Informatica A Journal of Computing and Informatics The Slovene Society INFORMATIKA Ljublana Informatica A Journal of Computing and Informatics Subscription Information Informatica (YU-ISSN 0350 - 5596) is published four times a year in Wmter, Spring, Summer and Autumn (4 issues). The subscription price for 1990 (Volume 14) is US$ 30 for companies and US$ 15 for individuals. Claims for missing issues will be honoured free of charge within six months after the publication date of the issue. Printed by Tiskarna Kresija, Ljubljana. Informacija za naročnike Informatica (YU ISSN 0350 - 5596) izide štirikrat na leto, in sicer v začetku januarja, aprila, julija in oktobra. Letna naročnina v letu 1990 (letnik 14) se oblikuje z upoštevanjem tečaja domače valute in znaša okvirno za podjetja DEM 16, za zasebnike DEM 8, za študente DEM 4, za posamezno številko pa DM 5. Ste^^lka žiro računa: 50101 - 678 - 51841. Zahteva za izgubljeno številko časopisa se upošteva v roku šestih mesecev od i^da in je brezplačna. Tisk: Tiskarna Kresija, Ljubljana. Na podlagi mnenja Republiškega komiteja za informiranje št. 23 - 85, z dne 29. 1. 1986, je časopis Informatica oproščen temeljnega davka od prometa proizvodov. Pri financiranju časopisa Informatica sodeluje Raziskovalna skupnost Slovenije. InformatiGa A Journal of Computing and Informatics Subscription Information Informatica (YU ISSN 0350 - 5596) is published four times a year in Winter, Spring, Summer and Autumn (4 issues). The subscription price for 1990 (Volume 14) is US$ 30 for companies and US$ 15 for individuals. Claims for missing issues will be honoured free of charge within six months after the publication date of the issue. Printed by Tiskarna Kresija, Ljubljana. Informacija za naročnike Informatica (YU ISSN 0350 - 5596) izide štirikrat na leto, in sicer v začetku januarja, aprila, julija in oktobra. Letna naročnina v letu 1990 (letnik 14) se oblikuje z upoštevanjem tečaja domače valute in znaša okvirno za podjetja DEM 16, za zasebnike DEM 8, za študente DEM 4, za posamezno številko pa DM 5. Številka žiro računa: 50101-678 - 51841. 2^teva za izgubljeno številko časopisa se upošteva v roku šestih mesecev od izida in je brezplačna. Tisk: Tiskarna Kresija, Ljubljana. Na podlagi mnenja Republiškega komiteja za informiranje št. 23 - 85, z dne 29. 1. 1986, je časopis Informatica oproščen temeljnega davka od prometa proizvodov. Pri financiranju časopisa Informatica sodeluje Republiški komite za raziskovalno dejavnost in tehnologijo, Tržaška 42, 61000 Ljubljana Informatica A Journal of Computing and Informatics EDITOR-IN-CHIEF Anton P. Železnikar Volaričeva ulica 8, 61111 Ljubljana ASSOCIATE EDITOR Rudolf Murn Jožef Stefan Institute, Ljubljana The Slovene Society INFORMATIKA Ljubljana Letnik 14 Številka 4 Oktober 1990 YU ISSN 0350-5596 Informatica Časopis za računalništvo in informatiko VSEBINA A System for Morphological Analysis of the Slovene .Language Understanding as Information II The Relation Scheme Extensions Avtomatsko učenje vodenja dinamičnih sistemov Uporaba slike v identifikacijskih postopkih Minimalne razdalje med geometrijskimi objekti Real-time Executives for Embedded Microprocessor Application Načrtovanje uporabniškega vmesnika Sodobni programski jeziki v sistemih realnega časa Dinamična nevronska mreža za klasifikacijo vzorcev Računalniška mreža in CATV The System of Knowledge Novice in zanimivosti (v angleščini) Avtorsko stvarno kazalo časopisa Informatica, letnik 14 (1990) T. Erjavec A. P. Zeleznikar 5 Svetlemu Kuzmanov 31 P. Mogin Tanja Urbančič 39 M. Bradeško 49 L. Pipan S. Žerdin 52 K Guid, B. Žalik Barbara Koroušić 58 J. M. Cooling, P. Kolbezen M. Debevc 64 R. Svečko, D. Đonlagić G. Godena V. Jovan 68 Jelena Ficzko 77 U. Rezar, A. Dobnikar D. Podbregar P. Zaveršek P. Kolbezen O. B. Popov 81 87 91 A SYSTEM FOR MORPHOLOGICAL ANALYSIS OF THE SLOVENE LANGUAGE Keywords: morphological analysis, Slovene language. Tomaž Erjavec NLU Lab., Department of Computer Science and Informatics Jožef Stefan Institute Jamova 39, 61111 Ljubljana ABSTRACT: Most computer programs dealing with natural language processing require access to a lexicon in computer readable format. Finding the entry in tfiè lexicon that corresponds to the word-form in the text being processed is the domain of the morphological component. The paper presents an integrated environment for decomposing input word-forms into their constituent morphemes (morphological analysis). The system comprises a morpheme oriented lexicon, a module for updating the lexicon, and a two-level morphological analysis/synthesis unit. The system is written to provide a coverage of Slovene inflectional morphology. The basic paradigms and lexical alternations of word forms are handled by the lexicon system, while the two-level component takes care of the phonologically induced alternations. 1. Introduction Natural language processing is a rapidly expanding field of-computer applications. Programs dealing in NLP range from spelling-checkers to automatic translation systems and natural language interfaces. A basic component of most such systems is the lexicon. The lexicon is needed to store the legal words of the language together with their morpho-syntaciic and other (depending on the application) properties. In this paper we deal with the problem of designing the lexicon and the program for accessing it in such a manner as to allow mapping from word-forms which occur in the text being processed to lexical entries (Erj90a). The problems that occur here are of a morphological nature. To take an example from Slovene: for the system to know that the word-form of the Slovene noun 'telesa' is the singular genitive or plural/dual nominative or accusative of the noun 'telo' (body), it must know (implicitly or explicitly) that 'teles' is a alternation of the morpheme 'tela' which occurs when it is followed by another (non-null) morpheme, and furthermore that the morpheme 'telo' denotes a noun of neuter gender, which has the ending '-a' in the above mentioned number/case combinations. A possible way around all this complexity could be to simply store all the word-forms of all inflecting words in the lexicon along with their (morpho-syntactic and other) properties. While this approach is readily applicable to morphologically impoverished languages, such as English, it is much more questionable for highly inflected Slovene. Lengthy paradigms of Slovene verbs, nouns and adjectives (e.g. about 50 word-forms for a typical verb) would make the lexicon much larger and highly redundant. With the dropping price of computer storage this alone may not sound like a sufficiently convincing argument. However, even if we were willing to pay the price of such an inflated lexicon, we would still only temporarily evade the problem of (Slovene) morphology. Alas, the morphological knowledge would only be removed form the 'output' module, but would still have to be in the system, namely in the 'input' module, that is, in the part of the system that deals with the input of new words into, the lexicon. Imagine otherwise the task of adding a new verb and all its fifty word forms with their specific properties into the lexicon. The problem of lexicon update is especially relevant for the Slovene language, as there still doesn't exist a lexicon of the language in computer readable (much less usable - i.e. properly structured) form. This is also the reason why our system also includes an input module, which is described in the last part of the paper. 2. The two-level model The two-level model of Kimmo Koskeniemmi (Kos84, Kos85) was selected as the fundamental scheme for our morphological analyzer. The morphological analyzer (MA) is the component, of the system that takes as its input the word-form as it appears in the text - the textual or surface representation - and matches it to an entry in the lexicon - the lexical or deep representation -which is composed of the constituent morphemes of the word-form. The change of form from surface to deep representation is described by two-level rules, which of course differ from language to language. Succinctly put, the two-level model (or program) checks whether a lexical string and a textual string are in a legal correspondence; the name of the model derives from the fact that the rules mediate directly between the two representations. Our choice of this model was influenced - among other things - by its prevalence in current (computer) morphological studies, which makes it well documented and thus easy to implement, as well as simplifying the task of writing the rules for (phonologically) induced alternations of Slovenian word-forms (Erj89, Erj90a). We will describe the model only briefly, as it has been extensively dealt with in other publications (e.g. Erj90b). The two-level model has its root in phonological rewriting rules and deals with phono-morphological alternations which occur in 'joining' the morphemes of a language. Thus, if we were to have in our lexicon the root of the noun 'ladj' (ship) along with the information that this morpheme can be followed by any morpheme for the endings of the first female declension of nouns ('-a" for nominative singular, '-e' for genitive singular and nominative plural, etc.), then a valid lexical word form would be for instance 'ladj-0', that is the root, followed by a morpheme boundary and the ending for genitive plural ('zero' morpheme/symbol). The surface word form for this case/number is, however, 'ladij', this alternation being caused by the phonology of Slovene; it isn't possible to pronounce a obstruent -I- 'j' in word final position. Our rule (somewhat simplified here) for such cases is: ■i epenthesis" 0:1 <=> Obst _ j -:0 #;0 ; To spell it out: the surface symbol 'i' should correspond to the lexical '0' if and only if it is preceded by any symbol from the set 'Obst' which is elsewhere defined as the set of obstruents, and followed by a morpheme boundary and word boundary (in other words a zero ending). We should point out, that the the lexical alphabet is in principle different from the surface alphabet. Let us conclude the description of the two level model by the description of the algorithm. The two-level rules are first compiled into finite automata (Erj90a, Kar87, Kos86), which take as their input pairs of lexical/surface symbols. On analyzing a surface word form, the automata are fed pairs of such symbols, the surface part of the pairs coming from the textual string and the lexical part from the lexicon. All the automata operate in parallel. If at the end of the surface string no automata have blocked and all are in their final states, then the correct lexical word was found in the lexicon. In such a way the automata actually 'guide' the search procedure in the lexicon into choosing the correct lexical entry, if one exists. The structure of the lexicon which supports such search is dealt with in the next section. 3. The lexicon A basic part of our system is the lexicon (Dal87, Erj89a), which is composed of sub-lexicons. We can, for instance, have a sub-lexicon for roots, another for endings of a male noun declension of Slovene, another for the conjugation endings of certain verbs, etc. A certain set of sub-lexicons is set as initial, meaning that a (recognizable) word can only start with a member of these sub-lexicons. The other sub-lexicons are connected to initial sub-lexicons by means of pointers, typically making them inflectional paradigms of various word classes. We shall call the elements of the sub-lexicons lemmas. Each lemma has three parU: the (tentatively speaking) morpheme, information about the possible continuations of the morpheme and its inherent morpho-syntactic (or other) properties, e.g.: bolezen decl_subst_f / eng=illness cat=subst g0n=f; This lemma contains the information that the morpheme 'bolezen', in order to make a legal lexical word-form, must be continued with a morpheme from the sub-lexicon named 'decl_subst_f2, and that its properties are that it has the English translation 'illness', and is a noun of female gender. For another example, let us look at a lemma from the sub-lexicon 'deci subst 12': -0 / nu=sg cas=nom; The lemma states that the nominative singular ending for the second female declension of nouns is a null ending '-0'; a morpheme boundary, followed by a 'zero' symbol. Furthermore, this morpheme has no continuation. A lexical word-form is thus constructed in the following manner: * start with the 'empty' lexical string and one of the initial lexicons as the 'active' lexicon; * choose a lemma in the active lexicon; * append the morpheme of the lemma to the lexical string; * mark the continuation lexicon of the lemma as active; * if the active lexicon is different from '#', then go to step two. Except in certain special cases (explained in Erj90a) the properties of a lexical word-form are simply the concatenation of the properties of constituent morphemes. Of course, the purpose of the system is to find a quite specific word-form in the lexicon; namely the one (or ones in cases of ambiguity) that correspond(s) to the surface word-form being analyzed. For this purpose, all the sub-lexicons are first compiled into an internal format. All the morphemes of each sub-lexicon are converted into a letter-tree, i.e. a tree with the name of the lexicon as the root, symbols (letters) of the lexical alphabet as internal nodes and (pointers to) lemmas as leaves. The nodes on a path from the root to a leaf constitute a morpheme of the leaf lemma. With such a data structure it is then a simple matter for the two-level automata to licence a path down the current tree, backtracking in cases of an ill chosen path. 4. Lexicon interconnection As we saw, the lexicon system takes care of regular paradigms of the inflecting words of the language, while the two-level rules handle phono-morphological alternations. The Slovene language, however, abounds in alternations that are lexically conditioned. That is not to say that no two-level rules can be constructed to cover these alternations, but rather that .they are not (purely) phonologically conditioned. There is for instance an alternation that affects only nouns of male gender which have the "animate" property, and another one which pertains only to the plural and dual of certain Slovene nouns (lengthening of the root with 'ov'). As two-level rules are sensitive only to the form of the word (string) they are processing, they are inappropriate for expressing such alternations. To handle lexically conditioned types of alternations, we have concentrated on the linking mechanism between the sub-lexicons. The "continuation" information belonging to a lemma can also include, along with a pointer to another sub-lexicon, a pointer to a description of how to modify the following sub-lexicon to express the desired alternation. To make the point clearer, we give a simple case of an alternation that affects some nouns of the male gender. In addition to the standard ending for genitive singular '■a', these nouns can also have the ending '-u' (e.g. 'sin-a" as well as 'sin-u' - (of) son): sin decl_subst_m1 (gen_u) / eng=son cat=subst gen=m; The continuation information for such nouns includes (together with the pointer linking them with the sub-lexicon of male gender declension endings) the name of a 'transformational' rule. On accessing the continuation sub-lexicon, it is first (locally) modified according to this rule. In our 'gen_u' case, a lemma '-u' with the morpho-syntactic attributes of genitive case and singular number is added to the continuation sub-lexicon. In addition to adding a lemma, these 'lexical alternations' are able to perform the following operations on sub-lexicons: * delete a lemma; * substitute a lemma; * prefix a sub-lexicon; * split a sub-lexicon according lo given criteria; * merge two sub-lexicons. 5. Input of new words , Computer lexicons for various applications should be of an "open" type, i.e. they should allow easy entry of new words into the lexicon by an inexperienced user. In our case this means that we cannot cxpcct the user to enter new lemmas as they will appear in the lexicon, since this would require knowledge of the implementation details of the system; e.g. two-level rules, the lexical alphabet, lexical alternations, etc. The user is therefore only expected to enter (Erj90): * the "canonical" form of the word (e.g. nominative singular for nouns) the "comparative" word-form of the word (e.g. genitive plural of the noun); * the basic paradigm; * certain vital properties of the word (e.g. noun, male, animate). Both word forms are input in the symbols of the surface alphabet. The lexicon input module then automatically determines: * the lexical root of the word (removing the suffix of the canonical word-form and mapping it into lexical symbols); * the continuation sub-lexicon(s); * lexical alternations (if any); * inherent morpho-syntactic properties of the word. The algorithm works roughly as follows: * compare the canonical word-form with the comparative word-form to determine (possible) lexical alternations; * nòndeterministically map the canonical word-form into the symbols of the lexical alphabet; * extract the root from the (lexical) canonical word-form; * from the paradigm and the morpho-syntactic properties of the comparative word-form determine which lexical ending the comparative word-form should have; * add this ending to the lexical root, getting the comparative word-form in its lexical form; * check the lemmatization by two-level matching of derived (lexical) and entered (surface) comparative word-forms; * if the match succeeds, finish, else redo any of the first three steps in a different way. 6. System structure We have so far discussed all the separate components of our system; now for a run down of the whole system. The system was implemented on VAX/VMS in Quintus Prolog and consists of the following parts: * the compiler, which takes as its input two-level rules and produces final state automata (transducers); * the lexicon module that provides a user interface for the creation and updating of the lexicon; * the lexicon output module, which is responsible for passing lexical word forms to the MA module. * the MA module itself, which, having access to the transducers and (indirectly) to the lexicon, is able to analyze Slovene word forms into their lexical counterparts, and also to synthesize word forms from lexical data. 7. Conclusions Our system is one of the first attempts toward the goal of producing a computer usable lexicon of Slovene words that could be used as a front-end to various natural language processing programs. So far we have concentrated mainly on the morphology of noun and adjective declensions (Top84), while the (very complicated) morphology of the verb still needs to be worked out. A morphological analysis system for "true life" applications should also be written with perfomance characteristics in mind; i.e minimizing its time complexity and storage reguirements for a lexicon with cca. ten thousand lemmas. Tlie task of keying in all these words and their properties also seems daunting. In my opinion, the effort would well be worth the trouble, as languages which will not become "computerized" in a very short time face a bleak future, since they could very well be rcduccd to the status of a local inferior dialect. References: (Dal87) Dalrymple M., Kaplan R., Karttunen L., Koskenniemi K., Shaio S., Wescoat M.: Tools for Morphological Analysis / Xerox Palo Alto Reasearch Center, Center for the Study of Language and Information, Stanford University, Report No. CLSI-87-108,1987 (Erj89) Erjavec T., Tancig P.: Dvo-nlvojska pravila za alternacije slovenskih samostalniških sklanjatev (Two-Level Rules for Alternations of Slovene Noun Declensions) / Proceedings of "V. kongres zveze društev za uporabno jezikoslovje Jugoslavije"; Ljubljana 1989 (Erj89a) Erjavec T., Tancig P.: Struktura računalniškega leksikona z morfemsklmi entitetami; primer slovenščine (The structure of a computer lexicon with morpheme entities for the Solovene Language) / "IV, Jugoslovanska konferenca o leksikografiji in leksikologiji 'Rječnik i društvo'" Conference Proceedings; Zagreb, 1989 (Erj90) Erjavec T., Tancig P.: Vnos samostalnikov v odprti leksikon slovenščine (Input of Nouns into an Open Lexicon for the Slovene Language) / "Informatička tehnologija u primenjenoj lingvistici" Conference Proceedings; ^greb, 1990 (Erj90a) Erjavec T.: Računalniški sistem za morfološko analizo in sintezo slovenskega jezika (A Computer System for Morphological Analysis and Synthesis of the Slovene Language) / Masters Thesis; Faculty for Electrical Engineering and Computer Sciences, University of Ljubljana; Ljubljana, 1990 (Erj90b) Erjavec T.: The Development and Description of the Two-Level model for Morpholologlcal Anolysis and Synthesis / Journal "Informatica" Vol. 14, No. 3.; Ljubljana 1990 (Kar87) Karttunen L., Koskenniemi K., R.M. Kaplan: A Compiler for Two-level Phonological Rules / in "Tools for Morphological Analysis"; Xerox Palo Alto Reasearch Center, Center for the Study of Language and Information, Stanford University, Report No. CLSI-87-108; 1987 (Kos84) Koskenniemi K.: A General Computational Model for Word-Form Recognition and Production / Computational Linguistics, Conference Proceedings; COLINO '84 (Kos85) Koskenniemi K.: A Generai Computational Model for Word Form Recognition and Production / Computational Morphosyntax, Report on Research 1981-84; University of Helsinky, Dept. of General Lingusislics Publications No. 13 (Kos86) Koskenniemi K.: Compilation of Automata from Morphological Two-Level Rules / Papers form the fifth Scandinavian Conference of Computational Linguistics 1985; University of Helsinky, Dept. of General Lingusistics Publications No. 15 (Top84) Toporišič J.: Slovenska slovnica (Grammar of the Slovene Language) / Založba Obzorja Maribor; 1984 UNDERSTANDING AS INFORMATION II INFORMATICA 4/90 Keywords: algebra, blindness, breakdown, cultural understanding, disorder, formalization, hermeneutics, idea of god as formalized information, information, informational expert system, informational algebra, informational logic, intelligence, intellectualism, loseableness, losing, misunderstanding, semiotics, understanding. Anton P. Železnikar Volaričeva ul. 8 61111 Ljubljana, Yugoslavia Understanding as information and its formalization can bring to the surface various formal informational concepts that could substantially change our knowledge of understanding as a basic as well as the most complex realm of informational entities. Understanding becomes a formal informational entity that informs spontaneously and circularly, in a hermeneutic (historically cyclic) or parallel-cyclic mode. Understanding can be formally decomposed into its constituting components, for instance, sensing, observing, perceiving, conceiving, and comprehending, all informing in their own and perplexed hermeneutic way. Formal axioms of understanding can be developed in a straightfoiward manner, constituting the basis of the algebraic theory of understanding. Intelligence and intellectualism can be treated as , distinct aspects which pervade other components of understanding. Meaning is the resulting information arising out of the process of understanding. Further, hermeneutics as a modus of understanding can be formalized by the introduction of the specific structure of informational cycles within understanding. Formalization of semiotics introduces the so-called semiotic type of understanding, considering syntax, semantics, and pragmatics as distinct, however perplexed informational entities. Several types of loseableness and losing of understanding can be formalized and perplexed, leading to the problem of the understanding blindness and breakdown of the performing blindness. Informational expert systems can be formalized in the form of dedicated formal informational systems of understanding applying hermeneutic as well as direct parallel-cyclic modes for production of expertise. Misunderstanding can be formally treated as an informationally incompatible system with regard to the ruling understanding. It is shown how disorder and cultural understanding (informational cultivation) could be formally decomposed. At the end of the essay an informational formalization of the idea of god (universal information) in an algebraic manner is presented. The aim of the formalization of understanding by algebraic means is to step vigorously on the way to an informational theory of understanding in the living as well as artificial. By the presented algebraic approach, artificial understanding can be directed into a new enlightenment and applicative perspective. On the other hand, the formal approach of understanding could prepare the ground from which various philosophical excursions into the domain of understanding would become possible. RAZUMEVANJE KOT INFORMACIJA II Razumevanje kot informacija in njegova formalizacija lahko izpostavita različne informacijske koncepte; ti lahko bistveno spremenijo znanje o razumevanju kot osnovno in kot kar najbolj zapleteno področje informacijskih entitet. Razumevanje postane formalna informacijska entiteta, ^ki informira spontano in cirkularno, na hermenevtičen (zgodovinsko cikličen) ali paralelno-cikličen ' način. Razumevanje je mogoče formalno razstaviti v sestavljajoče komponente, npr. v občutenje, opazovanje, zaznavanje, snovanje in doumevanje, ki vse informirajo na lasten in prepleteno hermenevtičen način. Formalne aksiome razumevanja je mogoče razvijati dovolj natančno, oblikujoč osnovo za algebraično teorijo razumevanja. Intelogenco in intelektualizem je mogoče obravnavati kot distinktna vidika, ki prežemata druge komponente razumevanja. Nadalje je mogoče formalizirati hermenevtiko kot modus razumevanja z vpeljavo specifične strukture informacijskega cikla v okviru razumevanja. S formalizacijo semiotike se uvaja t.i. semiotiCni tip razumevanja, ki upoSteva sintakso, semantiko in pragmatike kot distinktne, toda prepletene informacijske entitete. Formalizirati in preplesti je mogoče več vrst zgubijenosti in izgube razumevanja, kar privede do problema razumevajoče slepote in prekinjanja nastopajoče slepote. Informacijske ekspertne sisteme je mogoče formalizirati v obliki namenskih formalnih informacijskih sistemov razumevanja z uporabo hermenevtičnih kot tudi paralelno-cikličnih načinov produciranja ekspertize. Nerazumevanje je mogoče formalno obravnavati kot informacijsko nezdružljivi sistem glede na vladajoče razumevanje. Pokazano je, kako bi bilo mogoče formalno razstaviti moteno (patoloSko) in kulturno razumevanje (informacijsko kultiviranje). Na koncu spisa je prikazana Se informacijska formalizacija ideje o bogu (univerzalne informacije) na algebraičen način. Namen prikazane formalizacije razumevanja z algebraičnimi pripomočki je primerna usmeritev na pot k informacijski teoriji razumevanja, ki bi lahko zajela tako živo kot umetno (arteficialno). S prikazanim algebraičnim pristopom je prav umetno razumevanje mogoče nanovo osvetliti in aplikativno omogočiti. Razen tega pa formalni koncept razumevanja pripravlja ozadje iz katerega so mogoči raznovrstni filozofski izleti v domene razumevanja. Part Two ... There can be little doubt that Formalism is the most popular anti-Platonist position among practicing mathematicians. ... Penelope Maddy (RCP) 1122 A Formal Theory of Understanding as Information 15. On the Sense of Formalization of Understanding presentation and imagination? How does this theory root in the basic theory of information (for instance, in .the so-called informational logic and informational algebra) which was described in (IL-I, IL-II, IL-III, IL-IV, IIA)? Understanding as information is an ongoing, continuous informational process, that is, an implicit or explicit informational variable (entity, an acting or operating operand) which will be marked by the sj^ol 81. Understanding SJ as informational process understands or informs in an understanding way its own information (itself) and the arriving or to the understanding coming information, i.e., information coming into understanding. This style of explanation (expression) or of articulation of the problem of understanding is a specific style of linguistic expression and is necessary to bring the adequate formal apparatus into existence, to its semiotic appearance. In general, this, sometimes unconventional style of articulation, is characteristic for the entire realm of informational understanding of information, constituting the realm of informational philosophy and informational theory. ... according, to Goedel, mathematical intuition inspires us to build up theories which are then justified by their ability to systemize all of mathematics. ... Penelope Maddy (RCP) 1133 The question of the sense of formalization of understanding as information has to be put into consideration at the beginning of the discourse which follows, for understanding is the fundament of informational arising, from the most primitive informational form to the most complex informational processes of understanding. Is such formalization at all possible and, if it is, to which extent and practical (philosophical, mathematical, technological) relevance? What new quality of formal understanding does this theory bring to the surface and how can it impact our symbolic 16. Understanding as a Formal Informational entity . .. mathematics is neither a game with symbols nor pure metamathematics; rather, it is the study of logical consequences. Among philosophers, this view has been called If-thenism, and it has been embraced and ultimately rejected by three important figures in the philosophy of mathematics: Hilbert (before his program), Russel (before Principia), and Putnam (before his Platonism). ... Penelope Maddy (RCP) 1124 Let us paraphrase the above quotation in the following way: informational theory is neither a game with informational operands and informational operators nor pure metainformational theory; rather it is the study of informational (not only logical) consequences. Among philosophers, this view might be called Theoretic Informationalism, however, in cybernetics and informatics, it seems to be the way of postmodernistic understanding of information and the way of informational understanding of the cosmic (microcosmic and macrocosmic), the living (in biology, neural science) and the artificial (intelligent machines). By marking of understanding as information by Gothic symbol SI, we stress the ongoing informationally explicit or informationally implicit process of understanding. Thus, understanding SI will perform as regular informing of information, i.e., as a regular informational entity, variable, or operand, which informs itself and other information and is informed by itself and by other information. What is the meaning of this phenomenology in the case of understanding as information? Generally, understanding SI informs and is informed in all possible ways, i.e.. (SII) 31 « ((SI 1=) V (t= SI)) Some comments to this formula are necessary; (1) All operators [=, and V) in this or any other informational formula are "informational", if there is not explicitly defined that they are particular, for instance, mathematical. (2) Formula (SII) is completely open which is explicated by the right side of the operator That is, formula SI ^ informs freely, i.e. spontaneously and circularly to something symbolized by the empty place on the right side of the operator We can say that SI 'does not know yet' which informational entities somewhere could be informed by SI, with the exception of SI itself. This yields. (SI2) SI « (M 1= SI) (3) On the right side of operator ^^ in formula (SII) there is formula SI which symbolizes the completely free, i.e.> spontaneous and circular informing of something (yet undetermined) to SI, that is, in the way, where SI is informed by something not known yet. Formula SI hides the potentiality of SI to be informed by any information or the potentiality of any information to inform the understanding SI as information. Certainly, SI is informed by itself, thus, (SI2) is informationally valid also in this particular case. (4) Informational operator ^ is the most general informational operator (metaoperator, a kind of informational jockey operator) which can be replaced (particularized or, in a particular case, universalized) by any informational operator. Thus, operator represents the entire class of'possible informational operators which can be grouped as general =|, , parallel (|K 4. IK A\), cyclic (f-, [ji, yj), and parallel-cyclic operators (|f-, -||, of informing and non-informing, respectively. Instead of (SII), the so called system expression of understanding SI can be used, i.e., System SI, i.e. understanding as information, is a composition of two distinct processes, SI ^ and t= Sti which can inform anything and can be informed by anything, respectively. Thus, informationally, the following is implied: (814) (SI N) (t= (SI (SI ST); M) Informational operator ^ is read as 'implies/ imply' or 'is/are implied'. Processes SI |= and ^ SI imply informationally process SI ^ H or, one can say, process SI ^ SI is implied informationally by process SI [= as well as by process ^ SI. Informational operator ^ is not equivalent to the mathematical operator of implication, which is used, for instance, in an if-then clause. The next question is, what does understanding as an acting form of information produce. As any informing, i.e. processing of information, understanding SI as informing is a component of understanding entity (operand) a as information. In this case one says that SI is a particular implicit informing of a and that a as information of understanding uses SI as its own processing, which performs the function of understanding. Thus, the so-called general informing of a, i.e., 3(a), includes understanding SI(a), yielding (SI5) SI(a) C 3(a) Understanding SI of a is a specific process (subprocess) of informing 3 of information a. This means that a can also inform in a non-understanding, misunderstanding, or disordering way, where the non-understanding component of a within 3(a) is an informational difference occurring between St(a) and 3(a). This difference can be marked by S(a), i.e., (SI6) ®(a) = SI(a) \ 3(a) We see that understanding SI of a performs as regular information. Understanding SI as information informs and is informed as an autonomous informational entity, in a spontaneous and circular way, i.e., (517) SI « ( ... ((SI t= SI) St) . . . f= 4t) and within an informational realm a, by which it is impacted and which it Impacts, as (518) a, SI a, SI This yields (519) a|=a;at=SI;SIf=a;8I|=SI or according to (SI7) also, (SI7') SI(a) <=> ( ... ((2I(a) N SI(a)) N 81(a)) ... ^ SI(a)) The four processes of (819) can be expressed, for instance, in the sense of formula (SI7), in the form: (SI3) SI ((SI f=); (1= 81)) (aio) (1) (■ (2) ( (3) ( (4) ( ((a 1= «) 1= «) ... t= «); ((« il) 1= a, a) ... a, ((JI 1= a) h a, «) ■•■ N ((a SI) ^= SI) ... ^= M) a); System (aiO) can be marked simply by a system of two entities (operands) (a(a), a(a)), with the meaning that « and a create each other informationally and in themselves implicitly. 17■ Informational Components of Understanding . . . Platonism, of course, is the view that mathematics is the study of a mind-independent realm of abstract entities, that it is as much an objective science as astronomy, physics, or biology. At various points, these tenets conflict with each of the position considered so far - with Intuitionism on mind-independence, with Formalism on the meaningfulness of higher mathematics, with Logicism on the need for existence assumptions that go beyond pure logic - but the Platonist's traditional and starkest opponent is the Nominalist, who holds simply that there are no abstract entities.... Penelope Maddy (RCP) 1130 What could in general the components of understanding be: thought, spirit, sense, wisdom, mind, consciousness, reason, comprehension, perception, conception, understanding itself, etc.? We see that all these components have a common meaning which is the consequence of their understanding, the process of their semantic and pragmatic investigation. However, to each of these components the so-called intelligence - a property of information as information - can be attributed as a general principle. Thus, understanding a is a domain of informing of information joining all these components, which will be marked by a^^, a2, ... / aj,. What is the notion of informational component as an informational entity distinguished within another informational entity? What does the informational component constitute as an informational entity which arises within another arising entity? An informational component is a unity (informational unit) which can be distinguished (observed, investigated, comprehended) as the unity In informational sense within another informational entity. A component as informational unity can be informationally rounded up, closed, or bounded within a composed informational entity. Informational distinguishing of components means observing the informational differences, setting the informational boundaries, determining the informational oneness, and marking the united component informationally. However, an informational entity as unity is not an informationally isolated form or process of information: it informs and is informed within its master entity, but can also inform directly itself and other information and can be informed directly by itself and other information (arising inside and outside of its master entity). Let us mark canonically the possible components of understanding a by a^ (i = 1, 2, ... , n) and let these components be informationally active within a. Semantically, components of a can be, for instance, on the level of a being's metaphysics as intelligence, thought, spirit, sense, wisdom, comprehension, perception, conception, and understanding by itself. The question is how do these components perform informationally within a. Let us determine the performing of components ^21 ■■■ ' ^n understanding a, within a, in the following, basically parallel way: (ail) (ai t= M) 11= «i; (a2 ^ a) IN (a^ N a) IN a^; (a h M; (a N a2) 11= a; (»> a^) IN a This is a parallel system within which processes (ai ^ a) |t= ai,- (a t= a^) ||= a,- i = i, 2, ... , n inform in parallel and constitute the common or resultant understanding a. Simultaneously, components aj^, a2. «n inform particularly to understanding a, i.e., as a^ N= a within (a^ [= a) |i= ai; i = i, 2, ... , n and understanding a informs particularly to each component a^, a2, ... , a^, i.e., as a process a |= ai within (a |= ai) a,- i = 1, 2, ... , n. It can be imagined how this parallel system of understanding as informing of understanding components and understanding itself can become as complex as possible. Within this system, intelligence t)jj as information of understanding arises, characterized by an open and infinitely recursive formula of the form (ai2) ( ((a t= T)si) a) ... 1= r)a) We see how informational playing of entities a and r)5j can improve informational capabilities of both of them. Certainly, formula (ai2) is completely open for any other, outer informational impacting. 18. Axioms and Basic Rules of Understanding as Information (Formalization of Principles of Understanding) ... Logic is a matter of syntax, and syntax is a matter of convention. ... Penelope Maddy (RCP) 1126 18.1. Introduction. The aim of this section is twofold: to show the concept of understanding in the elucidation of the so-called general (redefined) concept of information (POI), i.e., as an informational entity (for instance, informational operand and operator, in fact, informational process) and to formalize this concept by means of a distinguished, so-called informational language (formulas of operands, operators, and parentheses). The theory developed by this language should proceed.from the general level, called metalevel of formulas; these can be particularized (decomposed) and again universalized (composed) stepwise, developing a formula or a system of formulas (for instance, a top-down and bottom-up design). At the beginning, a general syntax and semantics of operands, operators, and formulas of understanding as information is needed. Within this orientation it is possible to set some skeletal definitions. (18.1) DEFINITION. The formulas for formatting understanding as informational operands aré the following: (1) SI marks an informational operand (entity) of understanding as information; the Gothic letter SI is the symbol of the so-called implicit informing of SI, where SI is understood to be an implicit informational operator simultaneously, for instance, performing processes of understanding of information. Thus, the explicit role of understanding SI is the operand-like, while its implicit role is the operator-like. (2) (SI) is a parenthesized operand of understanding or parenthesized implicit operator of understanding. (3) SI t= reads "SI informs". (4) 1= SI reads "SI is informed". (5) SI SI reads "SI informs SI" or "SI is informed by SI" . □ (18.2) DEFINITION. The informationally semantic perquisites to the previous definition are the following: (1) SI can mark any operand or implicit operator of understanding, for instance, S, I, S, . . ., indexed or not. (2) SI can mark any multicomponent aggregate of understanding operands, i.e. SI]^, SI2, ••• , 8I„ or SIi; SI2; ... ; Sin or B, (C, ... , S, etc. (3) Symbol ^ is the general (explicit) operator of informing, called metaoperator; it can be replaced (particularized or universalized) by any. operator. For instance, SI ^ means "SI informs true". (4) Explicitly, ^ can be a unary, a binary, or a multiplex operator. In principle, ^ or its replacement is a multiplex operator-. The case SI ^ means that "SI informs to som^e yet unknown (or informationally"irrelevant) 'anđ~potentiaily unlimited number of operands'. Through this generalized concept, each informational formula (representing a simple or composite operand, i.e. operand entity) is open. It means, for instance, that in formula SI B, entity SI can additionally inform, that is, SI and ® can additionally be informed, that is, ^ B. (5) Operand SI by itself (as an autonomous entity) has the property SI and t= i-e., performs as an open operand entity. Thus, SI O (SI 1= SI) is an informational system where informational operator reads "means". (6) All operators in a formula are informational. They can also be standard mathematical operators, if. explicitly determined as such. Mathematical operators are particularizations of the informational ones. □ The previous two definitions constitute the formatting of the so-called operand formulas. In these formulas, operators ^ occur, for which it is possible to define the formatting formulas concerning operators. This might be senseful in processes of the so-called operator decomposition where the structure of an operator is becoming semantically relevant. (18.3) DEFINITION. In general, we can introduce the following operator formatting formulas: (1) ^ marks an operator (entity). (2) In an operand formula of the general form SI t=, operator can be replaced in such a way that the new operand formula becomes SI N (C 1=) or (SI 1= CC) |=. Thus, SI 1= ((SI ((£ t=)); ((f t= N)) (3) In an operand formula of the general form B, operator ^ can be replaced in such a way that the new operand formula becomes (1= I) B or t= (CC N «) • Thus, SI (((t= 0 1=»); (N (IS 1= B))) (4) As one can see from (2) and (3), the principle of ^'s replacement is the following: 1= is replaced by ((£ or by (^ \z (E) where ^ marks the existing operand (also an empty operand place) of the original formula. This may be more evident if ^ is replaced by "...", thus, the replacement is |= (CE ^ ... ) or (... N (3 (E).((SI t= «) t= »); (dl) SI (= B ^ (3 5:).(SI h (£ 1= B)) In these formulas, the particular informational operator 3 reads "informs the existence of oz 'is informed about the existence by'. For 3 <£ is an open formula, it reads as 'tC is informed of its existence by something undetermined yet ' . Traditionally, formulas (cl) and (dl) can be particularized even in more detail as follows: (c2) SI B :> ((V SI, B) 3 (E).((SI h «) 1= »); (d2) SI ^= B ( (V SI, B) 3 tC). (SI t= (CE B) ) In these formulas, aggregate (informational set) SI, B is an operand entity and V SI, B is read 'something informs each SI, B' . Further, formula (V SI, B) 3 C is read ' (V SI, B) informs the existence of C, or in short, 'each SI, B informs the existence of C . Operator ', ' has the meaning 'such that' or 'informs the possibility of. Thus, (V SI, B) 3 C as an operand entity informs the possibility of (SI S) (= B or SI (CC t= ®) • There is possible to understand in which way regular informational operators V and 3 differ from the the traditional logical quantifiers 'for all' and 'there exists'. Informational operators V and 3 are in principle multiplex operators, in a particular case they can be binary or unary (but still open) operators (not only quantifiers). (2.4) AXIOM. An operand entity, marked by SI, is divisible in an informational way. This yields, (f) SI ^ Sil, SIj, ••• SI SI^; SIj; ... In principle, the divisibility of SI is not limited or predetermined. Components SI^, SI2, ... of SI are informational parts (processes, forms) of SI as an operand entity, however, as operand entities within SI they can arise as any other operand entity. □ Instead' of (f), there is possible to" introduce (traditionally) (fl) Sil, Sl2' • • • C SI or SIi; sij; ... C SI and read 'entities SIi, SIj, ... inform to be parts (components, subentities) of entity SI'. Axiom (f) enables the introduction of a formula system, marked by SI, where SI can be a component of the system, it represents. This principle leads to the complexion of an infotmational entity, to its systematization (systemic structure and organization), in which components perform as autonomous, however, informationally connected entities within SI. 18.3. The internal structure (informational cycle) of an operand entity. The question of internal structure (or organization) of an operand entity SI concerns the right side of the open formula SI ^ SI t= ü/ in which informing of SI in (over) itself, i.e., SI 1= SI, is open in the sense that (SI 1= SI) |= and ^ (SI h SI). What could be the structure of the internal informing of SI in regard to the informational arising of SI? The concept we attempt to introduce with the last question is called the informational cycle. Within this cycle, each operand entity informs, as we say, spontaneously and circularly, that is, in a cyclically spontaneous way. The spontaneous has the meaning of a determined, undetermined, and/or unforeseen informational arising of SI, depending on informing of SI itself and on informing of operands informing SI, from the standpoint, i.e. from the informational circumstances and possibilities of entity SI. (18.3.1) AXIOM. The informational, cycle of an understanding operand entity SI is constituted by the following entities of SI: informing of understanding SI, marked by 3, Sjt, or 3(SI); counter-informing of understanding SI, marked by (£, (tjj, or C(SI) ; informational embedding or embedding of information into understanding SI, marked by Ejj, or ®(SI). Further, informing 3 of SI produces (generates, informs) understanding SI; counter-informing £ of SI produces counter-information y of understanding (for instance, misunderstanding, disordered understanding, information of counter-understanding); embedding ® of SI embeds in an informational way the arisen counter-information y and to SI arriving other Information ß, by producing the embedding information s of understanding, that is information, needed to understand an informational entity by understanding. Merely by e, the counter-arisen and the other, arrived information can be embedded into SI in an understanding way. We distinguish the following four cases of informing of entity SI within the so-called informational cycle: general, parallel, serial, and parallel-serial informing. □ , (18.3.2) AXIOM. A general interpretation of the informational cycle of understanding. Two general cases of informational cycle of an understanding operand entity can be distinguished. The general sequential (serial, cyclic) case can be expressed by the following, one-cycle formula: (gl) ((((((ß 1= SI) 1= 3) 1= £) Y) 1= ®) e) 1= M This formula is cyclic in regard to SI. For this formula it is also characteristic that the informing of the subsequent entity (3, CC, y, e, SI, etc. in the cycle) depends on all preceding entities, that is, that the history (memory) of the cyclic process is preset,ved. Formula .(gl) models only the so-called main cycle of an informational cycle within which all possible other subcycles can exist. For Instance, (g2) ((ß t= 41) 3) (= (SI, 3); (g3) (((ß M) 3) N 1) N (M. «); (g4) ((((ß t= 41) t= 3) 1= (ß, a, 3, I, Y, e, O 1= (a, 3, Y, 8); (k2) ß 11= a (ß, a, 3, c, Y, o 11= (a, 3, c, Y, e); (k3) ß I- a « ((((((ß 1= a) h 3) h- CC) h Y) h «) h o I- a; (k4) ß Ih a (ß 11= a) A (ß I- a) The right side of in (kl) is defined by (g8). The right side of ^ in (k2) is defined by (g8)# if operators ^ are replaced by operators The right side of ^^ in (k3) demonstrates à sort of pipeline principle in regard to a, that is, to components of its informational cycle. Parallel-serial informing (k4) is simply a mixture of parallel and cyclic informing. □ Definition 18.3.6 clears the sense of introduction of four informational operators, N/ IN, h, sJ^ti Ih- Operator A means 'informs in an "and" fashion in one and another direction' (in regard to the left and the right operand). Let us examine how an .operand does not inform and how does it inform in another (alternative) way. 18.4. Informing and non-informing of understanding What does in general the non-informing of an operand mean? In general, non-i,nforming is nothing else than informing in a non-informing way. For instance, if things inform (impact other things and itself) in a thing-characteristic way, then things do not or cannot inform in a thing-non-characteristic (non-impacting) way. By non-informing of an operand its property of non-informing can be explicated. We can explicate, for instance, that an operand does not inform true. If operand a does not inform in a certain way, we introduce a (li. The next question to this non-informing might be, in which particular way does a not inform. The antisyiranetric situation is the case of a, where a is not informed or cannot be informed (impacted) in a particular way. (4.1) DEFINITION. Operator ^ can be understood as a particularization of operator . The following equivalence can be introduced: (11) a fci ß (a ß) V (a t=aO|i^ß ß) V (a ß ) Operator V is an informational disjunction and thus, (12) (a ß); (a ß); (a ß) a fc^ ß This is an informational formula where each left multiplex operand of operator ^ implies o fet ß. □ 18.5. Alternative informing of understanding To make the discussion and construction of alternative cases of informing of understanding possible on the formal and the most general level, the so-called alternative operators can be introduced. It means that to each operator there exists an alternative operator with the following meaning: if the original operator, as we say, informs in one way, then the alternative operator, as we say, informs in another way. This approach enables to make the previous general definitions and axioms even more general. (18.5.1) DEFINITION. The operator pairs of one and another case of informing are the following: H) anđ (1^, f^) for general Informing and general non-Informing in one and another way, respectively; (11=, m) and ii||) for parallel informing and parallel non-informing in one and another way, respectively; (|-, -|) and H) for cyclic informing and cyclic non-informing in one and another way, respectively; and (II-, -il) and (It', 4) for parallel-cyclic informing and parallel-cyclic non-informing in one and another way respectively. According to the type of understanding these operators can be appropriately particularized (indexed), for instance, (Jiji, '^fei' Ni' Hii' '^M' Ihsi' HI»' where understanding 91 appears as a general parameter of understanding (for instance, hermeneutic interpretation, semantic explanation, particular comprehension, informational understanding, etc.) □ (18.5.2) AXIOM. Axiom 18.2.1 can now be broadened and generalized by considering Definition 18.5.1 in the following form; (ml) (m2) (m3) (m4) (m5) (m6) (m7) (mS) (m9) (mlO) (mil) (ml2) (ml3) (ml4) (mlS) (ml6) (ml7) (81 N; N M) (H SI; SI H) (SI IN; SI) (=il SI; SI =11) (SI h; f- SI) (H SI; SI H) (SI IH; II- SI) (HI SI; SI HI) (SI t=; (SI 1;^; (SI IN; (SI N SI; ^ SI; IN SI; IN SI; SI SI ^ SI SI (SI h; I- SI; (SI K- If SI; (SI IH; IH SI; (SI IH; IH SI; (SI t=; 1= SI; (=1 SI; SI =1; (SI IN; IN SI; (HI SI; SI 4; (SI h; I- SI; (H SI; SI -I; (SI IH; IH SI; (HI SI; SI HI, (SI |=; N SI; =i SI; SI H; (SI IN; IN SI; m SI; SI (SI h; H- SI H SI; SI -I (SI IH; IH SI HI SI; SI HI (SI |=; 1= SI =1 SI; SI H SI IN; IN SI 4 SI; SI 4 SI h; I- SI H SI; SI H SI IH; IH SI HI SI; SI HI ; SI ; SI : SI I St ; SI i SI : SI ; SI H SI; ^ SI; 4 SI; 4 SI; H SI; A SI; HI SI; 4 SI; St Ni; 4 SI; SI IN; (SI M; M SI) SI; SI (SI IN; IN SI) (^1 St; St HI) (SI |/; If SI) {A SI; SI 7^) (SI IH; IH SI) (HI SI; SI HI) SI H); SI 7i|); St HI) ; SI HI); SI H); SI tI); SI HI); SI HI); SI); SI Til); IN SI); HI SI; St HI) ; SI If; I/ SI); A SI; SI ; SI IH; IH SI) ; HI SI; SI HI) ; St Ni; Ni SI; 54 St; SI ' SI IN; IN SI; HI SI; SI HI) ; SI 1/; If SI; A SI; SI f|); SI IH; IH SI; HI SI; SI HI); SI t?i; tü SI; ^ SI; SI ,4; SI IN; l^' SI; HI St; SI HI; SI If; If SI; f| SI; SI f|; SI IH; IH st; HI SI; SI HI) Operand SI is in principle a completely open system, where operators |N, HI. ¥> HI, h/ H, h'l Al IH/ Hli IH, and HI symbolize the acts of various natures of informing and non-informing, that is, when entity SI informs and non-informs, i.e., SI N=, H SI, St N^, SI, SI |N, HI JI, SI IN, HI SI, St h, H SI, St If, f| SI, SI IH, HI si, si |H, and HI SI to yet unidentified entities and is informed and non-informed, i.e., N= SI, St =j, SI, SI IN SI, St HI, IN SI, St HI, h St, SI H, If SI, SI f|, |H SI, St HI, IH SI, and SI H| by yet unidentified entities (operands), in one and another way, respectively. In system (ml)-(ml6), components of the form St and N^ SI ([= is for N, H, IN, HI, IN, HI, h, H, k, 'I, IH, HI, IH, and HI) perform as distinguished entitles which can again inform and can be informed in the sense of (al) in Axiom 3.1. Inductively, the system of rank n of Si's informing and non-informing, in one and another way, can be constructed. This kind of recursion makes an operand entity completely open in transmitting (informing) information to and receiving (informing) information from yet unidentified operand entities by various kinds of informing and non-informing. □ It is evident how an understanding operand SI can inform and can be informed generally, parallel, cyclically, parallel-cyclically, and alternatively. All these possibilities can be expressed explicitly by the use of the listed operators. Thus, informational operands of understanding can become as complex as possible by the recursive decomposition of formulas, i.e., of operands as well as operators. 18.6. Informational theories of understanding Up to now we have presented a soft or general concept of the axiomatic base, giving the idea how in concrete theories these bases could be constructed according to the conceptualized needs. Theorems and their proofs will be informational formulas, where proofs are ways or scenarios representing the modes of getting theorems. A general informational theory of understanding can be developed ad infinitum, Bidding new and new theorems to the existing ones. Let us look at some examples. (18.6.1) THEOREM. Some very basic theorems are: (nl) Axiom (18.2.1) (a), that is, a (M N:; N St) yields also the opposite formula, that is, (SI t=; SI) St In this particular case, the meaning of an informational operand is SI « (SI N=; 1= SI); (n2) StN==^l=SI; N=SI=fStN; St|=!^t=SI; (n3) SI =1 <= =1 St; =j St C St H; SI H ÌÉ =i SI; (n4) SlN:=>=|SI;StN=<==HSI;Stt=?éH8I; (n5) StN==^St=|; SIt=4:St=^; SIJ=^SI=40 PROOF. Proofs of the listed theorems are the following: (nl') Operand SI informs and is informed and, if so, the informing of St in the sense to inform and to be informed can be the sufficient condition to say that St is informational operand. Thus, (St N= ; N St ) St. In this particular case, the consequence of the original and opposite implication is St <=» (21 (n2') If Jl informs, then it is at least informed by itself, that is, SI ^ (21 St). Thus, t= 21. If 21 is informed, then it informs also itself, that is, ^ SI ^ (SI ^ 21). Thus, ^ 21. To inform and to be informed can be completely different (particularized) operations. Thus, (21 ^ (1= SI). This theorem concerns the so-called informing in one way (operator f= or its particularizations, respectively). (n3') Theorem (n3) is the antisymmetric (alternative) case of theorem (n2) which concerns the so-called informing in another way (operator ^ or its particularizations, respectively). Operator (21 t=; t= f; « t= 21; 21 1= B; B |=; h »; » 1= »; (SI 1= B) (SI B); (St 1= ®) 1= (SI 1= B)) (02) ((B =1 21) =1 (» ^ 2t); (B =i 21) H; H (» =i 21); B =1 » H; H »; ® =t St; St =1; =j <= » =1 a (03) (2t, B t= 21, B) B t= St; SI =j » <= SI =f 51) (04) 21 [= B (21 t= SI, =); (B, St =j <= ® =1 ^ (05) SI 1= B (St, B t= B); (B =t B, St) B =i SI (06) SI t= B (...((St 1= B) t= B)...t= B); (B =|...(B =j (B =1 «I))---) B =1 SI (07) (SI ^ B) (B h 21); (St =1 ») (® =i S") PROOF, Proofs of the listed theorems are the following: (ol') In SI t= B, operands 21 and B are autonomous entities; thus, St implies St 21; 21 St and B implies B h; N »; B [= B. A similar implication exists for 21 |= B as an operand unity. Be.sides, 21 t= B in the narrower sense, where 21 informs (for instance, understands understanding) B, on the right side of a, is evident. So, 21 B on the left side of ^ has a broader, informational meaning, resulting from the previous axiomatic basis. (o2') This is a pure antisymmetric case of (ol) with the aim to remind us on the alternative nature of informational implication in this particular case. (o3') Since formula (St, B ^ SI, B) represents a system of four formulas, i.e., 21 |= 21; St B; B ^ SI; B t= B, the first part of theorem holds implicatively in one way. The second part of theorem holds similarly for informing in another way. (o4') There can be seen from previous definitions that formula 91 |= B implies in one way a ^"91; ÌI 1= S. In this case, M il is implied in one way by M itself. Similarly, formula S =j SI implies in another way the formula aggregate ® ^ SI; M H St- I" this case, SI ^ SI is implied in another way by SI itself. (o5') This theorem concerns B in a similar manner as theorem (o4) concerns SI. (o6') In formula ä )= B, operand SI performs as a thing which informs in a thing-characteristic way, operand B is in fact the observer of Si's informing in a B's characteristic way, and operator ^ is a mode of informational (operational) equipment (composition between SI and B. Since B can observe its own informing, it can observe the process SI B, thus, (SI ^ B) |= B, etc. On the other hand, SI is not necessarily aware of B's observing of operand SI, because SI is a completely open informing of SI in regard to possible observers. Therefore, (08) (St 1= B) feÉ^ (...((SI t= B) 1= Sl)...t= SI); (SI =!•••(« =1 (B =i «))•••) ^ (» =1 «J) is regular, if there is not an explicit informing on the sense of formula B ^ SI. The similar can be said for the case of informing in another way (operator =j). The proof of (o6) is an evident deduction on the basis of the previous definitions and theorems. In the discussed meaning of formula SI B a comment has to be added in regard to the meaning of operator The role of the observer is in fact on the SI-s side, if t= is read as 'understands'. In this case SI is observing B in an understanding way, that is, 91 understands B. (o7') Operators ^^ and are examples of particularized operators of non-informing with the meaning does not inform implicatively in one or another way, respectively. Thus, formula. SI (= B does not imply B 91 in the one way of informing and formula B ^ 91 does not imply SI H B In the other way of informing. Evidently, this can be deduced from the previous theorems. Instead of operator particularizations also operator compositions for ^^ and ^ could be introduced, for instance, fso^ and respectively. As we see, operator particularizations can always be understood as operator compositions and vice versa. In case with a composite operator, for instance, 5 influence of operands ^ and t) can be considered, for instance, in the form K ^^^^^ operator between operands ^ and t) is a complex composition. Another possibility of such operator composition would be other possible combinations, for instance. Q.E.D. We see how the discussed principles of particularization and universalization, of operand and operator substitution, of generic (also recursive) properties of operands and operators, for instance, SI (...((91 SI) ŽI)...N SI) constitute the realm of the understanding in an informational way. The principled (and advanced) generic recursiveness of informational formulas enables the conceptualization of operands (i.e. formulas) marking, for instance, discourse, time, intelligence, understanding, etc. as information in the "redefined" or "rethought" sense of information. Thus, the question arises: Where are the possible advances of this conceptualization (and universalization) fin the sense of mathematical disciplinarity? Would it be, for instance, permissive to say, that a mathematical function (formula) is intelligent, if it is capable to observe (inform, counter-inform, and embed) itself to some extent? (13.6.4) THEOREM. Some basic (systemized) theorems concerning informing and non-informing of understanding in one and another way are the following: (pl) (91 1= B; 8 =1 91) (SI N B; B =4 91; 91 B; B =i| 91; SI f- »; B H SJ; SI Ih B; B HI SI) (p2) (« ^ B; B 91) ^ (St B; B SI; SI ¥ 35; » A\ SJ; SI 1/ B; B rl SI; 91 Ij-- B; B 4 SI) (p3) (SI N B; B =1 SI) (SI M B; B ^ SI) (P4) (SI ^ B; B ^ SI) (SI 1= B; B =1 SI) PROOF. These theorems are only semantically grouped previous theorems with the aim to explicate some common and opposite properties of informing and non-informing of understanding 91, in regard to another understanding information B (that is autonomous or componential in regard to SI) , in one and another way. (pi') General informing in one or another way implies general, parallel, cyclic, and parallel-cyclic informing of understanding in one and another way. (p2') General non-informing of understanding in one or another way implies general, parallel, cyclic, and parallel-cyclic non-informing of understanding in one and another way. (p3') General informing of understanding in one or another way implies general, parallel, cyclic, and parallel-cyclic non-informing of understanding in one and another way. (p4') General non-informing of understanding in one or another way implies general, parallel, cyclic, and parallel-cyclic informing of understanding in one and another way. However, -,(($1 1= B; B =1 SI) (91 B; B 9É| 91)); there does not exist (symbol -i marks the operator of informational negation) an equivalent informational meaning between informing and non-informing of understanding in one and another way. Q.E.D. The phenomenon of informational recursion was already shown. An understanding informational operand SI informs and is informed. Its informing, 91 and ^ 91, informs too and is informed too, that is, (91 t=) (SI [=), etc., ad infinitum. Similar can be said for the case SI B, etc. Thus, informational recursion of understanding can be resumed by the following theorem: (18.6.5) THEOREM. A recursive series of recursive theorems of understanding is, for instance", the following: (ql) SI (SI (=; 1= M); (q2) SI t= ((SI 1=) l=; ^ (81 t=) ) ; (q3) M 51 => ((1= M) N; N (N SI)); ••• (q4) SI (SI ti; [i^ SI) ; (q5) SI tÉ ((SI ti) ti; (51 t^)); (q6) ti 51 ((t^ 51) t^; (I/ 51)); ... (q7) (SI =1; =1 SI) « 51; (q8) ( (SI =1) =1; =1 (51 ^) ) <= 51 (q9) ((=1 SI) =1 (=1 51)) <{: H 51; ... (qlO) (SI SI) C SI; (qll) ( (SI jI (SI ) C SI (ql2) ((Ti SI) SI)) C 51; ... Symbol '...' marks the possibility of theorem generation ad infinitum. Operators t^, and ^ can be replaced by their parallel 4> i), cyclic (h, V. H). and parallel-cyclic counterparts (|f-, Iji«, 4), respectively. Mixed cases of showed recursion among operator counterparts are possible. □ PROOF. The proof of these theorems is evident, for it proceeds from the previous definitions. For instance, we see how (q3) can be continued ad infinitum: ((= SI) t= :> (((t= SI) |=; t= (([= SI) t=)); ((t= SI) 1=) ((((N 51) t=) N) l=; 1= (((N 51) t=) etc. Q.E.D. 18.7. Informational algebras of understanding Already by the principle of particularization and universalization, different informational algebras of understanding can be generated from existing mathematical algebras. Thereupon, these algebras can be developed in a further manner by using other informational principles (POI), that is, axioms and theorems of informational logic, informational algebra (IIA), informational theory (redetermined theory of information), etc. Let us show a set of rules (axioms) for deduction of propositions as it appears within the classical mathematical logic. The rules of prepositional language (concerning understanding as information) are, for instance, the following: (rl) SI (8 ^ SI); (r2) (SI (SI »)) (SI ®) ; (r3) (SI ^ 8) ((B CC) (SI CO); (r4) (SI A 8) ^ SI; (r5) (SI A 8) 8; (r6) (SI 8) ((51 (E) ^ (SI (8 A C) ) ) ; (r7) SI (SI V 8); (r8) 8 (SI V 8); (r9) (SI I) ((8 CC) ((SI V 8) :>(£)); (rlO) (SI = 8) (SI 8); (rll) (SI = 8) (8 SI); (rl2) (SI 8) ((8 SI) (SI = 8)); (rl3) (SI ^ 8) ::> ((-1 8) (i SI)); (rl4) SI^ (-. (n SI)); (rl5) (-,(-. SI)) SI Let us use this set of rules to produce informational theorems by means of the previous informational axioms and theorems. Prepositional operands SI, 8, and CC pass over to informational operands; the similar happens to the prepositional operators A, V, =, and -i. This procedure is legal in an informational way, for operands and operators can be universalized from the prepositional into the informational ones. (18.7.1) THEOREM. As a consequence of rules (rl)-(rl5), seme informational axioms of understanding can be the following: (rl') .SI (8 ^ SI); (r2') (SI 1= (SI 1= 8)) (SI t= 8); (r3') (SI 1= 8) ((8 CC) ^ (SI t= <£)); (r4' ) (SI N ») 51; (rS- ) (SI t= 8) 8; (r6') (SI 1= 8) ((SI 1= (E) (SI t= 8, C)); (rV) SI (SI t= 8); (r8') 8 (51 8); [eq. rl'] (r9') (SI N «) ((» 1= «) (51, = (= CC)); (rlO') (SI = 8) ❖ (SI 1= 8); (rll') (SI = 8) :> (8 1= SI); (rl2') (SI 1= 8) ((8 h 51) (51, 8 t= SI, =)); (rl3') (SI 1= 8) ((ti 8) (SI t^)); (rl4') SI (ti (f 51)); (rl5') (ti (ti SI)) SI □ PROOF. The set of listed theorems of understanding is not informationally systematic, for it is directly deduced from the mathematical (prepositional) algebraic system. Proofs are the following: (rl") There is SI (t= SI) and 8 t= 51 is one of the possibilities of [= SI, which is an open formula. (r2") If SI informs the informing SI t= »» then SI informs 8 in an informational way. (r3") The informing SI 8 implies also informing SI ^ CC, if 8 informs C. (r4") Informing SI |= 8 implies SI as an informational operand. (r5") Informing SI . 8 implies 8 as an informational operand. (r6") Informing SI |= 8 implies SI 8, E, if SI informs C. (r7") There is SI (SI t=) and SI 8 is one of the possibilities of SI t=, which is an open formula. (r8") This theorem is equivalent to (rl'). (r9") Informing SI |= C implies SI, 8 t= «f/ if ® informs CC. (rlO") Particular informing SI = 8 implies SI t= because SI = 8 is only a particular informing of other possible informing of SI to 8. Formula SI = 8 means that SI informs 8 in an equivalent way (or to be equivalent) and vice versa. On the other hand, operator = can be universalized into operator (rll") Because of informational equivalence between SI and 8, formula SI = 8 implies also 8 ^ SI; in this respect, a joint theorem of (rlO') and (rll') would be appropriate, for instance : SI s (a t= » 1= M) (rl2") Informing SI t= » implies $1, » h St/ ®» if » ^ SI; this is a trivial consequence proceeding out of basic definitions. (rl3") Informing 81 » implies SI fcii », where operators and ^ are general operators of informing (understanding) and non-informing (non-understanding). Further, informing SI 95 implies B and SJ thus, (t?i S) (SI |?i) for the case SI 8. (rl4") Informing of SJ implies |= SI (SI cannot inform in all possible ways). Non-informing ^ SI can always (by definition) be expanded into (1= SI). Thus, SI (1= (h SI)). (rl5") The reverse informational implication to (rl4') is evident. As soon as SI appears in a formula, this formula implies the occurring (arising) of operand SI. Q.E.D. The presented set of theorems is in no way an adequate selection for the demonstration of the appropriateness of an informational theory of understanding or any other general theory of informing. Implication as a form of mathematical inference belongs- to the most primitive means in informational sense. The so-called higher informational modi (rules of inference) are, for instance, not only modus ponens and modus tollens, but also modus rectus, obliquus, procedendi, operandi, vivendi, necessitatis, possibilitatis, etc. which are described into more detail in (IL-IV). 18.8. Conclusion On the basis of the exposed definitions, axioms, and theorems, informational theories of understanding for various purposes, needs, and applications can be generated in(de)finitely. Mathematical theories appear to be only particular cases of informational theories. The essential distinction between mathematical and informational theories of understanding might be the following: (1) The nature of mathematical operators is informationally static, determined firmly, that is, functional-deterministically, algorithmically, set-theoretic-definitely, algebraically, arithmetically, etc. The nature of informational operators of understanding is dynamic (arising, developing, generative). This principle sets an informational operator of understanding as a variable, similar to the operand variable. In the case of the notion of explicit and implicit informational operator we are confronted with the so-called "large" transcendence in regard to the mathematical notion of operator and function, respectively. This transcendence brings a new perspective into understanding of the multiplex nature of operators. (2) The nature of mathematical operands is either constant (a value) or variable (a member of a set of values of different nature). The nature of informational operands is like an arising formula which in its cycles of arising is spontaneously developing as a consequence of operand's own and of other operands impacting. In this way, formulas, representing operands, are dynamic, spontaneous structures, behaving as informational entities by themselves. (3) Mathematical expressions (symbols, formulas, functors, etc.) are meant to be fixed formal aggregates (symbolic compositions, functional structures). Informational formulas of understanding are fixed structures only exceptionally, particularly, otherwise they are dynamic, performing as arising informational operands. (4) The entire mathematical knowledge can be incorporated into the informational concept in a senseful way and, maybe, some new understanding and possibilities of mathematical advance in informational sense can be found. It is to believe that informational concept can offer new ways of mathematical thinking and, particularly, of the informational rethinking of the realm of mathematics. (5) Although the exposed thinking in this essay will cause a demonstrative opposition of mathematicians, it will, maybe, stimulate also the necessary rethinking of some mathematical foundations, bring a new spirit, and open the possibilities of a new constructivism on the way to living (neuronal) processes in the future scientific investigation. 19. Intelligence as Understanding and vice versa At the beginning, let us state the question concerning intelligence and intellectualism as understanding. Intelligence becomes a technical term (for instance, as understood in the artificial intelligence community) which stresses the meaning of intelligence as knowledge, ability to learn, information, news, technically bounded expertise, and that like. Sometimes, to be intelligent means to be informed in the usual, however, not Informationally redetermined way (POI) and not predominantly in the sense of the highest possible and autonomous human creativity, but first of all in the manner of socially integrative ideology. If we exclude the meaning of intelligence as the basic eternal quality of divine Mind, intelligence becomes a regular (artificial, autopoietic) system performance, irrespective of the place of its occurrence, for instance, within technological systems as well as in the domain of regular animal and human behavior, thinking, and acting. It may be said, that intelligence as information appears as a component of understanding as information on the lowest possible level of understanding, on the border separating the so-called intelligent and non-intelligent systems. Intelligence as information becomes just the criterion of this border where real understanding begins. On the other side, intellectualism as information pervades the realm of understanding as its autonomous and mainly creative power in understanding information. Intellectualism as information appears as a higher form of intelligence, with the emphasized cognitively creative component of understanding. While in its regular states of informing, understanding perforjns intelligently, so in its breakdown phases, for instance discovering its own loseableness, understanding performs in an intellectual way. If intelligence is a regular performance of understanding as information. then intellectualism is an exceptional informational capability, by which the ruling intelligence of understanding is broken down, proceeding into a new state (or essentially different form) of intelligence within the informationally arising understanding. Thus, intellectualism as information appears as an informationally regulating power over intelligence. Intelligence and intellectualism can be-understood to be distributed within understanding as information, that is, they can arise within any component of understanding. So, let us present the formal side of the concept. Let SI mark understanding, 3 intelligence or informing of SI in an intelligent way, and S"*" intellectualism or informing of SI in an intellectual way. Then, (31) 3 G 3"^; g, C SI At the first glance, intelligence 3 is subordinated to intellectualism 3"*" and both are subordinated to understanding 81. In some cases, intellectualism 3"*" can be subordinated to intelligence 3, thus, (32) 3'^ C 3; 3, 3"^ C SI The domination of intellectualism S"*" over intelligence 3 in case (31) constitutes the so-called intellectual mode of understanding, while the domination of intelligence 3 over intellectualism 3"'' in case (32) constitutes the so-called pseudo-intellectual or simply intelligent mode of understanding. In the second case, intellectualism 3"*" is in fact suppressed by intelligence 3, thus, it can be omitted from the further investigation. The general (or simple) intelligent model of an understanding informing SI becomes (33) (((SI t= ix) N 3) N rj) 1= SI where ^ reads 'informs in an understanding way', tx is the meaning informed (produced) by understanding SI, and t) is intelligent information as produced in the cycle of understanding (33). If necessary, this model can be particularized in detail according to the needs or concrete requirements. For instance, engineering or artificial intelligence models will proceed from these simple circumstances. The general intellectual model of an understanding informing SI is an informational expansion of the general intelligent model (33), that is, (34) (((((SI li) t= 3) 1= t)) 1= 3"^) t)+) t= SI In this intellectual model, intelligence (3, r)) is the condition sine qua non, where intellectualism (3"'", t)"*") arises out of intelligence as its necessary informational background. In this model, t)"'' marks the so-called intellectual information, produced by intellectualism 3"''. Thus, for instance, a primitive intellectual model of the form (34') (((SI N tx) 1= 3+) t= T)+) SI either implicitly assumes the existence of intelligence or is simply only an intelligent system of the form (33), that is, a quasi-intellectual system. In the informational circle of intellectual understanding (34), intellectualism arises out of intelligence, that is, is informed by intelligence, however, via understanding SI controls or Informationally impacts intelligence within the understanding cycle. In a similar way, intellectualism informs understanding, however, it is essentially controlled by the informing of understanding within the understanding cycle. The mechanism of the informational cycle enables that intellectualism and intellectual information impact any informational component of- the understanding cycle, for instance, additionally in a parallel way, that is. (35) 3"^/ 1^= SI, lA, 3, T), 3+, T)-' Intellectualism can be understood to be the highest level of informing within understanding SI, that is, the most creative, decisive, and actual process of informing which can control, impact, and influence all in the understanding involved informational components of understanding. Let us analyze two partial cycles (loops) within the informational . cycle of understanding, concerning intelligence and intellectualism, that is, their possible mutual impacting and the dominance of the one over the other. For instance, (36) (((3 N r)) N 3+) N t)+) ^ 3; 3+, 7)+ C 3, r) or (37) (((3+1= T)+) 1= 3) N T)) N 3; 3, r) C 3+, T)+ The case (36) might be called quasi-intellectual, while the case (37) is a proper intellectual case, in which intellectualism dominates over intelligence. It may happen that both partial cycles exist within understanding and are instantaneously dominating, depending from the circumstances. In the case of the intelligent dominance (36), understanding loses its essential creative, critical, breakdown, also radically changing informing, so it can proceed into a stable, blind state of loseableness of understanding. If intellectualism interrupts the state of blindness or loseableness of understanding, intelligence strengthens the blindness or loseableness and performs in a counter-informational way against the intellectualism. In the intelligent case, intelligence dominates over intellectualism, preserving the informing of intelligence dominant in regard to intellectualism. This is the well-known case of system intelligence (systemic uniformity, stupidity) in which intellectual breakdowns are minimal and exceptional. Intelligent system is lowly creative and weakly critical, and thus intelligently regular. Intellectual system senses the loseableness of understanding by breakdowns of system blindness and changes essentially the performance of intelligence. As mentioned, both systems can exist within understanding, opposing each other and performing counter-informational against each other, improving each other Informationally, that is in an intelligent and intellectual way simultaneously. Thus, the informational conservatism of intelligence is pervaded by the informational radicalism of intellectualism and vice versa. Understanding as information is the framework of this intelligent and intellectual game. 20. Meaning as a Result of Understanding ... in one case we take language as conveying information and instruction; in the other case we leave aside the individual participant and see the process of conversation and understanding as a distributed, coherent events shared among participants, where meaning and understanding is relative to the recursive process of interpretation within the conversational unit. ... Francisco J. Varela (PBA) 269 What is meaning in regard to understanding? How does meaning arise by understanding in a complex, distributive, and distributed informational cycle of understanding? How the complexity of understanding can be considered in a cyclic and a parallel way simultaneously? Is understanding as an arising information the most complex form of informing which can be imagined, although such a form of understanding cannot be performed physically by a human autopoietic system? Does it mean that a much more complex artificial informational machine could beat a human communicating community in regard to the quality and complexity of understanding? Why we are putting such questions right at this place of our conversation? Let us look at the following formal informational concept of understanding. At the beginning of our understanding discourse or discourse of understanding we have to put the question of the complexity of an ongoing understanding. What are the informational components of understanding? They can arise and disappear in a spontaneous and circular way. In this point we can turn to ourselves, to the image we have on that how our minds build up the informational phenomenology of understanding. The very classical and basic components of understanding 21 can be, for instance: the sensing S producing the information of sensing (or sense), marked by 6(tx, (Xß) Of course, this difference can be considered in the process of understanding in a more complex manner, for Instance, as > (H5) ((((lA, ((ß h ») 1= l^) N «) N tx) 8(ix, tXß)) N SI) (= tx where (H4) becomes a subcycle of the main cycle of understanding. The main cycle considers the arising difference 8(ix, within the subcycle, and so forth. In fact, any cycling of understanding of some information - ^and understanding of something is always an informational cycling -is hermeneutical in itself. For instance, understanding within a being's metaphysics is always hermeneutical. For instance, if a is understood to be a regularly arising (on-line, real-time) information, then its understanding automatically concerns also its history, that is, (H6) (a, n a) IX, where S = (S, s"*", p), where S marks syntactic informing. S"*" semantic informing, and P pragmatic informing of understanding. Semiotics s in a narrower sense belongs to the category of meaning ix and the oemiotic informing S to the understanding M. It is supposed that an input information « has its own semiotic structure, for instance, s^). By means of semiotic understanding S the input information a is analyzed syntactically, semantically, and pragmatically and then by semiotic synthesis, by means of semiotics s, the meaning (x(a) of a is generated in a syntactical, semantical, and pragmatical way. In this process, the standard components of understanding (for instance, sensing, observing, perceiving, conceiving, and comprehending) are controlled in a semiotic way, within two cyclically successive steps of understanding: by a semiotic way of informing S and then by a semiotic standardization step of the produced meaning. Through this process, semiotics performs the control over the function of understanding within an semiotic understanding system, that is, (51) («, (x(a) 1= S) |=g ix(a) In this system, S performs analytically and then together' with s synthetically, constructing the meaning of «. ' System (SI) understands a syntactically as s(a), semantically as s'''(a), and pragmatically as p(a), where a appears syntactically as Sgj(a), semantically as and pragmatically as Poj(o), that is, as a = (S5j(a), p(a)) or (52) ((a 1= S^(oc)) s^(a)) |= a; ((« ^ S+„(a)) s+^(«)) t= «; ((a (= P„(a)) 1= Pa(a)) |= a Formula (SI) can be decomposed in the following way, for instance: (53) ((s„(a), s+^(a), p„(«)), (tx(s^(a)), ix(s+„(«)), tx(p«(a))) [= (S, S+, P)) p) (ix{s„(a)), (x(s+„{a)), ix(p„(a))) "guarantees" the history of a. In this process, the meaning tx produced by a is informationally approaching to the informing (that is, to the essence. Being, entity, information) of a. 22. Formalization of Semiotics as a Particular Form of Understanding Semiotics is a concept of understanding and, in this way, it can be seen as a part of understanding, that is, the so-called semiotic understanding. Semiotic understanding is structured (constructed artificially) in the following way: semiotics s is an informational set of perplexed semiotic components, that is, s = (s, s"*", p), where s marks the syntax, s"*" the semantics, and p the pragmatics. The informing part of semiotic understanding is the semiotic informing S, which is an informational set of perplexed informing components, that is. By this formula, semiotic kinds of meaning and semiotic components of informing and semiotics in narrower sense become diversely perplexed. Thus, syntax can influence semantics as well as pragmatics, semantics can influence syntax as well as pragmatics, and pragmatics can influence syntax as well as semantics on the level of semiotic components of meaning. We recognize how semiotic informational entities can be understood in a semiotic way. 23■ The Loseableness and Losing of Understanding The loseableness and loosing of understanding are basic informational phenomena for ascertaining of deficiency occurring during the process of understanding. Loseableness and losing of understanding can concern in principle all distinctions, that is, components of understanding and, thus, for instance, sensing,- observing, perceiving, conceiving, and comprehending within the process of understanding may become insensitive, deficiently observable, insufficiently perceivable, inadequately conceivable, and miscomprehensive, respectively. In course oi its arising, understanding can lose or fail certain essential informing, so it cannot progress in some possible domains or open new horizons of distinct understanding. Thus, after and during the informing of understanding, its deficiencies can be sensed, observed, perceived, conceived, and comprehended, constituting the so-called loseableness and losing of understanding as information. ■ The components of loseableness and losing of understanding can be marked by indexed 2 (loseableness) and 3ft (losing) for their occurrences as informing and, further, by indexed X (loseableness) and p (losing) for their occurrences as information, resulting from such informing. For instance, according to Heideggerian concepts of loseableness (UAI), we can introduce: and X^^ for informing and information of understanding lost in what is encountered within-the-world; 2gq and Xgg for informing and information of understanding lost in equipment; 2g(j and Xg^ for informing and information of understanding lost in everydayness; and Xfg for informing and information of understanding lost in factual circumstances; and X^j. for informing and information of understanding lost in irresoluteness; 2jg and Xjg for informing and information of understanding lost in just-always-already-alongside; 2i[,p and Xjj^p for informing and information of understanding lost in the making-present of the "to-day"; 2pt and Xp^ for informing and information of understanding lost in possibilities which trust themselves upon one; 2py and Xp^ for informing and information of understanding lost in publicness; and Xgj^ for informing and information of understanding lost in something with which information is concerned; 2ty and X^y for informing and information of understanding lost in the "they"; and X^g for informing and information of understanding lost in the they-self; B^^g and X„g for informing and information of understanding lost in the world of equipment; and 2q„ and Xq„ for informing and information of understanding lost in one's world. Further, according to Heideggerian concepts of what understanding can lose (UAI), we can introduce : and for informing and information of understanding losing its aroundness; Sjjj^ and pjjg for informing and information of understanding losing its basis; »Bg and pgg for informing and information of understanding losing its Being; Sßt and pgj. for informing and information of understanding losing the Being of its "there"; and pg^, for informing and information of understanding losing its Being-in-the-world; Sgj, and pgj, for informing and information of understanding losing its equipmental character; Rfg and pf^ for informing and information of understanding losing its force; Kgg and pgg for informing and information of understanding losing its genuineness; Kj^jj and p^^, for informing and information of understanding losing its indigenous character; Sj^y and p^^ for informing and information of understanding losing its involvement-character; Sj^g and p^g for informing and information of understanding losing itself; and p^.^^ for informing and information of understanding losing its readiness-to-hand; and p^^^ for informing and information of understanding losing its time; etc. How can now these Heideggerian notions concerning understanding be involved in a formal way? How to build up formulas considering this sort of informational phenomenology within understanding? If loseableness can be comprehended " as a state of lostness of understanding within itself, then loosing is a process of diminishing, vanishing, or disappearing of appropriate understanding. The loseableness is similar to informational cycling, for instance, in its own and outward insensibility, non-conditionality, uncommonness, unreasonableness, hopelessness, and unsuccessfulness . The loseableness of understanding can be constituted by a closed, firm, and self-sufficient informational pycling, being informationally isolated for the acceptance of other, different, and distinct information. In this way, loseableness of understanding supports itself and arises into its own realm of loseableness. Let us introduce Cj^, Xj^ for i £ {ww, eq, ed, fc, ir, ja, mp, pt, pu, si, ty, ts, we, ow) and Rj^, p]^ for k e {ar, ba, Be, Bt, Bw, ec, fo, ge, ic, iv, is, rh, ti). Let us see how a particular loseableness X^ and a particular losing Sftjj, p]^ is/are involved during the process of understanding il. Let us suppose the basic informational cycles within understanding 31 in the following form: (21) («1) (M 1= 2i) 1= 31; (31 1= «j^) 1= 81 or in a more complex form (22) ((81 1= 2i) 1= Xi) h SI; («2) ((81 ^ Kj,) N Pk) 1= 21 Considering meaning ^ produced by understanding 31, we can introduce (23) (((IX N M) N 2i) Xi) t= li; (R3) (^(y. M) «jt) ^ pjj) ^ VI etc. In (Sj) and (R3) meaning tji dominates over its understanding M, that is, M C pi. In the other case these roles can b'e changed, for instance, (S4) ((((SI ^ IX) 1= 2i) N \i) 1= M) (IX 1= St); (S4) ((((SI 1= IX) %) pj^) SI) (IX $1) Thus, we can resume these cases in a more complete form in the following sense: (2S1) ((21), («1)) (2i, «H C St); ((22), (»2)) ^ ((2i, Xi, Sjj, Pk C St); . (2i C Xi); (Rj^ C Pit)); (aftS) ((23), (R3)) ((St, 2i, Xi, Pit C IX); (St, Si C Xi); (St, »k C pjj); (St C 2i); (SI C %)) Cases (24) and (S4) imply more detailed consequences, that is, (S»4) ((24), («4)) ((IX, 2i, Xi, «H, pjt C St); (ix, 2i C Xi); (IX, Sjt C pjj); (tx C 2i);- (ix C Rjj)) in which roles of (x (SSRS) and St are exchanged (mutually replaced). It seems that in the 'intelligent case, meaning subordinates understanding to meaning (2S3) and that in the intellectual case, understanding subordinates meaning to understanding (S2SÌ4). We recognize how the dominance of components of meaning and understanding can be varied and become embedded in the one way or another. Through the losing of understanding some informational distinctions disappear from the horizon of understanding and thus SI is becoming poorer and poorer in an understanding way. How can this disappearing of understanding be sensible by a formula? For instance, losing Jlj^, producing the information of losing pj^, diminishes (reduces) into a steady form of losing "Rj^, producing "pj^ as a linear process of diminishing understanding, that is, («5) (t= (((% 1= pjt) -%) t= -pj,) 1= within an informational cycle. By a careful observing, even the beginning of this diminishing process can be sensed, for instance, in the form («6) (h (((«3^1= pjj) L N -pj,) N where L marks the operator of informational beginning and where entities "5)3^ and "p]^ come into existence and begin to inform within the entire informational cycle of understanding. Linear cyclic segments (R5) and (»6) bring to the surface another considerable phenomenon of the concept of informational cycle in general. These segments hint to the following general philosophy of a cycle: in general case, linear segments of a cycle can be formulated as (si) (s2) (N «) N N (a N) or Both formulas are open in the sense to be informed and to inform, however, in the essentially different ways. In (si), understanding SI informs as a consequence of the entire history of SI, of all the informational impacts onto St. In fact, formally, St does not inform directly, but through the entity (^ SI), that is, as (f= St) |=. And this formula says explicitly that (t= SI) informs. In case (s2), St informs from itself other entities by the open formula (SI t=) and only that what was informed by St can be informed by something else. Within a cycle, this informational segments become (tl) (t2) (B SI) t= « « (= (St «) or These are two essentially different concepts of informational cycle. The first formula can imply SI C and the second formula, 9 C St, and so forth. 24.' A Formalization of the Blindness and the Breakdown of Understanding ... Indeed the three schools of thought with which most of us began our official philosophizing about mathematics Intuitionism, Formalism, and Logicism - all stand in fundamental disagreement with Platonism. Nevertheless, various versions of Platonistic thinking survive in contemporary philosophical circles. ... Penelope Maddy (RCP) 1121 Blindness of understanding is a course of the process of understanding which lacks the perceiving of external informational disturbances, performing as breakdowns of this course and causing essential changes which can arise into new orientations. A blind understanding is an informationally steady process pervaded, for instance, with its specific truth, knowledge, belief, myth, worship, and cult. This course of understanding considers the care of its improvement and reality only in an unrevealed and unexplained way, being safely closed into its own spontaneous circulation, so that outward information, irrespective of its character, intensity, importance, contents, or meaning is predominantly sensed as informational noise. By definition, informational noise v performs as inessential information for its observer (understanding) . In this way, informational noise cannot inform the concerned information in a way other than a noisy way. Thus, the main axiom concerning informational noise v in regard to understanding St becomes (HI) (V 1= St) ^ SI The analysis of the left side of this implication yields (B2) ((V SI) 1= (V (=51 a) ) (Njl JI) In the case of informational noise, there is (B3) a) :> (hji SI) explicating the informational closeness of blinded understanding. A blind understanding as information performs insensible to its outward informational domain. Blindness B as an implicit informational component of understanding SI is a hidden and partly unaware component of understanding and can be implicitly distributed property of other understanding components, thus, each of them is insensitive to an essential extent in sensing, observing, perceiving, conceiving, and comprehending. The blindness of understanding as an informational phenomenology contributes to the closed spontaneity and circularity in the form of firm, solid, insensible informing of understanding. On the other side, a certain extent of blindness of understanding guarantees the stability of the informational cycling, keeping the produced meaning ^ as a safe, reliable, and believable information. However, blindness B appears always as an implicit, unsensitized, distributed, unaware, and unconscious informational component of understanding SI. Therefore, breaking the blindness of SI means always breaking down (or interrupt) the ruling, the dominant orientation of SI by an outward informational impulse. While circulating in an instantaneously stable cycle, understanding SI can be broken down by the arising of the broken-down blindness B, caused by the breaking impulse, called the breakdown information ß. Thus, (B4) ß L (((B> B) t= b) C SI); (B L B) 1= 8 describes that ß begins (informational operator L marks the beginning) the process in which broken-down blindness B arises out of blindness B, producing the information of breakdown b within understanding SI. At the beginning, the broken-down blindness B becomes a weak, but already distinctive component of SI, which begins to destroy cyclically the ruling blindness B more and more into the sense af B. 25. Formalization of an Informational Expert Systems Informational expert systems (lES's) constitute a new, the so-called informational approach to the projection (in German, der Entwurf, i.e. in the sense of throwing out; in EnglisÄ, project, design, construction) of living and artificial expert systems. An lES can be classified as a system containing the property of the so-called expert understanding of an outward information by some (or all) of its components. According to this informational containment, lES's can be in the range of the primitively informational to the most complexly and perplexedly informational ones. Now-a-days, expert systems do not reach the level of the most primitive lES at all. Let a be information coming into the expert consideration (understanding) by an lES. Operand a can certainly be only the marker of a complex, multi-componential. set of informational operands, for instance, a^, ... , ocjj,. Let E]^, ... , E^ be already available but also arising expert modes of understanding of an lES. Further let e^^, . . . , ejj be expert informational entities concerning the input Information a and produced (informed) by the expert modes of understanding Ej^, ... , Ej^. Further, let E be the master (integrative) expert mode of understanding of an lES and let e mark the master (integrative) expert information (resulting expertise) concerning a. Under these circumstances various lES's can be determined, for instance, marking them by lES's of rank 0, 1, 2, and so forth, according to their informational power, that is, according to the extent of their informational (expert) complexity, perplexity, and hierarchical structure. Within this classification, linear and circular lES's can be distinguished. The most primitive, linear lES would be the one with the empty set of the expert modes of understanding Ej^, ... , E^j. This is the linear lES of rank 0: (El) a h E; E 1= e Rank 0 marks a kind of improper linear lES. The next step is the linear lES of rank 1 (E2) a t= El, ... , Ej,; El 1= e^; ... ; E^ t= e^; ei, ... . E; E |=,e The choice and arising of E^, ... , Ej, of an lES depends on the informational nature of a, that is, lES is informationally sensitive to the nature of input information in respect to its organization. That means that an implicitly or for the observer not visible structure of lES exists which decides on the choice and arising of the expert modes of understanding El, ... , Ejj. Further, it is to stress that Ei, , Ej, and E are proper informational entities which inform, counter-inform and embed information according to the principles of information. So, Ei, ... , E^ and E are proper modes of understanding in an informational way. In case (E2), a.is an outward information, however, the produced ei, ... , e^^ and e are data (messages), that is passive informational entities (products). The next step of development of linear lES's would be to let ei, ... , ejj and e co-inform in the system, but still in a linear informational way, that is, without explicit informational expert cycles. The most powerful or, in fact, the proper lES's can be constructed by the introduction of the expert (conceptual) circularity into the game of producing a real expertise of the accessed outward information a. Thus, a linear IBS (El) can be expanded into the circular lES of rank 0 only by the assumption that entity E within the system a^ E; E ^ e performs in an informationaliy circular way. This is certainly true, for E is a regular informational entity. Similarly as in the linear case, rank 0 marks a kind of improper circular lES. Thus, the next step is the circular lES of rank 1, that is, (E3) « N El, ... , E^; t= e^; ... ; En 1= e^; ®1.....®n N E; E N E^, ... , E e The expert cycle in (E3) is a distributed form of cyclicity in regard to E, that is (E4) Ei e^; e^ 1= E; E E^; i = 1, . . . , n This type of cyclicity differs essentially from the case of the so-called continuous (a history concerning or hermeneutical) cyclicity, which would be (E5) ((E^ t= e^) N E) h E^; i = 1, ... , n The distributivity of the master expert understanding E is reflected in i = 1, ... , n. The next innovative step in the direction of the circular, on-line (real-time) lES's of the second rank would be the following: let in (E3), '1' and be proper informational entities which inform, counter-inform, and embed information. In this case, these informational entities can inform in a proper way to the particular, first-level expert modes of understanding, that is. (E6) -1' ■• , e^ 1= El, This system represents a complete mode of cross informing among the left and the right entities of operator ^ in (E6), that is. (E7) e t= Ei; , N ^l! ! e 1= E„, ■ ; ®1 N ^n! ®n N Ei; ... ; ejj 1= Further, because of the attribute on-line or real-time, the input information a, considered in an expert way, must be observed continuously and in distributed manner, for instance, as (E8) (a N Ei) h i = 1.....n Thus, the complete formula of the on-line, circular, and distributed lES of rank 2 becomes (E9) (« Ei) 1= e^; i = 1, ... , n; ®i' ••• ' ®n t= E N El, ... , E„; E ^ e; e, ei, ... , e^ =|i, ... , E^ However, we recognize how this formula is still reduced in the sense of understanding of a. For proper understanding of a more and more historicity of the previous (historically conditioned) understanding of a has to be taken into consideration. A higher rank of the continuous and distributed understanding of a in the described way would be, for instance, (ElO) (((a N Ei) t= ei) 11= E) It: e; . i = 1, ... , n where n is certainly a independent variable! The last formula is an approach to a parallel lES with much more proper (historical) understanding of a. By the previous discussion, the feeling of possibility of the projection of multi-level lES's is coming into the consciousness. A multi-level lES of rank i would be simply in introducing the particular modes of expert understanding, for instance, in the form (Ell) Eijj; i = 1, ... , m; k = 1, ... , n This introduction would yield, for instance, (E12) Ell, ••• ' Eix; [first level] E21, ... , Ejyj [second level] E il' E iz' [i-th level] In this system, i, x, y, and z function as constructively independent variables. Their concrete occurrence can be conditioned by the nature of « and by the mode of understanding (the step of development) of an lES. Complexity of components, their perplexity, and number of levels are not foreseeable in advance. This limitlessness could lead to lES's of unforeseeable power which could exceed the today imagined or expected possibilities. 26. The Formal Notion of Misunderstanding In the realm of understanding, the problem of misunderstanding can arise as an informational difference within understanding among two or more parallel processes of understanding. If a is information which comes in understanding consideration of two processes of understanding, say M pnd then the resulting meanings ix^C«) IJ^s^") ^ incompatible in an informationaliy understanding way, that is, the informational (understanding) difference 8{n5j(a), becomes an informationaliy distinctive (sensible, observable, perceivable, conceivable, and comprehensible) entity. The distinction in understanding of a arises out of the understanding processes, for instance, (Stl) ((a, lAsi, lis äl) t= l^) ««(l^' t^)); (Slt2) ((a, tijj t= ®) N t= l^)) where jx^j = txji(ot), ng = ix