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BASIC INFORMATIONAL AXIOMS*
INFORMATICA 3/92
Keywords: axiomatization, circularity, externalism, functionalism, informational axioms, intelligent informational entity, internalism, metaphysics, parallelism, phenomenalism
Anton R Zeleznikar Volaričeva ulica 8, 61111 Ljubljana
Slovenia
This essay brings an approach to the possibilities of axiomatization in the realm of the informational. The axiomatization begins from the informatio prima which says that entities inform or, formally, a (=. This primary postulate causes logical consequences which are comprehended as extensions of the primary axiom of informing of entities (informing as externalism, informedness as internalism, circularity as metaphysics, and informational openness as phenomenalism). A consequent axiomatic deduction (together with operational particularization) is shown for general, alternative, negative, alternative negative, functional, metaphysical, parallel, and intelligent informational cases. Informational axiomatization is a part of the informational logic in which the notion of information is extended to the phenomenal extremity. Axioms are expressed by the formalized symbolism which considers the concept of the informing/informedness (active/passive, subjective/objective, operand/operator) nature of informational entities [TIL].
Osnovni informacijski aksiomi
Ta spis prinaša nekatere možnosti aksiomatizacije v domeni informacijskega. Postopek aksiomatiziranja se začenja pri t.i. informatio prima, ki govori, da bivajoče informira, ali formalno a \=. Ta prvi postulat povzroča logične posledice, ki jih razumemo kot razširitve prvotnega aksioma informiranja bivajočega (informiranje kot eksternalizem, informiranost kot internalizem, cirkularnost kot metafizika in informacijska odprtost kot fenomenalizem). Pokazana je dosledna aksiomatska dedukcija (skupaj z operacijsko partikularizacijo) za splošni, alternativni, negativni, alternativno negativni, funkcijski, metafizični, paralelni in inteligentni informacijski primer. Informacijska aksiomatizacija je le del informacijske logike, kjer je pojem informacije razšiijen do skrajnih pojavnih meja. Aksiomi so izraženi s formaliziranimi simboli, ki upoštevajo koncept informiranja/informiranosti (aktivne/pasivne, subjektivne/objektivne, opernadne/operatorske) narave informacijskih entitet [TIL].
*This essay is a private author's work and no part of it may be used, reproduced or translated in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles.
0. Introduction
...das Mathematische will sich selbst, im Sinne seiner eigenen inneren Forderung, begründen; es will sich selbst ausdrücklich als Maßstab allen Denkens herausstellen und die daraus entspringenden Regeln aufstellen.
—Martin Heidegger [FND] 78
Nowadays, the background of informational reasoning seems to be more or less forgotten, fallen into oblivion, but also unrevealed and still subconscious. The broadened notion of information [Owi] was the initial step to this recognition in which formal principles and their consequences have been brought to the reader's consciousness.
To develop a broadened and complete theory of information and informational phenomenology,
an exhaustive axiomatic background of informing of entities is needed, on which the arising phenomenalism, scenery, processibility and, lastly, formalism can be built up. This step is necessary not only with the aim to broaden the scope of informational cognition, but also to prepare the basis for the future development of that what we call the informational machine. The reader must be aware that informational axiomatization, in opposition to the known mathematical one, is a continuum in which a new axiom or theorem can always arise between two already existing ones. Informational continuum appears as an infinitesimal realm of information and an informational theory is never developed to an end, staying open for further change, improvement, and broadening. In this essay we shall discuss these assertions among several others.
Axioms are formal principles which arise from the principal position and attitude of the logical mind. Informational axioms—among others, for instance, mathematical ones—are the most general and can absorb any thinkable axiomatic concept and include it appropriately into the observed informational system. This is a hypothetical assertion which can always be proven from one concrete case to another. The informational seems, at best, to be the most adequate principle of the universe. We shall try to give a principled explanation of this assertion at the end of the essay.
Our way to the realm of axiomatic scenery of the informational must begin with some very basic assertion, which could be called informatio prima.. Afterwards, axioms, logical consequences, and theorems can arise in a logically spontaneous, however, systematized way. Let us open this realm of cognition in the next section of the essay.
1. Basic Axioms
Der Grundzug muß in jenem bestehen, was die Grundbewegung der Wissenschaft als solcher gleichursprünglich maßgebend durchherrscht: es ist der Arbeitsumgang mit den Dingen und der metaphysische Entwurf der Dingheit der Dinge. Wie sollen wir diesen Grundzug fassen?
—Martin Heidegger [FND] 52
As in any axiomatic system, some basic questions prevail: Where is the origin of the informational
axiomatic system? Which is the first, most natural axiom, from which other axioms and consequences arise in a possible, logical, and systematic way?
As the so-called informatio prima of any logic the following could be asserted: Entities inform. If something (an entity figuring as the operand) is marked by a, we can say that a informs. This statement, consciously or unconsciously, pri-mordially constitutes the first principle of any theory and can be expressed symbolically by
(1) «=»(«*=)
This axiom is informatio prima and pertains to any entity, that is, operand, marked by a. The axiom is expressed in the implicative form, using informational operator of implication, which is read as »implies (that)« from the left to the right, and as »is implied by« from the right to the left of formula (1). In the same way, operator \= is read »informs« in the one, and »is informed by« in the other direction. Thus, for formula (1) we have the following explanations: Entity a implies that it informs. Entity a implies that something (the empty place on the right of operator f=) is informed by it. Anytime and anywhere we have or there is an entity a, it informs (a
One of the very essential comments to axiom
(1) is that its formula is circular in respect to entity a (operand a occurs on the left and on the right side of operator =». The other comment is, that a as an informing entity, transmits (operator (= on the right of operand a) its phenomenality into the informational universe.
The first axiom induces consequences which can still be understood as axiomatic. Namely, if we adopt the fact of this axiom, that »if a, then a informs«, then probably something—some other or the same entity—must and can be informed. To inform without to be informed would not have any particular sense in the realm of the informational which is the universe of informing (phenomenaliz-ing) of information (phenomena). Thus, the second most fundamental axiom is coming to the surface, as
(2) a (|= a)
We can show the basic way to this axiom from the first axiom and vice versa by the following asser-
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tion: if a informs, then something (a, (B, ...) would be informed. This fact yields
(3) (a|=)=>Q=a)
which is a consequence of axiomatic character. Further by a similar logical reasoning, which says, that if something is informed, there exists, sometime and somewhere, an informing entity a, thus,
(4) .0=c0 =>(«[=)
The last two axiomatic consequences can be certainly interpreted more universally, by introduction of different entity markers a, (3, y, ... . So,
(3') (a[=)=»(l=a.P.T.-) (4') (|=a)=>(a,p,T,
These consequences explicate the openness of the informational universe in respect to the informing and informedness of an entity a, where it can inform and be informed by several entities (operands), simultaneously.
To remain strictly consequent, axiom (2) follows from axiom (1) in the way
(5) (a=>(o|=))=>(l=o) or logically, even more exactly,
(6) (a (a [=)) (a => fl= a))
As already mentioned, in axiom (1) and its consequences the property of a's circularity is hidden. Formula (1) is evidently implicatively circular in respect to a, since operand a stands on the left and on the right side of operator =>. Together with formula (1), also formulas (2) to (6) are perplexedly a-circular. This circularity is of especial interest in case of formulas (3) and (4), since they enable to conclude on a's particular informing which we call metaphysics of entity a. The initial state of a's metaphysics is expressed by formula
(7) a (a a)
This formula reads: a implies that a informs a and is informed by a. Or, in general, an entity informs (impacts) itself and is informed (impacted) by it-
self. Formula (7) is a consequence of both formulas (3) and (4), that is,
(8) ((a |=)=»(|= a)) =»(a|= a)
where the so-called metaphysics follows from the circularly tied informing and informedness of a and
(9) (0= a) => (a (=))=> (<*!=<*)
where metaphysics is a consequence of simultaneous, cyclically tied informedness and informing of a. Last two formulas conceptualize the circular or cyclic nature of an informational entity (operand, also informational formula) a.
The next essential consequence in the framework of informatio prima follows directly from both axioms (1) and (2). This consequence speaks that an entity informs and is informed simultaneously, that it is the informational system of an entity's informing and informedness. Thus, at the end of such concluding, there is,
(10) af=(a^a)
This formula is not only the most general, but also the most open expression pertaining to entity a and its environment, to a's phenomenality in space and time. It makes evident that a will inform some, yet unidentified entities and itself in the form a (= and that it will be informed by some, yet unidentified entities and itself in the form f= a. It is evident that both formulas a |= and |= a are completely open, having the operand empty place on the right and on the left side of operator [=, consequently.
The next question concerns the informational spontaneity of an informing entity. How does spontaneity concern the basic axioms?
Within a f=, entity a informs in a spontaneous way. Spontaneity is the axiomatic property of operand a as well as operator [=. Informing of any informational entity a is spontaneous within the a's -informational cycle which is constituted by informing, counter-informing, and embedding of information. Further, in a operator \= is particularized in a concrete case and marks only different possibilities of a's operating potentiality.
Formulas a \= and f= a depict phenomenalit-ies of one and the same entity a. They express
possibilities of a's informational impacting and impactedness, that is, to impact the world and to be impacted by the world. The impacting and impactedness of an entity is spontaneous, uncertain and, in general, indeterminable. The spontaneity of a's informing can be understood through its metaphysics, a\=ot, which can be informationally decomposed by different observers in different (spontaneous) ways.
After this discussion we can summarize the four basic implicative axioms (informatio prima) pertaining to an informational entity a, as follows:
(11) (1 °) a => (a \=) [externalism of a] (2°) a =£> (f= a) [internalism of a] (3°) a (a ¡= a) [metaphysics of a] (4°) a (a |=; |= a) [phenomenalism
of a]
Let us have the following dictionary of the primary four informational cases of informing:
(12) (1 °) a external informing of a;
a's informational impacting; a's informing (informingness) (2°) (= a: internal informing of a; a's informational impactedness; a's informedness (3°) a |= a: metaphysical informing of a;
a's metaphysical circularity; a's informational metaphysicalness (4°) a }=; a: phenomenal informing and informedness of a; a's informing and informedness phenomenality; a's complex phenomenalism
parallel negative general informing:
operator 1^; parallel alternative negative general informing: operator 7%
cyclic general informing: operator \—; cyclic alternative general informing:
operator —|; cyclic negative general informing: operator
cyclic alternative negative general informing: operator /];
parallel-cyclic general informing:
operator ||-; parallel-cyclic alternative general
informing: operator -j|; parallel-cyclic negative general informing: operator
parallel-cyclic alternative negative general informing: operator /||
To all these cases the so-called subscribed informational operators can be added. A subscript (for instance, subs) means an operational particulariza-tion or universalization. Thus, such operators are:
Nsubs' Hsubs» Msubs» ?Nsubs' INsubs» Hlsubs» t^subs» 7^lsubs> f~subs> Hsubs» l"/subs> /Hsubs» II-subs» Hlsubs» Ib^subs' /llsubs
and, certainly various operators of the type
I—subs> —lsubs> ~'subs» I subs' ^subs' ^subs» ~'subs» ^subs» ^subs» <*z^subs' =subs» ^subs> —subs' ^subs» ^subs> 3subs> "subs' ^subs» ^subs» '-subs» ^-subs» wsubs> °subs»
There are different modes of general informing and it is possible to make a rough classification in the following sense:
general informing: operator (=; alternative general informing: operator negative general informing: operator f^; alternative negative general informing: ^j;
parallel general informing: operator |f=; parallel alternative general informing: operator =]|;
etc. with particular meanings.
We may not forget that all operators (arbitrarily universalized or particularized) used within informational logic are informational. Informational operators are attributes (properties, phenomenalities, happenings, predicates) belonging to the informing and informed entities (operands, formulas, formulas within formulas) and connecting them informationally. The direction (operational orientation or directionality) of operators depends on the verbal definition of a particular informational operator. For instance, in their
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simple present tense, some verbal forms as »informs«, »inclüde«, »operates«, etc. are oriented from the left to the right operand, while others, for instance, »observes«, »understand«, »conceives«, etc. are oriented from the right to the left operand. Thus, we can distinguish externally and internally oriented operators as determined by simple present tense of verbs. Operands are marked informational entities which are single Greek or Fraktur symbols or formulas composed of operands, operators, parentheses, commas, and semicolons. We shall show the syntactic structure of informational formulas later on.
2. Alternative Basic Axioms
Zum Wesen des Mathematischen als Entwurf gehört das Axiomatische, die Ansetzung von Grundsätzen, aus denen alles Weitere in einsichtiger Folge gründet. Wenn das Mathematische in Sinne einer mathesis universalis das gesamte Wissen begründen und gestalten soll, dann bedarf es der Aufstellung ausgezeichneter Axiome.
—Martin Heidegger [FND] 79
We are interested to have explicit alternative possibilities to the discussed basic axioms with the aim to be able expressing explicitly the possible alternatives in each particular case as well as"on the most general level. Alternative axiomatic alternatives pertain also to the principle of informational spontaneity and to other possibilities of informing, for instance, to negative informing or non-informing, parallel, serial, cyclic, parallel-cyclic informing, etc. How it is possible to express alternative informing in cases of the most general and particular informing?
The alternative possibility of informing speaks in favor of the ability to express simultaneously the case of basic (in general ^-operational) and alternative (=foperational) informing. We say that an entity informs in one (basic or general) or another (alternatively basic or general) way. How to mark the so-called other case?
In general, entity a informs and is informed in this and that way. It informs alternatively and such situation can be expressed symbolically
(1A) a=>(=ja)
The differences between formula (1) and (1A) are the following:
(i) Operators (= and =) are counter-oriented, for instance, operator \= is the left-right and operator the right-left symbol. Through this we can read formula a f= as a informs and formula =| a_as a informs alternatively (in respect to the case a. f=).
(ii) Simultaneously, operators |= and =| are dual to each other in the sense of informing and informedness of an entity. For instance, if a means a informs, then a =j would mean a is informed (alternatively). We can read operators (or formulas) strictly from the left to the right side. This possibility of reading is important for the distinguishing of verbal cases in which something impacts (is impacting) something informationally and something is impacted by something informationally. The following meanings of informational operator duality can be used:
operator type f= operator type =j
informs is_informedC_by)
is informing is_being_informed(_by)
informed has/have_been_
_informed(_by)
has/have_been_ has/have_been_
„informing „informed (by)
was/were_informing had_been_informed(_by)
will/shall_inform will/shall_be_
_informed(_by)
etc. In short, the left side of operator and the right side of operator =\ behave as informing entities irrespective of the tenses. On contrary, the right side of operator |= and the left side of operator =j behave as informed entities irrespective of the tenses.
A
(in) In regard to formula (1 ), to be alternatively consequent, one could introduce the alternative basic implicative axiom in the form
(1A) (=]«)<=«
This formula is read from the left to the right as: =j a is implied alternatively by a. From the right to the left, this formula can be read »normally«, that is, a implies alternatively =) a. The complete possibility of meaning is the following: »a informs
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alternatively« is implied alternatively by a. In the opposite direction one can read: a implies alternatively that a informs alternatively.
As we see, the alternative axioms can now be obtained automatically from the basic, non-alternative ones, that is,
(2*)
(3*) (<)
(5*)
(7*) <8a>
(9 ) (10*) (11A)
(12A)
a=>(a=0
(a=0=>(H«)
(a (=j a)) (a (a 4» a (a a)
((=]a)=>(a=D)=>(a=jcc)
((a =|)=>(H <*))=>(«=!«)
a : (1'
(2°) (3°)
(4°)
(1°)
(2°)
(=| a; a =0
a => (=| a) [alternative externalism of a] a (a =|) [alternative internalism of a] a4(a=ja) [alternative metaphysics of a] a (=| a; a =0 [alternative phenomenalism of a] =\ a: alternative external informing of a; a's alternative informational impacting;
a's alternative informing
(informingness)
a =j: alternative internal
informing of a;
a's alternative informational
impactedness;
a's alternative informedness a=|a: alternative metaphysical informing of a; a's alternative metaphysical circularity;
a's alternative informational metaphysicalness =| a; a =\: alternative phenomenal informing and informedness of a; a's alternative informing and informedness phenomenality; a's alternatively complex phenomenalism
In regard to the informational, the alternative informational means to have (possess) another pos-
(3°)
(4°)
sibility of informing to the already existing or identified one. Informational alternativeness is one of the basic principles of informational spontaneity, where in each particular case of informing an unforeseen, uncertain, indeterminable alternative possibility can arise and come into existence.
3. Axioms of Negative and Negative Alternative Informing
Negative (negating, negational) informing, called also non-informing, is an alternative mode of informing in comparison to basic and alternative basic informing of an entity. Negative informing is, for instance, from the observer's point of view, something which is missing informationally in the informing activity of an entity. On the other side, the informing entity can inform by itself the deficiency of its certain informational impacting and, in this way, cannot influence the informationally tied entity in some other way. Negative informing belongs to the oldest cultural forms of communication and language. Within this tradition a clear distinction between informing (or informedness) and non-informing (or non-informedness) of entities is possible and senseful.
In general case, where \= marks the (positive) informing, the negative case is marked by ^ with the meaning »does or do not inform« or, on the other hand, »is or are not informed (by)«.
For any imagined entity it can be said that it does not inform in a certain (particular) way. This seem to be a natural situation of things which certainly cannot inform in all possible ways. The assertion »entity a does not inform« is expressed symbolically
dN)
a
Axiom (1 ) is a particular case of informatio prima, is in the realm of non-informing of entity a. And certainly, if a does not inform in a certain way, then something cannot be informed in a certain way. According to axiom (2) for informing of an entity, we have the negative axiom
(2N)
a :
(M<*)
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This axiom reads: if a is an entity, then it is not informed in a way. As we recognize, the negative axioms can now be obtained automatically from the basic, non-alternative ones, that is,
3n 4n}
A sK
X &
9 1
11")
(12N)
«)=>(« bO;
(a =>(«£)) =>(£«);' (a (a M)) =>(«=> (£«)); a => (a a);
((a f?9 =>(£«)) =>(«£«);
a =>(a^;^a); (1 °) a => (a [negative
externalism of a] (2°) a fl^ a) [negative
internalism of a] (3°) a (a a) [negative
metaphysics of a] (4°) a =Ma ^ a) [negative
phenomenalism of a] (1°) a f^: negative external informing of a; a' s negative informational impacting;
a's negative informing (informingness) (2°) a: negative internal informing of a; a's negative informational impactedness; a's negative informedness (3°) a ^ a: negative metaphysical informing of a; a's negative metaphysical circularity;
a's negative informational metaphysicalness (4°) a ft a: negative phenomenal informing and informedness of a; a's negative informing and informedness phenomenality; a's negative complex phenomenalism
The negative informational in regard to the informational means the lack of certain possibility of informing in regard to already existing or identified one. Informational negativism is one of the basic principles of informational spontaneity,
where in each particular case of informing a negative unforeseen, uncertain, indeterminable possibility can arise and come into existence.
The alternativeness of the negative informing can be expressed explicitly. It is important to have the possibility of alternative negative informing and informedness on a universal and particular level. Thus, we can repeat the entire system of formal expressiveness and its meaning for alternative negative informing as follows:
an
(2an (3an
(4AN) 5an
(7 ) (8AN) (9AV (10 ) (11AN)
(12AN)
a => (A «)
a (a jty a)^>(a#>;
(a => a);
(a => (^ «)) (a (a
a => (a A a);
((^a)=^(a7£j))=^(a^a); ((a7tD=>(a?£0)=>(a^a); a (^ a; a
(1 °) a G^ja) [alternativenegative
externalism of a] (2°) a => (a 7^) [alternative negative
internalism of a] (3°) [alternativenegative
metaphysics of a] (4°) a => a; a [alternative
negative phenomenalism of a] (1 °) A a: alternative negative external informing of a; . a's alternative negative informational impacting; a's alternative negative informing (informingness) (2°) ft a: alternative negative internal informing of a; a's alternative negative informational impactedness; a's alternative negative informedness (3°) a A a: alternative negative metaphysical informing of a; a's alternative negative metaphysical circularity; a's alternative negative informational metaphysicalness (4°) alternative negative
phenomenal informing and informedness of a;
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a's alternative negative informing and informedness phenomenality;
a's alternative negative complex phenomenalism
4. Axioms of Basic and Basic Alternative Parallelism
Parallelism belongs to the most significant philosophical and technological approaches of modernism. Communication and cooperation among parallel processes constitute the kernel of mastering of complex computing machinery which architecture is built-up in a massively parallel structure and organization together with signal processing and networking. And this is the reason why we shall strictly repeat the scenario of axiomatization for the case of parallel and alternative parallel informing.
Informational parallelism means that entities inform freely, that is, spontaneously and circularly in parallel ways. An operand as an informationally open entity (phenomenalism) informs unforeseea-bly parallel and is in this way informed. It means that from the informational point of view, there are no limits for a concrete parallelism: it can be extended or suppressed in case of any informational entity (operand) in question. So, let us discuss this particular situation in some axiomatic details.
For any imagined entity it can be said that it informs in a parallel way, that its »signals« are spread (broadcasted) in space and time where they can impact entities physically, observationally, or in other ways. This seem to be a natural situation of things which inform in all possible ways to different places and sites. The assertion »entity a informs in parallel« is expressed symbolically
(2P)
a ■
(lb cO
d")
a
(a IN
Axiom (1 ) is the parallel case of informatioprima, is in the realm of the parallel informing of entity a. And certainly, if a informs in parallel ways, then entities are informed in parallel. According to axiom (2) for informedness of an entity, we have the parallel axiom
This axiom reads: if a is an entity, then it is and can be informed by several (parallel) entities simultaneously. From this point on, the parallel axioms can be produced automatically from the basic ones, that is,
CO
(4)
(5P} (6P}
(8P) (9^
(11P)
(12P)
(a \\=) => (||= a);
(N «)=>(« IN;
(a => (a \\=)) => (\\= a);
(a
(a=>(|t=o0);
(a IN a); (a IN«);
(2°) (3°) (4°) (1°)
(a N) = a (a |[= a);
((«N^t«)) ((IN a) (« IN) a => (a IN; IN a); (1°) a (a IN) [parallel externalism of a] a (|N a) [parallel internalism of a] a (a |N oc) [parallel metaphysics of a] a (a |N; IN °0 [parallel phenomenalism of a] a |N: parallel external informing of a; a's parallel informational impacting;
a's parallel informing (informingness) IN a: parallel internal informing of a; a's parallel informational impactedness; a's parallel informedness a |N oc: parallel metaphysical informing of a; a's parallel metaphysical circularity;
a's parallel informational metaphysicalness a |N; IN parallel phenomenal informing and informedness of a; a's parallel informing and informedness phenomenality; a's parallel complex phenomenalism
The parallel informational in regard to the informational means the possibility of informing addi-
(2°)
(3°)
(4°)
tionally in regard to already existing or identified modes, that is, simultaneously in many different modes. Informational parallelism is one of the basic principles of informational spontaneity, where in each particular case of informing a parallel unforeseen, uncertain, indeterminable possibility can arise and come into existence.
The alternativeness of the parallel informing can be expressed explicitly. It is important to have the possibility of alternative parallel informing and informedness on a universal and particular level. Thus, we can repeat the entire system of formal expressiveness and its meaning for alternative parallel informing as follows:
(2AP) a=>(cHI) O (a HD => (HI a);
(5*p (a=>(=fla))=>(a=|D; (6~) (a (=j| a)) => (a => (pc =j|));
O a=»(a=l|a);
(8AP) ((HI a) => (a =|D) => (a =j| a);
(9AP) ((a=|D=>(=||a))=»(a=l|a); (10AP) a=»(=j|a;aHD;
(11AP) (1°) a (=j| a) [alternative parallel externalism of a] (2°) a =3> (a =\\) [alternative parallel
internalism of a] (3?) a (a =]| a) [alternative parallel
metaphysics of a] (4°) a => (=j| a; a =j|) [alternative parallel phenomenalism of a] (12ap) (1°) =\\ a: alternative parallel external informing of a; a's alternative parallel informational impacting; a's alternative parallel informing (informingness) (2°) ||= a: alternative parallel internal informing of a; a's alternative parallel informational impactedness; / a's alternative parallel informedness (3°) a =|| a: alternative parallel metaphysical informing of a; a's alternative parallel metaphysical circularity; a's alternative parallel
informational metaphysicalness (4°) =j| a; a =||: alternative parallel phenomenal
informing and informedness of a; a's alternative parallel informing and informedness phenomenality;
a's alternative parallel complex phenomenalism
The parallel informational phenomenality concerns the most sophisticated metaphysical and intelligent conceptuality (phenomenology) of the informational (mathematical, mathetical in the sense of the Greek mathesis). In an informational situation, parallel informational phenomena pertaining to this situation can come into existence. The general and particular operators of informational parallelism (|(=, =)|, express this faculty of parallel phenomenality in a positive or negative way. Therefore, the following parallel circularity can be considered as the most significant in the context of parallel informational system composition and decomposition:
p
(13 ) )Parallel circular externalism:
(aH=>(a|h); (a|K)=>(aN •■•;«!=);
Parallel circular internalism:
0= a) => (\\= a);
(|f=a)=>0=a; ... ;>a);
Parallel circular metaphysics:
(a f= a) => (a |(= a);
(a|(= a) => (a|= a; ... ; a|= a);
Parallel circular phenomenalism:
(a |=; f= a) => (a |(=; ||= a);
(a |f=; |= a) (a f=; |= a; ...; a \=; \= a)
As one can understand, informational paral- • lelism brings into the focus the so-called parallel circular mode of informational arising of entities, systems, formulas, etc.
5. Axioms of Informational Function
Wir bringen diesen gesuchten Grundcharakter der neuzeitlichen Wissenshaltung auf einen Titel, wenn wir sagen: Der neue Wissensanspruch ist der mathematische. Von Kant stammt der oft angeführte, aber noch wenig begriffene Satz: »Ich
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behaupte aber, daß in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist.«
—Martin Heidegger [FND] 52
A function 9 is an acting or producing informational entity in the realm of several, the function concerning entities. Therefore, it is senseful to speak about a functional scheme in which different entities can be involved when functional activity is coming into existence. It means that the functional scheme should retain sufficiently general perquisites enabling it to be embedded and connected into and with distinguished and essentially different informational entities, respectively.
In general, a function is never something without its roots from where it arose or is still arising. Several essential and general questions can come into the discourse before we list the axioms pertaining to informational function 9. Let us start with some basic questions.
(J) Which entity (informational operand) c[) produces 9?
(2) For which purposes, marked by it, function M)b
|=70h
The last formula can be particularized in an operand and operator way, that is,
(15) ((func .prod Nwith (^purpose Nconsider
^sensc ^significance» 1 intention)) Nproduce ^function) Nrefer 4,func_prod
The operand and operator entities of the last formula have the following particular meanings: ti)func_prodis a functional producer; 7tpurpose marks purposes which consider (operator Consider) °P~ erand entities as crsense, ^significance» t intention*» these purposes are with (operator (=wjth) entity
^func.prod» now» the functional producer with its purposes produces (operator (=produce) function ^function and the whole process is in reference to (operator f=refer) the functional producer
4)func_prod- this way, the function arises cyclically, influencing backwards its producer with certain purposes.
The next question is: (3) From which distinguished informational domain does junction 9 depend and how does it produce a result p for purposes 7t?
Through this axiomatic question, function 9 becomes together with domain ^ the producer of result p. In this point we introduce a shortened form of formal expression of »function (»
The particularized form of the last formula is
(18) ^function(2,domain)
((^domain Ninfluence function) Nproduce ((presult Nfor ^purpose) ^of (J,func_prod))
In the last formula, one can see how the result of the function and its domain within certain purposes become a function of the functional producer as well. It seems that the functional producer predetermines, in the realm of its own purposes, not only the function which it produces, but also to some informational extent the functional result itself. This comment may concern specifically any con-
11
struction of the so-called scientific functional dependencies, where the role of the functional producer is left to the scientist.
The next question pertains to the functional domain: (4) To which informational domain a a functional domain does belong and by which selecting entity |3 it is identified as a significant domain of the function a> to pNof mean?
Questions (1) to (10) spread the possibilities of the functional axiomatization enormously. In concrete cases, different axioms pertaining to specific situations and attitudes come to the surface.
At the end of this section, let us mention the following evident example of a recursive informational functional structure: the mind is a function of a concrete living brain; a thought is a function of the mind; a distinguished thought item is a function of the thought; the distinguished thought item, the thought, and the mind impact the arising of the brain, for instance, the appearance of new neurons and neuronal interconnections; thus, the brain is a function of its own phenomenality, that is, of thought items, thoughts, and the mind in its entirety; etc.
Further, the functional expression ((a |= $(«))(= a); (a^a)=>((3(a)|=a)f=3(a)); Counter-informing axioms: (a \= a, r h T) £(«); (T|=T)C(at=a); S(a) C S(a);
S(a)=»((a,T^(S(a))t=a, T); (a a, y (= y)
(( s)l (a [= a, y N T. s N s) ((@(a)|=a, T,s)|= E(a); (a \= a, y |= T. s N £) => ®(a); Metaphysical hierarchy:
(Y |= Y> e N s) c (« N <£(a), g(ci)C^a); Decomposition of a, y, and s:
<£(a) =>((«, J=(£(a))|=a,Y); ®(a) => ((a, y,e|= ©(a)) |= a, y, s); Decomposition of^(ot), (Sfa), anJ ©fa);
(a|=a)=>(CS(a)f=a)f=S(a));
(a (= a, y N T)
((S(o)|=a, Y)N®(a)); (a a, y |= Y. s N s) => ((<5(a)|=a, Y,s)h@(a>)
To the parallel metaphysical axioms, the cyclic or serial axioms can be introduced in which the occurring components a, 3(a), y, E(a), s, and @(a) äre positioned into middle-long and long metaphysical cycles. One of the axioms pertaining to the long metaphysical cycle is, for instance,
(23) (a |= a) =>
((((((a |= 3(a)) (=T)f= W) \= e) |=e(a))|=o)
Other long axioms can be obtained by the permuting of operand components in the last axiom.
6. Axioms of Informational Intelligence
Wo der Wurf des mathematischen Entwurfs gewagt wird, stellt sich der Werfer dieses Wurfes auf einen Boden, der allererst im Entwurf emtworfen wird. Im mathematischen Entwurf liegt nicht nur eine Befreiung, sondern zugleich eine neue Erfahrung und Gestaltung der Freiheit selbst, d.h. der selbstübernommenen Bindung. Im mathematischen Entwurf vollzieht sich die Bindung an die in ihm selbst geforderten Grundsätze.
—Martin Heidegger [FND] 75
Entities inform according to axiom (1) which represents informatio prima from which axiomatic consequences can follow in a mathematically projective way. How can entities inform intelligently and what are the perquisites of intelligent type of informing? Intelligence seems to be merely a pred-
13
icate (property, faculty) belonging to an informing entity. Thus, the way to intelligent informing of entities must proceed from the axiom (1) which has to be particularized for the case of intelligent informing.
An informational entity can have such or another intelligent position or attitude in the general and particular sense. The majority of readers would agree that an intelligent entity (IE) informs some of the facts, concepts, or fiction in the following six initial points:
(1) IE informs (expresses, describes) something (an entity) in an evaluating, observing, or understanding way;
(2) IE produces the so-called meaning, sense, spirit, and/or significance of something as information;
(3) IE forecasts, analyses, and synthesizes information concerning something under dynamic, unforeseeable, uncertain, and indeterminate circumstances;
(4) IE brings unknown, unconscious, invisible, impossible information concerning something to the surface;
(5) IE is spontaneously and circularly oriented, intentional, and goal-directed when it observes itself and evaluates various possibilities for its own development;
(6) IE behaves as a living entity in the world where it plays the struggle for its own benefit, delight, survival, etc.
Following the axiomatic route demonstrated for the general informing of entities (operator |=) in formulas (1) to (12), for an intelligently informing entity i the following particular and significant axioms can be considered:
Oj) (■=>(!- t= intelligent)! (2^) i- ((^intelligent l); (3^) (l< ^intelligent) ^ ((^intelligent l)> (4^) ((^intelligent L) (L (^intelligent)^ 0 ) L => it (^intelligent Oi (10 ) l => (t, (^intelligent ^intelligent l)> (ll1) (1°) i => (i |=in,eiiigent) [intelligent externalism of i ]
(2°) t ((= intelligent 0 [intelligent internalism of i ]
(3°) i (t ^intelligent 0 [intelligent metaphysics of i ]
(4°) l (l (^intelligent! (^intelligent 0
[intelligent phenomenalism of i]
Let us have the following dictionary of the primary four informational cases of intelligent informing:
(121) (1°) i ^intelligent- intelligent external intelligent informing of i; i 's intelligent informational impacting;
i 's intelligent informing (informingness) (2°) ^intelligent ^ intelligent internal informing of i; i 's intelligent informational impactedness; i 's intelligent informedness
(3 ) I (^intelligent
intelligent metaphysical informing of l;
t 's intelligent metaphysical circularity;
i 's intelligent informational metaphysicalness
(4 ) i ^intelligent» (^intelligent •
intelligent phenomenal informing and informedness of i; i 's intelligent informing and informedness phenomenality; i 's intelligent complex phenomenalism
There are different modes of intelligent general informing and it is possible to make a rough classification in the following sense:
intelligent general informing: operator
(^intelligent» intelligent alternative general informing:
operator ^intelligent intelligent negative general informing:
operator ^intelligent intelligent alternative negative general
informing: operator ^intelligent intelligent parallel general informing:
operator ^intelligent intelligent parallel alternative general
informing: operator ^intelligent intelligent parallel negative general informing: operator Intelligent
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intelligent parallel alternative negative
general informing: operator ^intelligent; intelligent cyclic general informing:
operator [-intelligent; intelligent cyclic alternative general
informing: operator -intelligent; intelligent cyclic negative general informing: operator ^intelligent; intelligent cyclic alternative negative
general informing: operator ^intelligent; intelligent parallel-cyclic general informing:
operator ^intelligent; intelligent parallel-cyclic alternative
general informing: operator -¡intelligent; intelligent parallel-cyclic negative general
informing: operator ^intelligent; intelligent parallel-cyclic alternative negative general informing: operator /¡intelligent
We see how intelligent informing of an entity i is nothing else than any other particular informing of this entity. Intelligence is merely a specific perquisite of the entity which produces »intelligent« information concerning something which this entity evaluates, observes, or understands. Thus, to the basic axiomatic system of an intelligently informing entity i the parallel system can be added, considering the previous six initial points (1) to (6) of intelligent informing. The markers and to them corresponding meanings of the Greek and Fraktur operands and operators are the following:
^benefit benefit
3benefit(a) something's own benefit; benefit of
something Tcircumst circumstance(s) ^delight delight
6deiight(°) something's own delight; delight of something
^development
development
^dynamic dynamics
^dynamic(Ycircumst)
dynamic circumstances
"•indeterm indeterminacy
1 indeterm(Tcircumst)
indeterminate circumstances i intelligently informing entity
something
1 impossible impossible, the
^ impossible^)
impossible concerning something, the
1 invisible invisible, the
1 invisible^)
invisible concerning something, the
h.
ive
liveliness
^live(£entity)
living entity
^meaning meaning
l^meaningO7)
meaning of something ^possibility possibility/possibilities
^possibility^ (^development))
possibility of intelligent entity
development
a something (an arbitrary entity to be
»understood« by t)
'sense
sense
CTsense(a) sense of something ^significance
significance
^significance^)
significance of something
^spirit sPirit
aspirit(CT) sPirit of something
^struggle struggle
CTstruggIeObenefit(a)> ^delightC^)» CTsurvival(a)> • ••)
struggle of something's own benefit, delight, survival, etc. ^survival survival °survival(CT)
something's own survival; survival of something "uncertain uncertainty "uncertain(Ycircumst)
uncertain circumstances
u
i(o)
entity intelligently concerning
unconscious
unconscious, the
"unconscious
unconscious concerning something, the
"unforeseeable
unforeseeable, the
"unforeseeable(Ycircumst)
unforeseeable circumstances
"unknown unknown, the
"unknown(0) unknown concerning something, the
®world world
15
Nanalyze analyze(s)
Nbehave behave(s)
Ncircular is/are_circular
Nevaluate evaluate(s)
Nforecast forecast(s)
Ngenerate generate(s); bring(s)_to_the_surface
Ngoal_directed
is/are_goal-directed (=¡n is/are_in; inform(s)_in Nintentional
is/are_intentional Nobserve observe(s) Nplay
play(s)
hproduce produce(s)
^spontaneous
is/are_spontaneous
Nsynthesize
synthesize(s)
Nunder is/are_under; inform(s)_under
^understand
understand(s)
The six additional paragraphs pertaining to IE can now be formalized by the corresponding informational formulas as parts of the informational system of IE i. For the first sentence and its formula we have:
(131.1) IE informs (expresses, describes) something (an entity) in an evaluating, observing, or understanding way; \= (<• ^intelligent 0) => devaluate a> VNobserve ^ Nunderstand
In this formula, IE i, that is, its metaphysics i ^intelligent i-, is the intelligent observer of a, so, it evaluates, observes, and understands a. The second sentence and its formula is:
(D1^) IE produces the so-called meaning, sense, spirit, and/or significance of something as information;
Nproduce (^meaning (a)> asense(CT)' aspirit(a)> ^significance^))
The third sentence together with its formula is
(131.3) IE forecasts, analyses, and synthesizes information concerning something under dynamic, unforeseeable, uncertain, and indeterminate circumstances;
(*- Nforecast Nanalyze ^ (CT)>
^ Nsynthesize ^ (a)) Nnnder (^dynamic(Tcircumst) » uimforesceabIe(Tcircumst) » uuncertain(Ycircumst) « ^ indeterminate(Ycircumst))
The fourth sentence and formula are:
(13'.4) IE brings unknown, unconscious, invisible, impossible information concerning something to the surface;
1 Ngenerate
O-Hinknownfa) > ^inconscious(CT)' ^ invisible^) » impossible^))
The fifth sentence and its formula are
(n1^) IE is spontaneously and circularly
oriented, intentional, and goal-directed when it observes itself and evaluates various possibilities for its own development;
((k ^spontaneous» 1 Ncircular'» 1 ^intentional' Ngoal-directed) Nobserve 0 Nevaluate ^possibility C1, (^development))
The sixth sentence and its formula are
(13!.6) IE behaves as a living entity in the world where it plays the struggle for its own benefit, delight, survival, etc.; ((** Nbehave ^live(£entity))
Nin ®world) Nplay ^truggleObenefit^). ôdelight(a)>
'surviv
al(CT)> •••)
7. The Syntax of Informational Formulas
In this essay we have used various informational formulas from the beginning, in the most basic axioms. In parallel, we developed a foundation of the so-called operand-operator (0-0) theory which could be understood also outside the so-called informational realm as a mathematical theory. We
16
could simply speak about a self-sufficient operandoperator theory which does not concern only the so-called informational theory. This would mean to ignore the semantics pertaining to the informational phenomenality and introduce operand-operator objects as the only ones into the O-O theory. However, such theory would be nothing else than a particular case of an informational theory. The syntax delivering syntactically correct informational formulas is simple and straightforward. If a represents any possible operand entity, the formation rules of well-formed informational formulas 9 are the following:
(24) (1°) (p=>a;
(2°) (a',' («•;»;
(4°) (9 f=):
(6°) (9 ^9);
(8°) );
(9°) a=>(a|(3|r| ... a>|81|®|<£| ... S);
(10°)a=>asubs;
(11°)(= => (MHIMITNIINNIII^MII •••
L|jnir|A|vh|'=»'|<=|^M-
5É|=|^|V|b|-|€|çî|C|c|o|
(i2°)N=»NSubS;
(13°) Ncomp*» a4°)KomP=>0=«»NI,(,N«l=,),l
In the last formula system we have marked symbols V, '('. ')'» '=>', and T to avoid the ambiguity with the equally marked symbols appearing in syntactic system (xx). In formula (11°) operator entities appear as informational entities. Entity a is a simple operand, that is, operand marker in the sense of formulas (9°) and (10°).
Conclusion
The informational seems to be the most adequate principle of the universe. Things, entities, phenomena, information, minds, machines, and the entirety of the universe inform and are informed, that is, impact and are impacted phenomenally, perform as informational entities, as formations concerning formations; and, that is, speaking roughly, information. In the essay, we have un-
folded this situation and attitude through the con-ceptualism of informational externalism, internal-ism, metaphysics, and phenomenalism which all can be accepted in a general and in every particular case. Thus, any other axiomatic approach can be derived from the informational as a specific axiomatic particularization. On the other hand, an informational (verbal and symbolic) language [TIL], that is, the new formalism, came into the foreground and made the progressing into the domain of thé informational (intelligent) machine possible.
Various forms of the verb to inform obtained the power of the verb to be and the role of the most primordial (simple and composed) verbal forms. Thus, any operator connection between informational operands could be expressed dynamically in a general or particular way, widening the scope of the verbal phrase which includes informing to arbitrary other words and word groups with particular, composed meanings. The informational approached to that what we can call the understanding way of informing, living the atomized, strictly linguistically structured and organized conceptual-ism in the background, unconscious, however not excluded from the informational realm. In any case, the verb to inform became the central happening between entities and enabled to broaden the informational semantic scope to any dynamically understood relation, operation, and phenomenality, occurring between informational entities (operands).
On this way, the informational axiomatiza-tion can encounter the most problematic, semanti-cally complex, intelligent, and real situations, offering e new, flexible, and promising apparatus in the development of nowadays and future theories, systems, and machines.
References
[FND] Heidegger, M., Die Frage nach dem Ding, Max Niemeyer Verlag, Tübingen (1987). [Owi] Železnikar, A.P., On the Way to Information, The Slovene Society Informatika, Ljubljana (1990). [TIL] Železnikar, A.P., Towards an Informational Language, Cybernetica 35 (1992) 2, 139-158.