–90%) complian t with the categorical-design principle. Part of the goal f or phase 3 is to bring these “ inner” namespaces to w ard 100% compliance. The basic a rc hitecture of term languages is illustrated in Figure 8, whic h consists of three sub-diagrams — the term sub-diagram (upp er left), the term tuple sub-diag r a m (lo w er left) and the La wv ere sub-diagram (righ t). The term sub-diagram illustrates the ﬁxp oin t solutio n for terms, the term tuple sub-diagram illustrates the em b edding struc- tures for term tuples, a nd t he La wv ere sub-diagram illustrates the La wv ere category la w ( L ) of a term la nguage L , whic h has indicia as ob jects, term tuples as morphisms, and substitution as comp osition. The La wv ere construction is a collection o f (small) categories and functors indexed b y term languag es and term language morphisms. Abstractly , the La wv ere construction is a (small) category ob ject in the (large) category of functors and natural transformations b et w een the categories T rm-Lang and Set . In Figure 8, the nodes represen t basic functors from T rm-La ng to Set , and the edges represen t natural transfor- mations b et w een these basic functors. There are ﬁv e simple functors var , ftn , case , trm and tpl , f or v a r iables, function sym b ols, v ariable cases, terms and term tuples, resp ec- How ever, we us ed v ariables for a t least four reasons. (1) Standar ds suc h as the common logic standard (CL) use a set of v ariable s as part of their lexicons. (2) W e did not wan t to r ely upon sequences and a natural nu mbers a xiomatization. (3) The use of v ariable-type pairs for ar guments in function declaration is c o mmon in progra mming languages. (4) W e wan t the ex pression op era tor, which requires v ariables and arities a s part of the formulation of ex pr essions, to be a mo nad. It is very e asy to add natural num b ers, v alence and ter m sequences, th us prese nting tr aditional ﬁr s t or der logic. 27 The functor ftn and the natural transformation # are explicitly represented in Figure 8, wherea s T rm-Lang is the am bient categ ory , and la w is represented by the La wvere sub-diagr am. 21 trm.la ng : “ Ther e is a (lar ge) c ate gory T rm-Lang of term languages. The obje ct class of T rm-Lang is t he class of al l term languages. The morphism class of T rm-Lang is the class of al l term language morphisms. Comp osition i n T rm- Lang is c omp osition of te rm language morphisms. ” (cat:cat egory languag e) (= (cat:obje ct langua ge) trm.la ng.obj:objec t) (= (cat:morp hism language ) trm.lang .mor:morphis m) (= (cat:sour ce langua ge) trm.la ng.mor:sourc e) (= (cat:targ et langua ge) trm.la ng.mor:targe t) (= (cat:comp osable langua ge) trm.la ng.mor:compo sable) (= (cat:comp osition language) trm.lang.m or:compositio n) (= (cat:iden tity language ) trm.lang .mor:identit y) trm.la ng : “ Ther e is a functi on symb ol functor fr om t he (lar ge) c ate gory of term languages to the (la r ge) c ate gory of sets f tn : T rm - Lang → Set . Ther e is an arity na tur al tr a nsformation # : f tn ⇒ var ◦ ℘ : T rm - Lang → Set , whose L th c omp onent for any t erm language L is the arity funct i on for L . ” (func:fu nctor functio n) (= (func:sou rce function) languag e) (= (func:tar get function) set:set ) (= (func:obj ect function) trm.lan g.obj:functi on) (= (func:mor phism functio n) trm.lan g.mor:functi on) (nat:nat ural-transfor mation function-ari ty) (= (nat:sour ce-category function- arity) langua ge) (= (nat:targ et-category function- arity) set:se t) (= (nat:sour ce-functor function-a rity) functio n) (= (nat:targ et-functor function-a rity) indicia ) (= (nat:comp onent functio n-arity) trm.lang .obj:functio n-arity) trm.la ng : “ The L awver e c onstruction is a functor la w : T rm - Lang → Cat fr om the (lar ge) c ate gory of term languages to the (lar ge) c ate gory of smal l c ate gories: for any t erm language L , law ( L ) is a smal l c ate gory with c opr o ducts and for any term language morphism f : L 1 → L 2 , law ( f ) : law ( L 1 ) → law ( L 2 ) is a functor b etwe en smal l c ate gories that pr eserves c opr o du cts. ” (func:fu nctor lawvere ) (= (func:sou rce lawvere) language ) (= (func:tar get lawvere) sml.cat: category) (= (func:obj ect lawvere) trm.lang .obj.tpl:law vere) (= (func:mor phism lawvere ) trm.lang .mor.tpl:law vere) T able 5 : Co de f or trm-lang 22 Term case trm ftn ⊗ tpl trm trm ⊗ tpl elem ⇒ subst ⇐ subst ⇐ = ǫ ⊗ 1 tpl ⇓ Tuple case ⊇ ftn va r va r ◦ ℘ va r × trm trm tpl va r ◦ ℘ π 0 ⇐ π 1 ⇒ p roj ⇒ ⇒ indic ǫ ⇐ ⇐ # § ⇐ # ⇒ { - } ⇓ { - } ⇓ # ⇓ La wvere va r ◦ ℘ = va r ◦ ℘ tpl tpl ⊗ tpl tpl va r ◦ ℘ tpl va r ◦ ℘ va r ◦ ℘ # ⇑ § ⇑ 0 th ⇐ 1 st ⇒ § ⇐ # ⇒ § ⇓ ◦ ⇓ # ⇓ = 1 ⇑ = ǫ = function as term embedding # = function/term/tuple arity § = tuple i ndex Figure 8: T erm Lang uage Architec ture: Set -v a lued f unctors and natura l transformations on the catego ry T rm-Lang tiv ely . The ﬁrst three are basic and the last t w o are inductiv ely deﬁned. Ba sed on these, there are ﬁv e comp osite functors: var ◦ ℘ , var × trm , ftn ⊗ tpl , trm ⊗ tpl , tpl ⊗ tpl , for in- dicia (v ariable su bsets), indexed terms (v ariable-term pairs), substitutable function-tuple pairs, substitutable term-tuple pairs and comp osable tuple-t uple pairs, resp ectiv ely . The ⊗ sym b ol refers to a matc hed Cartesian pro duct — the arity of the ﬁrst (function, term or t uple) matc hes the index of the second (tuple). 10. The IFF Meta langua ges T able 6 lists selec ted examples of IFF terminology org anized along metalev els. In gen- eral, the terminology intro duced at an y leve l uses the terminology in the same or higher metalev els. O b j ect lev el on tologies use low er metalev el terminology and functionality . F or example, t he t erminology and functionality introduced and a xiomat ized in t he lo w er metalev el IFF theory namespace ( trm ), w hic h is part of the IFF First Order Logic (meta) On tology ( IFF-FO L ), could b e used to organize and axiomat ize a n E-commerce o n tology in the ob ject lev el. Low er metalev el namespaces and meta-ontologies use upp er lev el termi- nology and functionality . F or example , the terminology and functionalit y introduced and axiomatized in the upp er metalev el IFF Category Theory (meta) On tology ( IFF-CAT ) is used to organize and axiomatize the low er metalev el IFF la nguage na mespace ( lang ) t hat is part of the IFF-FOL meta-on tology . Upp er metalev el namespaces and meta-ontologies use top metalev el terminology and functionality . F or example, the terminology and func- tionalit y in tro duced and axiomatized in the top metalev el IFF T op Core (meta) On tology ( IFF-TCO ) is used to organize and axiomatize the IF F-CAT meta- o n tology . There are thousands of terms in the IFF. T erms are divided into tw o classes, whic h w e can call “ usable terms” and “supp orting terms”. An IFF term, whic h is deﬁned in a part icular namespace on a particular metalev el, is a usa b l e term when it is used by at 23 Co re P eriphery metashel l ‘ and’, ‘or’, ‘implies’ , ‘iff’, ‘forall’ , ‘not’, ‘forall’ , ‘exists ’ meta-ur ‘ objec t’, ‘morphism’, ‘rel ation’, ‘subobje ct’, ‘restric tion’, ‘abridg ment’, ‘composi tion’, ‘identity’ meta-top ‘ collecti on’, ‘subcoll ection’, ‘fun ction’, ‘composi tion’, ‘identity’ , ‘restric tion’, ‘relatio n’, ‘abridgme nt’ ‘ collection -pair’, ‘parallel -pair’, ‘opspan’ , ‘binary- product’, ‘equali zer’, ‘pullbac k’ meta-upp er ‘ class’, ‘functio n’, ‘relation ’, ‘subclas s’, ‘restrict ion’, ‘abridgm ent’, ‘composi tion’, ‘identity’ ‘ category’, ‘functor’, ‘natural -transformati on’, ‘adjunction’, ‘monad’, ‘graph’, ‘partial- order’, ‘total-o rder’, ‘equivalen ce-relation’, ‘ unary-func tion’, ‘binary-fu nction’, ‘curry’, ‘hom’, ‘exponent’ ‘ classifica tion’, ‘infomorph ism’, ‘concept ’, ‘concept-l attice’, ‘bond’ meta-low er ‘ set’, ‘function’ , ‘relatio n’, ‘subs et’, ‘restric tion’, ‘abridgmen t’, ‘composit ion’, ‘identit y’ ‘ entity’, ‘relation’, ‘arity’, ‘language’, ‘theory’ , ‘logic’, ‘language- morphism’, ‘theory- morphism’, ‘logic-mor phism’ ‘ diagram’, ‘colimit’, ‘direct-flo w’, ‘inverse -flow’ ob ject leve l ‘ person’, ‘company’ , ‘employ’, ‘salary’, ‘date’, ‘real-number’ , ... T able 6 : Example IFF T erms ‘ abridgment ’ ‘ antisymmet ric’ ‘ bijection’ ‘ binary coprodu ct’ ‘ binary interse ction’ ‘ binary product ’ ‘ binary union’ ‘ bond’ ‘ bonding pair’ ‘ categorica l equival ence’ ‘ category’ ‘ class’ ‘ classifica tion’ ‘ cocone’ ‘ coequalize r’ ‘ colimit’ ‘ colimit injection ’ ‘ collection’ ‘ compositio n’ ‘ concept lattice’ ‘ cone’ ‘ constant function ’ ‘ coproduct’ ‘ currying’ ‘ diagram’ ‘ disjoint’ ‘ domain’ ‘ empty’ ‘ epimorphis m’ ‘ equalizer’ ‘ equivalence relation’ ‘ evaluatio n’ ‘ exponent’ ‘ function’ ‘ functional relation ’ ‘ functor’ ‘ hom’ ‘ hypergraph ’ ‘ identity’ ‘ inclusion ’ ‘ infomorphi sm’ ‘ initial’ ‘ injection’ ‘ institutio n’ ‘ institutio n morphis m’ ‘ inver se currying’ ‘ isomorphic ’ ‘ isomorphi sm’ ‘ language’ ‘ limit’ ‘ limit projectio n’ ‘ mediator’ ‘ monad’ ‘ monomorphis m’ ‘ mor2rel’ ‘ morp hism’ ‘ natural transform ation’ ‘ nothing’ ‘ null’ ‘ object’ ‘ one’ ‘ opspan’ ‘ order’ ‘ packing’ ‘ pair’ ‘ parallel pair’ ‘ partial funct ion’ ‘ partial order’ ‘ pfn2ftn’ ‘ pfn2rel’ ‘ product’ ‘ pro duct with’ ‘ pullback’ ‘ pushout’ ‘ reflexive’ ‘ relation’ ‘ restrictio n’ ‘ set’ ‘ signature’ ‘ source’ ‘ span’ ‘ spangraph’ ‘ subclass’ ‘ subcollectio n’ ‘ subordinat ion’ ‘ subset’ ‘ surjection’ ‘ symmetric ’ ‘ target’ ‘ terminal’ ‘ ternary copro duct’ ‘ ternary product ’ ‘ thing’ ‘ three’ ‘ total order’ ‘ total relation ’ ‘ transitive ’ ‘ triple’ ‘ two’ ‘ unique functi on’ ‘ unique morphis m’ ‘ unit’ ‘ unpacking’ ‘ vocabulary’ ‘ zero’ T able 7: Basic IFF T erms 24 vlrg.s et : “ Two (very lar ge) set s (aka c ol le ctions) ar e isom orphic when they ar e c onne cte d by a (very lar ge) bije ction. ” (forall (?c0 (collectio n ?c0) ?c1 (collectio n ?c1)) (and (iff (isomorph ic ?c0 ?c1) (exists (?f (vlrg.f tn:bijection ?f)) (and (= (source ?f) ?c0) (= (target ?f) ?c1)))) T able 8 : metashell co de least one other term in another namespace on that metalev el or on a lev el b elow that one. An IFF term, whic h is deﬁned in a particular namespace on a par ticular metalev el, is a supp orting term when it is used b y another term in that same namespace. Because of conceptual warran t, all IFF terms should b e usable or supp orting , and p erhaps b oth. Hence, all IFF terms are necessary , but most IFF terms are “conceptually derive d”. This means that they are a conceptual comp osite of tw o or more ba sic IFF terms. An IFF term is a b asic IFF term when it is not the conceptual comp osite of tw o or more other IFF terms. T able 7 lists some basic IFF terms. There are four IFF metalev els: low er, upp er, top and ur. Eac h metalev el services the lev els b elo w: the ur metalev el services the top metalev el, the ur and top metalev els service the upp er me talev el, the ur, top and upp er metalev els service the low e r metaleve l, and the ur, top, upp er and low e r metalev els service the ob ject-lev el. There is one metalanguage based at each metalev el. An y metalangua g e can b e used by the meta-ontologies and on tologies at a ll lo w er lev els. These metalang ua ges can b e deﬁned in terms of their top- do wn nesting. Eac h metalev el k has an asso ciated metala ng uage meta ( k ), whic h is the union of “old” terminolog y , “ sp ecialization” terminology and “new” terminology . The old terminology is the terminology of the metalanguage meta ( k +1) asso ciated with the metalev el k + 1 immediately ab o v e, the sp ecialization terminology is the terminology of t he sp ecialization of the k + 1 axiomatization to the k th metalev el, and the new terminology is from the non-sp ecialization axiomatizatio n in the v arious meta- on tologies at metalev el k . Pro ceeding in a top-dow n fashion, the metalanguage hierarch y starts with a logical shell called the metashell . The metashe ll enables a lisp-like ﬁrst-order expression using connectiv es and restricted quan tiﬁcation. T able 8 con tains an example expression from the collection namespace of the IF F-TCO meta-ontology that uses m uc h of the me tashell : it uses b oth univ ersal and existen tial restricted quantiﬁcation, plus the conjunction and equiv alence connectiv es . In order to deﬁne the metalanguage for a particular metaleve l, the metashel l language is t ypically used in tw o w a ys. First, the t yping of the terminol- ogy for a metalanguage uses the metalanguage at the next higher metalev el, where the metashel l expression is restricted to mem bership f o r sets, application for functions, and holds for binary relations [this is strictly category-theoretic]. Second, the meaning of the terminology for a metalang uage uses the metalanguage at the next higher metaleve l [this includes unrestricted meta shell (quan tiﬁers, connectiv es, etc.)]. In addition t o the metashe ll , there results a hierarch y of metalang ua ges (Figure 9) 25 metash ell meta-u r meta-t op meta-u pper meta-l ower Figure 9: The IFF Metalanguage Hierarc h y co ordinated with the the metastack. Eac h metalanguage includes the ones ab o v e, with the metas hell ﬁrst order logical expression a s innermost: the metash ell is con tained within the m eta (4) = meta -ur core metalanguage, whic h is con tained within the meta (3) = meta- top me talangua g e, whic h is contained within the meta (2) = meta-uppe r metalan- guage, whic h is con tained within the meta (1) = meta-lower metalanguage. An y ob ject- lev el langua g e uses t he meta- lower metalanguage in it s a xiomat izat io n. The meta-ur metalanguage is sp ecial — it axiomatizes the en tire metastac k. Visualizing the meta- language nesting in terms of the metastac k, the meta-ur metalangua ge acts as a cen tral spindle orien ted v ertically (see Figure 2), around whic h the metalanguages for the three other metalev els ar e ringed. 11. The IFF Appl icati on Similar to softw are dev elopmen t, there are tw o approaches to the standardization of con- ceptual kno wledge (ontology dev elopmen t): the monolit hic a pproac h and the mo dular approac h. But standardization strongly aﬀects main tenance, and in the main tenance of on tologies [8 ] the modular approac h t o conceptual kno wledge represen tation ha s adv an- tages o v er the monolithic. In a dditio n, imp orta n t distributed en vironmen ts, suc h as the Seman tic W orld-Wide W eb and organizational intranets, represen t t heir information in a mo dular fa shion with mu ltiple ontologies and sc hemas. F or these reasons, the SUO adv o cates the mo dular approac h in their lattice of theories (LOT) pro ject. The goal of the LOT pro ject is to create a f r amew ork “ whic h can supp o rt an op en-ended n um b er of theories organized in a la t t ice to g ether with systematic metalev el tec hniques for mo ving from one to ano ther, for testing their adequacy for an y giv en problem, and for mixing, matc hing, combining, and transforming them to whatev er form is appropriate for what- ev er problem an y one is trying to solv e” (John F. So w a, SUO archiv es , see also [18]). A c hallenge to the mo dular approac h is the need for a seman tic integration of o n tologies. The main application o f the IFF is institutiona l, inv olving o ntology dev elopmen t and seman tic in tegratio n. T o b e lo gic independent, the IFF represen ts and manipulates o n to- logical structures within the metatheory of institutions [4], whic h allo ws t he formalization, 26     ❅ ❅   ❅ ❅ ❅ ❅ F OL Language (with equalit y) F OL Language Equational Language (universal algeb ra) Exp ression Language T erm Language Figure 10: The IFF-FOL Mo dule Hierarch y represen tation, impleme ntation and translation of log ics. The metat heory of institutions is b eing a xiomatized in the upp er metalev els of the IFF. Its institutional approach t o logical seman tics prov ides a principled framew ork for the mo dular design of ob ject-lev el on tologies in general, and for the “lattice o f theories” approach to ontological organiza- tion in particular. Within institutions, the la t t ice of theories is the ﬁbring or indexing of the category of theories ov er the category of languages (ak a signatures). Institutions formally express seman tic in tegration as a n on tological fusion pro cess [9 ]: align ontologies within a diagram of theories and fuse aligned on tologies via the colimit of this dia gram. The IFF has w ork-in-pro gress axiomat izatio ns f or the institutions and connecting institu- tion morphisms of information ﬂo w ( IF ), eq uational log ic ( EQL ), order-sorted ﬁrst order logic ( F OL ) and the common logic standard ( CL ). Although semantic in tegration can b e represen ted within a n arbitrary institution [9], seman tic in tegration in FOL-related institutions is of particular in terest to the SUO pro ject. The arc hitecture of F OL inv olv es a fundamen tal exomorphic-endomorphic distinction 28 . The exomorphic asp ect of FOL is needed for a principled formulation of the “ lattice of theories” concept. The IFF First Order Logic (meta) On tology ( IF F-FOL ) giv es a traditional axiomatiza- tion for F OL. The IFF-FO L is em b edded within a very mo dular arc hitecture as illustrated in Figure 10. This describ es sev eral institutions a nd institution morphisms, including EQN the institution for equational logic and F OL the institution for ﬁrst order lo gic (with equality). The cen tral bifurcation in Figure 10 is b et w een terms and expre ssions. F OL languages are the pullbac k of expression languages and term languages ov er (bi- jections of ) v ariables. FOL languages with equality are the pullbac k of FOL lang uages 28 Exomor phic r e fers to the arc hitecture outside the notion o f an FOL la ng uage, whereas endo morphic refers to the ar chitecture within an FOL langua ge. The exomorphic asp ect of FOL is concerned with the institutional approach to F OL; this is a n indexed categ ory a rchitecture bas ed o n substitution along language morphisms. The endomorphic aspe c t of F OL, which is formalized in v arious treatments of categoric al logic (see [15] and [17]), is concerned with the h yp erdo ctrine appr oach to F OL; this is an indexed ca tegory architecture based o n substitution along tuples in the L awvere co nstruction. 27 and unive rsal algebra (equational languages) o v er t erm languages. Expression languages consist of relation (type) sym b ols and v ariables. T o get FOL lang uages with equalit y , pullbac k relatio ns along functions and equations. T erm languages consist of function (t yp e) sym b ols and v ar ia bles. The La wv ere construction is deﬁned here. Equations can b e added giving equational langua ges (equational presen tations) as an extension of term languages. They deﬁne a quotien t of their Lawv ere category . Express ion la nguages consist of relation (t yp e) sym b ols and v ariables. P eircian exis ten tial graphs can b e included here. F OL langua ges consist of function (t yp e) sym b ols, relation (type) sym b ols a nd v ariables. F rom the mo dular p ersp ectiv e of Fig ure 10, an FOL langua g e is a term language and an expression languag e that share a common set of v ariables. The imp ort a n t submo d- ules in the FOL axiomatization ar e the follo wing: a term/tuple ﬁxp oint axiomatizat io n, an expre ssion/arity ﬁxpoint axiomatization, the La wv ere construc tion axiomatization, an axiomatization for term-tuple copro ducts, the term monad axiomatization and the expres- sion monad axiomatizatio n. Eac h edge in the diagram of ( Figure 10) is asso ciated with institution morphisms in bo t h directions, pr o jection do wn w ard and inclusion up w ard. 12. Summ ary and F uture W ork In this pap er, we ha v e presen ted the Informat ion Flow F ra mew ork (IFF) as a descriptiv e category metatheory b y discussing it s design g uidelines, dev elopmen t phases , arch itecture and institutional application. Meta-ontologies (i.e., metatheories), as w ell as dictionar- ies, are classiﬁed b y the prescriptiv e-descriptiv e distinction. Th e dev elopmen t of meta- on tologies has strong analogies with the dev elopme nt of dictionaries. The IFF, whic h w as the ﬁrst of sev eral pro jects accepted for on tology dev elopmen t under the auspices of the Standard Upp er Ontology (SUO) w orking group, fo rms its structural asp ect. The design guidelines mak e apparent that the IFF is b eing dev elop ed b y following the intuitions of the w orking category-theorist. The IFF architec ture, whic h consists of metalev els partitio ned in to namespaces, is structured by its namespace mec hanism. The IFF metastac k, whic h forms the heart o f the IFF architec ture, represen ts the small, the large, the very large and the gene ric. This is formalized in terms o f four metalev els of sets, link ed by f unctions and binary relations. The sp ecialization b et w een metaleve ls is eﬀected b y set subset, function restriction and relation abridgmen t. Three principles disco v ered during the dev elopmen t phases of the IFF are conceptual w arrant, categorical design and institutional logic. The linguistic asp ect of the IFF is represe nted b y four nested metalanguag es, with eac h meta- language link ed to a metalev el. The metashell pro vides standard logical nota tion for the metalanguages with restricted quan tiﬁcation for t yping. The main application of the IF F uses the metatheory of institutions to a bstractly express the mo dular mo deling tec hnique of the la ttice of theories fr a mew ork used in the dev elopmen t and seman tic in tegration of ob ject-lev el on tologies. W e anticipate that ﬁbrations and indexed categories will b e cen tral to the fourth phase of IFF dev elopmen t. The large categories associated with many low er lev el meta- on tologies are ﬁb ered. F urthermore, indexed catego ries, ﬁbratio ns and the Grothendiec k 28 construction are ve ry imp ortant to the metatheory of institutions, whose axiomatization is curren tly under dev elopmen t in the IFF. W e also w an t to axiomatize sk etc hes, and incorp orate these with the axiomatization for institutions. F urthermore, it w ould b e in teresting to in v estigate the connections b etw een hig her or der logic a nd the ﬁrst order expression at the v arious IFF metalev els . Finally , we mak e tw o observ ations. The IFF regards category theory as a meta-on tology , whic h is exactly the spirit of the pap er [2]. Ho w ev er, in a sense the IFF has outgro wn its origins and institutional applications. It w as orig inally based up on information ﬂo w and formal concept analysis with a slender category theory core. But now it has an increasingly strong categorically o rien ted sup er- structure. W e prop ose to split the old IFF in to (1) the descriptiv e catego r y metatheory prop er, calling this meta , a nd ( 2 ) a new smaller IFF that co v ers only the institutional application to on tology dev elopmen t and semantic in tegration. In additio n, w e prop o se the initiation of a w orking group for dev elopmen t of a catego r y theory standard, as currently manifested in meta . Conten ts 1 In tro duction 2 2 On tologies 2 3 Descriptiv e Metatheories 4 4 The Standard Upp er O n tology (SUO) 6 5 The IFF Design Guidelines 8 6 The IFF Dev elopmen t Phases 10 7 The IFF Arc hitecture 14 8 The IFF Metastac k 18 9 The IFF Low er Metaleve l 19 10 The IFF Metalanguages 22 11 The IFF Application 25 12 Summary and F uture W ork 27 References [1] Jon Barwise and Jerry Seligman. Information Flow: The L o gic of Di s tribute d Sys- tems , v olume 44 of Cambridge T r acts in The or etic al Computer Scienc e . Cam bridge Univ ersit y Press, 19 9 7. [2] C.N.G Dampney and Mic hael Johnson. 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The Information Flow Framework: A Descriptive Category Metatheory

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